Embed Size (px)
Citation preview
Thesis for the degree לתואר) תזה(עבודת גמר
דוקטור לפילוסופיה
למועצה המדעית שלתמוגש מכון ויצמן למדע
ישראל, רחובות
מאתסורוקר-חמוטל ברי
Doctor of Philosophy
Submitted to the Scientific Council of theWeizmann Institute of Science
Rehovot, Israel
By Hamutal Bary-Soroker
בטבעות מזוסקופיותיםדיתממ םזרמיPersistent currents in mesoscopic rings
:יםמנח יוסף אמרי' פרופ
וולמן-אורה אנטין' פרופ אהוד אלטמן' דר
Advisors: Prof. Yoseph Imry
Prof. Ora Entin-Wohlman Dr. Ehud Altman
א"תשע, אדר February, 2011
Abstract
The fundamental phenomenon of persistent currents (PCs) in mesoscopic normal (i.e.
non-superconducting) rings is still today challenging both theoreticians and experi-
mentalists of condensed matter physics. In this work we analyze, theoretically, the
PC of non-interacting and of interacting electrons. In the latter case we consider the
effect of pair breaking on the current.
The main part of this work considers the contribution of superconducting fluc-
tuations to the mesoscopic PC of an ensemble of normal metallic rings. The rings
are made of a superconducting material whose low bare transition temperature T 0c is
much smaller than the Thouless energy Ec = ~D/L2, where D is the diffusion coeffi-
cient and L is the circumference of the ring. The effect of pair breaking is introduced
in this study via the example of magnetic impurities. We find that over a rather broad
range of pair-breaking strength 1/τs, such that T 0c . ~/τs . Ec, the superconducting
transition temperature is renormalized down to minute values or zero while the PC is
hardly affected. The PC is determined by the renormalized interaction on an energy
scale given by the maximum of Ec, ~/τs, and the temperature. Our work may provide
an explanation for the magnitude of the average PCs in copper and gold, as well as
a way to determine their T 0c ’s. The measured PCs in copper (gold) correspond to T 0
c
of a few (a fraction of) mK. The dependence of the current and the dominant super-
conducting fluctuations on Ecτs and on the ratio between Ec and the temperature
is analyzed. We also discuss the renormalization of the effective interaction in the
presence of magnetic impurities.
The PC of non-interacting electrons in one, two, and three dimensional thin rings
is thoroughly studied. We find that the results for non-interacting electrons are more
relevant for individual mesoscopic rings than hitherto appreciated. The current is
averaged over all configurations of the disorder, whose amount is varied from zero up
to the diffusive limit, keeping the product of the Fermi wave number and the ring’s
circumference constant. Results are given as functions of disorder and aspect ratios
of the ring. The magnitude of the disorder-averaged current may be larger than the
root-mean-square fluctuations of the current from sample to sample even when the
mean-free path is smaller, but not too small, than the circumference of the ring. Then
a measurement of the PC of a typical sample will be dominated by the magnitude of
the disorder-averaged current. We also develop an alternative statistical approach to
approximate the current in the zero-disorder limit.
xivwzzeraha micinzn minxf ly mneiw `id mihpeewd zwipkna zeiceqid zertezd zg`mb ,dkiynn ef drtez .(mikilen-lr mpi`y mixneg) milnxep mixnegn zeieyrd zeitewqefnmicinzn minxfa zwqer ef dfz .wvend avnd ly dwiqitd megzn mi`wiqit xbz`l ,meik.mipexhwl`d oia divw`xhpi` lk oi` xy`k ode ,mipexhwl`d oia dkiyn zniiw xy`k od
.mipexhwl` ly zebef mixaeyd mipepbpnl yiy mxfd lr drtyda mipc ep` oey`xd dxwnalanqp`a micinznd minxfl zekilen-lr zeiv`ehwelt ly dnexza wqer dfzd xwir,T 0
c ,zekilen-lr zf`tl "diwpd" xarnd zxehxtnh mipc ep` ea lcena .zeilnxep zerah lyep` .Ec = ~D/L2 ,qle`z zibxp`n izernyn ote`a dphw ,zerahd zeieyr epnn xnegaep` ea lcena .diqeticd reaw z`e zrahl ipiite`d swidd z` ,dn`zda ,D-ae L-a mipnqnmegz miiw ik mi`xn ep` .mihpbn mindfn i"r znxbp mipexhwl` ly zebef zxiay ,miwqerizernyn ote`a ohw epi` mxfd ea ,T 0
c < ~/τs < Ec-y jk ,1/τs ,zebef zxiay avw ly agxp,zlnxepn efyk ,divw`xhpi`d itl rawp mxfd .mihpbn mindfnn diwpd zkxrna ekxr znerldf megza ,jci`n .dxehxtnhde ,~/τs ,Ec oian dlecbd `idy dibxp`d zl`wql ,aexwaozip df lcen zxfra .xzeia dphw e` zqt`zn jilen-lrd avnl df`td xarn zxehxtnhT 0
c ik mikixrn ep` .adfe zyegp zerah ly milanqp`a micinzn minxf ly zecicn xiaqdlly zelzd z` migzpn ep` .adfa oielw-ilin zeixiyr dnke ,zyegpa oielw-ilin xtqn epid.dxehxtnhl Ec oia qgiae Ecτs xhnxta zeihppinecd zekilen-lrd zeiv`ehwelt lye mxfd
.mihpbn mindfn zegkepa dkiynd zivw`xhpi` ly divfilnxepxa mipc ep` ,ok enkmicnlp zeicnn-zlze ,-ec ,-cg zeraha divw`xhpi` ixqg mipexhwl` ly micinzn minxfxy`k ze`vezd ,zeccea zeraha micinznd minxfd xear ik mi`ven ep` .ef dfza dagxdaep` .dk cr jxredy itkn xzei zeihpeelx opid ,zegpfen mipexhwl`d eia zeivw`xhpi`dzcin xy`k ,zraha zeixyt`d xcqd-i` zeivxebitpewl qgia mxfd rvenn z` miaygnswid ly dltknd z` mixney ep` jk jeza .iaiqeticd leabd oial qt` oia dpzyn xcqd-i`zihqn ddeab zeidl dieyr rvennd mxfd zcehiltn` .dreaw inxt ly lbd xtqne zrahd,icn ohw `l j` ,ohw mipexhwl`d ly iytegd jldnd xy`k elit` ,(rms) mxfd ly owzdi"r xwira x`ezz zipite` zraha cinznd mxfd ly dcicn ,df avna .zrahd swidl qgia`ll zekxrna mxfd xe`zl ,ziaihpxhl` ,zihqihhq dyib mibivn ep` ,sqepa .rvennd mxfd
.xcq-i`
iii
This thesis is dedicated to my grandfather - Opi - Alexander (Axel) Ze’evi.
iv
v
Acknowledgments
I warmly thank my advisors, Prof. Yoseph Imry and Prof. Ora Entin-Wohlman, for
many hours of discussion on physics and for their guidance and support. I learned
a lot from their intuition for physics, and thank them for introducing me to the
fascinating subject of persistent currents.
I wish to thank Dr. Ehud Altman, Prof. Moty Heiblum, Prof. Yuval Oreg, and
Prof. Ady Stern for fruitful discussions.
It is my pleasure to thank my friends in the condensed matter department: Ariel
Amir, Gilad Barak, Rafi Bistritzer, Assaf Carmi, Emanuele Dalla Torre, Eran Gi-
nossar, Lilach Goren, Eytan Grosfeld, Sebastian Huber, Roni Ilan, Karen Michaeli,
Dganit Meidan, Izhar Neder, Maoz Ovadia, Zohar Ringel, Eran Sela, Georg Schwiete,
Vadim Shpitalnik, and Marija Vucelja.
I thank my family for being there for me: my parents Nachum and Dganit, and
my sisters Avigail and Achinoam Alexandra. A special thank to Shlomit Bary for her
huge help.
I thank my loved ones, Lior for his endless support, and Tiltan and Avishai for
the joy they bring us.
vi
vii
Publication list
[P1] H. Bary-Soroker, O. Entin-Wohlman, and Y. Imry, Effect of pair breaking on
mesoscopic persistent currents well above the superconducting transition tem-
perature, Phys. Rev. Lett. 101, 057001 (2008). (Viewpoint in Physics [1])
[P2] H. Bary-Soroker, O. Entin-Wohlman, and Y. Imry, Pair-breaking effect on
mesoscopic persistent currents, Phys. Rev. B 80, 024509 (2009). (Editors’
Suggestion)
[P3] H. Bary-Soroker, O. Entin-Wohlman, and Y. Imry, Persistent currents of non-
interacting electrons in one-, two-, and three-dimensional thin rings, Phys. Rev.
B 82, 144202 (2010).
viii
CONTENTS ix
Contents
Abstract i
Acknowledgments v
1 Introduction 1
2 Background 5
2.1 Equilibrium currents in 1D normal rings . . . . . . . . . . . . . . . . 5
2.2 London equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Persistent currents in superconducting rings . . . . . . . . . . . . . . 9
2.4 Pair breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Renormalization of the electron-electron interaction . . . . . . . . . . 13
3 Effect of pair breaking on mesoscopic PCs well above Tc 19
3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Pair-breaking effect on mesoscopic persistent currents 27
4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.3 Derivation of the persistent current . . . . . . . . . . . . . . . . . . . 31
4.4 The dominant fluctuations . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4.1 The dominant imaginary time fluctuations . . . . . . . . . . . 38
4.4.2 The dominant spatial fluctuations . . . . . . . . . . . . . . . . 38
4.4.3 The dominant harmonics . . . . . . . . . . . . . . . . . . . . . 39
4.5 The temperature dependence . . . . . . . . . . . . . . . . . . . . . . 40
4.5.1 High-temperature regime, T À max1/τs, T0c , Ec . . . . . . . 40
4.5.2 Low-temperature regime, Tc ¿ T ¿ 1/τs, Ec . . . . . . . . . 41
4.6 Renormalization of the effective interaction . . . . . . . . . . . . . . . 42
4.7 Comparison with experiments . . . . . . . . . . . . . . . . . . . . . . 43
x CONTENTS
4.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Persistent currents of non-interacting electrons in one-, two-, and
three-dimensional thin rings . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Expression for the persistent current . . . . . . . . . . . . . . . . . . 52
5.4 Approximations for the PC harmonics . . . . . . . . . . . . . . . . . 54
5.4.1 Dimensionality of the system . . . . . . . . . . . . . . . . . . 54
5.4.2 Contributions of consecutive channels to 〈Im〉 . . . . . . . . . 55
5.5 Uncorrelated-channel regime . . . . . . . . . . . . . . . . . . . . . . . 57
5.6 Correlated-channel regime . . . . . . . . . . . . . . . . . . . . . . . . 59
5.7 rms fluctuations versus 〈I〉 . . . . . . . . . . . . . . . . . . . . . . . . 59
5.8 Discussion of experimental data . . . . . . . . . . . . . . . . . . . . . 61
5.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Appendices 63
A Supplementary background on superconductivity 65
A.1 The first experimental observations of superconductivity . . . . . . . 65
A.2 Microscopic description of the superconducting phase . . . . . . . . . 66
A.3 Ginzburg-Landau theory . . . . . . . . . . . . . . . . . . . . . . . . . 68
B Gauge invariance of second order perturbation theory 71
C Derivation of the renormalized interaction 73
D Derivation of the partition function of interacting electrons in the
presence of magnetic impurities . . . . . . . . . . . . . . . . . . . . . 77
E An alternative statistical approach for the description of the
current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Bibliography 85
1
Chapter 1
Introduction
Consider an electron encircling a static magnetic flux, where the magnetic field is not
present in the area that the electron may occupy. Will the electron be affected by the
flux? According to classical mechanics – No. As found by Y. Aharonov and D. Bohm
in 1959, quantum theory provides a different answer [2]. If the electron can move
coherently around the ring, its wave function will have a phase shift which depends
on the flux it encircles. This phase shift can be observed in several realizations of the
Aharonov-Bohm (AB) effect.
The persistent current (PC) is a realization of the AB effect [3]-[6]. Both nor-
mal and superconducting rings exhibit these flux-dependent currents. Macroscopic
superconducting rings can exhibit PCs, since superconductivity is a bulk coherent
phenomenon. In contrast, in normal rings (made of, e.g., a metal or a semiconduc-
tor), PCs occur only when the rings are of a mesoscopic size (. 10µm) and are kept
at very low temperatures (. 10mK). These conditions are required to ensure that the
electrons can maintain a coherent wave function while going around the ring at least
once. In other words, observing PCs in normal rings is possible when the circum-
ference of the ring, L, is comparable or smaller than the dephasing length, Lϕ, and
the thermal length, LT . There is no requirement that L À `, where ` is the elastic
mean free path, PCs do not decay in time, even in a system that has a finite electrical
resistance, since they characterize the ground state of the system [7]. The orbital
response of a ring-type structure to magnetic fields was observed and discussed very
early [8], mainly in the context of the response of aromatic molecules. However, it
was not before 1990 that PCs were observed in rings of a mesoscopic size (∼ 10µm).
Even today, the phenomenon of PCs in normal rings challenges both theoreticians
[9]-[23] and experimentalists [24]-[31] of mesoscopic physics. The main challenge is
to explain the large amplitudes – compared to theoretical predictions – of the PCs
measured in ensembles of normal rings [24]-[27]. It was shown by V. Ambegaokar and
U. Eckern [20, 21] that the PC of interacting electrons is much larger than the one of
non-interacting electrons. This can be explained by the suppression of fluctuations of
2
the local density of the electrons due to interactions [32]. However, the calculations
of Ambegaokar and Eckern [20, 21] yield a current whose amplitude is too small (by
almost an order of magnitude) compared to the ones measured in several experiments
[24]-[27].
In Chapters 3 and 4 we propose an explanation for this puzzle. We show [33,
34] that by taking into account the effect of pair breaking, the calculated PC of
attractively-interacting electrons can agree with the PC measurements in copper [24]
and gold [25] rings. The contribution of superconducting fluctuations to the PC may
be significant even when the transition temperature of the material is suppressed by
pair breaking. The above situation appears in the accessible regime
T 0c . ~/τs . Ec . (1.1)
Here T 0c is the bare transition temperature, at which a material that is perfectly
clean from pair breakers turns into a superconductor. The pair-breaking rate and the
Thouless energy are denoted by 1/τs and Ec, respectively. The transition temperature
is suppressed when the left inequality in expression (1.1) is satisfied. We show that
when the right inequality in expression (1.1) is satisfied, the PC is determined by
the renormalized interaction on the scale of Ec, and is almost insensitive to pair
breakers. We suggest that once copper and gold are purified from pair breakers,
these materials would exhibit a superconducting phase transition, and we estimate
their T 0c . We consider magnetic impurities as the pair breakers in our analysis. A
purification of these materials from magnetic impurities is difficult experimentally
[35]. Therefore, a superconducting transition has not been yet directly observed in
these metals [36]-[38]. The transition temperatures of realistic samples of metals such
as copper and gold are expected [36]-[38] to be extremely small or zero. In Ref. [21]
an attractive interaction between the electrons was considered but the effect of pair
breaking was not considered. Therefore, Ambegaokar and Eckern [21] have employed
in their estimates small values of the attractive coupling. Consequently, they came
up with a magnitude for the PC which is smaller by a factor of order 5-8 than the
measured one [24].
In Chapter 5 we study the disorder-averaged PC of non-interacting electrons in
one- two- and three-dimensional thin rings [39]. We consider various amounts of
disorder in the ring – from zero up to the diffusive limit. By this we generalize and
correct previous results in the field [9]-[12]. The magnitude of the disorder-averaged
current is compared to the root-mean-square (rms) fluctuations of the current. The
former may be larger than the rms fluctuations even when the ring is slightly diffusive.
In the following chapter we give the background of our work. The results of our
study are published in the journals of the American Physical Society [33, 34, 39].
Chapters 3, 4, and 5 are copies of Refs. [33], [34], and [39], respectively. Appendices
A, B, and C contain supplementary background material. Appendices D and E, are
3
copies of the appendices of our papers [34] and [39], respectively. Appendix D, gives
a detailed derivation of the partition function of the system discussed in Chapters
3 and 4. In Appendix E, we give an intuitive approximation for the PC of non-
interacting electrons in clean rings. This approximation relies on the probabilities of
the transverse channels in the ring to be occupied by an even or an odd number of
electrons.
Analogous effects in normal-superconducting proximity systems, will be discussed
in Ref. [40] (mentioned also on page 13), which is in progress and not included in this
thesis.
4
5
Chapter 2
Background
This chapter begins with a summary of the arguments given in the work of M.
Buttiker, Y. Imry, and R. Landauer [7], showing that a current characterizes the
ground state of an AB normal ring. Then, we summarize the London theory and
compare the PC of a superconducting ring to the one of a ring of non-interacting
electrons. Finally, we discuss the renormalization of the coupling constant of the
interaction between the electrons, and the effect of pair breakers on this renormal-
ization. Specifically, we consider pair breaking due to an AB flux in a ring geometry
and due to magnetic impurities. Appendix A contains a brief summary of results of
the microscopic theory of superconductivity and the Ginzburg-Landau theory used
in this section.
2.1 Equilibrium currents in 1D normal rings
A major difficulty in the theory of PCs was that electron scattering was expected to
eliminate the currents in any realistic system. In a typical mesoscopic metallic ring, `
is much shorter than L. Thus, it was expected that a particle will lose coherence before
it completely encircles the ring. This expectation is not correct since elastic scattering
due to a static non-periodic potential influence but does not eliminate interference
effects. This potential, which determines `, may result from, e.g., impurities in the
crystal or imperfect boundaries. Inelastic scattering, where, e.g., a phonon is absorbed
or emitted by an electron, does destroy the coherence. However, Lϕ can be made
longer than the circumference of the ring.
Buttiker, Imry, and Landauer [7] have shown in 1983 that PCs do not vanish due
to static potentials. They pointed out the equivalence of the Schrodinger equation in
the following two cases: a one dimensional periodic crystal and a one dimensional AB
ring with a static potential. The Schrodinger equation of a one dimensional periodic
6 2.1
crystal is [−~
2∇2
2m+ V (x)
]ψ = Eψ , V (x + a) = V (x) , (2.1)
where the potential has the periodicity of the crystal. In this case the wave function
is a Bloch wave [42]
ψk(x + a) = eikaψk(x) . (2.2)
The Schrodinger equation of a one dimensional AB ring is[− ~
2
2m
(∇+
2πiφ
L
)2
+ U(x)
]ψ = Eψ , (2.3)
where φ is the flux in units of the magnetic flux quantum, Φ0 = h/e, and the potential
satisfies
U(x + L) = U(x) , (2.4)
similarly to V (x) in the one dimensional periodic crystal. We can now gauge out the
flux using the transformation
ψ(x) = ψ(x)e−2πiφx/L . (2.5)
The Schrodinger equation for ψ is given by Eq. (2.1) with V → U and ψ → ψ.
Therefore, ψ is also a Bloch wave
ψk(x + L) = eikLψk(x) . (2.6)
Moreover, the spectrum of the ring is equivalent to the spectrum of an electron in a
periodic potential. The periodic boundary condition on ψ yields
ψ(x + L) = e2πiφψ(x) . (2.7)
Comparing Eqs. (2.6) and (2.7) one sees that for the AB ring
k =2π(m + φ)
L, (2.8)
where m is an integer. Thus, the spectrum of an AB ring varies with the flux. The
spectrum is periodic in the flux, and its periodicity is given by Φ0. The role of Φ0 is
analogous to the role of the reciprocal-lattice constant in the picture of the reduced
Brillouin zone of the one dimensional crystal.
A weak aperiodic potential removes the degeneracies of the energy levels at φ =
±1/2, and a stronger potential changes the spectrum even further. The key point
is that the spectrum depends on φ in a non-trivial way. The velocity of an electron
in the nth level at a given flux, φ, is proportional to the slope of the energy of the
nth level, En(φ), at φ = φ. The population of states in the normal ring plays an
important role. For example, close to the Fermi energy, EF , an energy level may
cross EF at a finite flux. Thus, at this flux the population in a ring in the grand-
canonical ensemble is changed [43] Summing the velocities of all the particles in the
ring yields the equilibrium flux-dependent PC.
2.2 7
2.2 London equation
In the Drude model the motion of a particle of a mass m and charge e in an electric
field E is given by
mdv/dt = eE−mv/τ , (2.9)
where v is the drift velocity and τ is the phenomenological relaxation time. The
latter describes the time it would take the scattering from defects to bring the drift
velocity of the particle to zero. For a normal metal, τ is the mean-free time between
collisions, τ . Thus, the steady-state current obeys Ohm law J = nev = (ne2τ/m)E.
On the other hand, taking 1/τ = 0 for a density of n∗ particles of charge e∗ and mass
m∗, yields
dJ/dt = (1/4πλ2L)E , (2.10)
where
λ2L =
m∗
4πn∗e∗2. (2.11)
(We take c = 1 for the speed of light.) Taking the curl of Eq. (2.10) one finds that
the microscopic value of the magnetic field in this case obeys
∂
∂t
(∇2B− 1
λ2L
B
)= 0 . (2.12)
We use here Maxwell equations
∇×B = 4πJ +∂E
∂t(2.13)
and
∇× E = −∂B
∂t, (2.14)
ignoring the last term in Eq. (2.13).
In 1935, the brothers F. and H. London [44] employed the equation
∇2B− 1
λ2L
B = 0 (2.15)
to describe the Meissner effect in superconductors. Namely, they chose the constant
solution of Eq. (2.12) to be zero. Equation (2.15) results in an exponential decay of the
magnetic field from the surface of a superconductor into the bulk, where the London
penetration length is given by λL, in agreement with the experimental observation of
Meissner and Ochsenfeld [45]. The above derivation is carried out for a clean system.
Later it was found that not too strong disorder just modifies the penetration length,
see Sections 2.4 and A.2.
London theory assumes a smooth variation of the current and density of the su-
perconducting electrons. Thus, in the London theory the penetration length, λs, must
8 2.2
be much larger than the coherence length, ξ. Namely, this theory is not appropriate
for type I superconductors. Type I superconductors do exhibit the Meissner effect
but here λs is calculated using a method suggested by Pippard [46], which is based
on a non-local description of the magnetic response of the superconductor. For type
II superconductors the London equation is valid for weak fields, H < Hc1.
As was pointed out by F. London in 1935 [47], one can obtain Eq. (2.15) by
assuming a rigid superconducting wave function. We begin with the expression for
the current density
〈J〉 = −n∗e∗
m∗ 〈p + e∗A〉 , (2.16)
and assume a rigid wave function which keeps the ground-state expectation-value of
the momentum at its value for A = 0. By time-reversal symmetry, the ground-state
expectation-value is 〈p〉 = 0. This yields
〈J〉 = −n∗e∗2
m∗ A . (2.17)
Taking the curl of Eq. (2.17) and employing the Maxwell equation (2.13), results in
Eq. (2.15). Equation (2.17) is valid only in the London gauge
∇ ·A = 0 . (2.18)
This gauge is required to ensure charge conservation, ∇ · J = 0, and that the current
will not flow outwards from the surface of an isolated simply connected superconduc-
tor [48]. In the London gauge A has no longitudinal component. Therefore, the as-
sumption imposed on the wave function used below Eq. (2.16) can be weakened. The
wave function is assumed to be rigid only with respect to transverse perturbations.
In the following paragraph we discuss the derivation of the current using perturbation
theory in the vector potential for an arbitrary gauge at zero temperature.
In the absence of A the current vanishes, and thus there is no contribution from
the first order term in perturbation theory. The diamagnetic term which is second-
order in the vector potential is given by
E(2)dia =< i|H2|i > , H2 =
e∗2A2
2m∗ . (2.19)
Here |i〉 stands for the ground-state wave function of the superconductor in the ab-
sence of A. Employing
J = − 1
V
⟨∂H
∂A
⟩= − 1
V
∂E
∂A(2.20)
gives
Jdia = − 1
V
∂E(2)dia
∂A= −n∗e∗2
m∗ A . (2.21)
2.3 9
Here V is the volume of the system. The diamagnetic part of the current, Eq. (2.21),
looks like the London expression for the current, Eq. (2.17). However, Eq. (2.21)
holds for an arbitrary gauge. The remaining paramagnetic term in second-order
perturbation theory in the vector potential is given by
E(2)para =
∑j
|〈i|H1|f〉|2Ei − Ej
, H1 = − i~e∗
2m∗ [∇ ·A + A · ∇] . (2.22)
This term is negative since the unperturbed ground-state energy, Ei, is smaller than
the unperturbed energies of the excited states, Ej. Thus, quite generally, this term
leads to a paramagnetic response. When the London gauge is used, we want E(2)para to
vanish in order for the current to agree with Eq. (2.17). This leads one to suspect that:
(1) the matrix elements for creating transverse excitations in the superconductor by
a slowly varying magnetic field vanish, and (2) the excitation energies Ei − Ej are
finite.
The longitudinal component of A results in a non-vanishing paramagnetic cur-
rent. However, the contribution of the longitudinal component of A cancels when
summing the diamagnetic and the paramagnetic terms. The above derivation as-
sumes that the unperturbed Hamiltonian is time-invariant and its ground state is
non-degenerate. In Appendix B we show that in this case the sum of the diamagnetic
and the paramagnetic terms is gauge invariant.
2.3 Persistent currents in superconducting rings
In this section we obtain the mean field expressions for the PC of a superconducting
ring from the Ginzburg-Landau theory. The latter is summarized in Appendix A.
Expressions for the flux quantization, and the Little-Parks effect are also obtained. We
consider a thin superconducting ring enclosing a magnetic flux Φ. Using A = Φ/Lϕ
in Eq. (A.17) results in a set of solutions for the order parameter
Ψn = Ψ0
[1− 4π2(n + φsc)
2ξ2/L2]1/2
e−inϕ , n = ±1,±2, .. (2.23)
Here ξ is the coherence length and φsc = Φ/Φ0SC is the flux in dimensionless units,
where Φ0SC = (h/2e). The equilibrium state of the superconductor is given by Ψn,
where n is the integer that minimizes (n + φsc)2. The PC density at equilibrium is
obtained by substituting Ψn, Eq. (2.23), in Eq. (A.18)
J = −ϕ4π~e|Ψ0|2
mL
[1− 4π2(n + φsc)
2ξ2/L2](n + φsc) . (2.24)
When ξ ¿ L, the equilibrium current density, Eq. (2.24), has a sawtooth shape as a
function of the flux, with a periodicity of Φ0SC [49].
10 2.3
Well below Tc the order parameter of a bulk superconductor is not modified when
the magnetic flux is changed. Thus, changing the AB flux can result in high currents.
When the system is cooled below Tc in the absence of a flux, the order parameter is
given by Ψn=0. In this case (also when |φsc| > 0.5) the current is given by Eq. (2.24)
with n = 0. When L À ξ, see Eq. (A.22),
J = −ϕ16πe
~Lξ2N (0)∆2φsc . (2.25)
Here ∆ is the gap at φsc = 0, and N (0) is the density of states around the Fermi
energy. The dimensionality of the density of states is one over energy. For clean and
for dirty systems, ξ is given by Eqs. (A.13) and (A.14), respectively. The gap, ∆,
does not change due to weak disorder (Anderson theorem [50]). Therefore, at zero
temperature Eq. (2.25) yields the currents
Iclean ∼ −NtotevF
Lφsc , Idirty ∼ Iclean
`
ξ0
∼ −Ntot∆`
Lφsc . (2.26)
Here Ntot is the number of transverse channels in the ring [51]. For finite temperatures
T . Tc, the temperature dependence of ξ2∆2, see Appendix A, yields that the current
is given as in Eq. (2.26) with an additional prefactor of (T − Tc)/Tc. The Ginzburg-
Landau mean field theory gives zero PC at T > Tc. Note that we have used this
theory in Eq. (2.26) although it is strictly valid only for temperatures around Tc.
The PC in a superconductor is much larger than the one of a normal ring. The
pairing correlations in the superconducting phase on one hand, and the change in
the occupation of the single-electron states in the normal phase on the other, result
in qualitatively different PCs in these phases. For a normal ring, the sign of the
current is highly sensitive to the exact value of kF L. The magnitude of this current
is calculated [9] in two regimes in which the analytical expression for the current can
be simplified. In the case of zero temperature and zero disorder, these regimes are
defined in the first and second rows of Table 5.1 [39, 52], and the magnitudes of the
corresponding currents are given by
∼√
NtotevF
Land ∼ Ntot
kF L
evF
L, (2.27)
respectively. These values for the magnitude of the PC of a normal ring are smaller
than the one in Eq. (2.26) for a clean superconducting ring by
√Ntot and kF L , (2.28)
respectively. In the diffusive limit, ` ¿ L, the typical magnitude of the current in a
normal ring is given by the rms fluctuations of the current
δI =eEc
~sin(2πφ) =
evF
L
`
Lsin(2πφ) . (2.29)
2.3 11
The ratio between the current of a dirty superconductor and the typical magnitude
of the current in a diffusive normal ring is very high
∼ NtotL/ξ0 = ∆/δ , (2.30)
where δ = ~vF /NtotL is the level spacing. The differences between the PC of normal
and of superconducting rings are further discussed in Section 5.2 and Appendix E
[39].
The transition temperature of a superconducting ring decreases in the presence
of an AB flux because of the kinetic energy associated with the current in the super-
conducting phase. W. A. Little and R. D. Parks [53] measured the dependence of
the transition temperature of small cylinders on the flux. They observed a periodic
parabolic-like behavior of Tc(Φ), with a period of Φ0SC. Substituting the equilib-
rium order-parameter, see Eq. (2.23), in Eq. (A.15) yields a similar behavior of the
free-energy density
f − fN0 = −1
2n(0)∆2
[1− (2πξ)2(n− φsc)
2
L2
]2
. (2.31)
Here N (0) divided by the volume of the system is denoted by n(0). The normal-
superconductor transition occurs when f − fN0 = 0. In short rings with ξ > L/π
the flux may fully suppress Tc, see Eq. (2.31). In thin and narrow rings and near the
transition temperature, the fluctuations of the order parameter become important
[54]-[56]. Then, for example, the sawtooth shape of the equilibrium current mentioned
above is rounded [4], [57]-[59].
G. Schwiete and Y. Oreg [59] studied the effect of pair breaking due to an AB flux
on PCs in mesoscopic superconducting rings. They show that superconducting fluc-
tuations result in a measurable PC in the regime where the flux-dependent transition
temperature vanishes. Their results are relevant to recent transport measurements
of the strong Little-Parks oscillations [60] and to PCs measurements in Al rings, see
Ref. [59]. The studies in Refs. [33] and [59] were carried on independently around the
same time.
Evaluating∮
J · dl around a line encircling the ring and going through its interior
results in the fluxoid quantization [47, 61]
Φ′ = Φ +4πλ2
s
c
∮J · dl = nΦ0SC . (2.32)
When the width of the ring, W , satisfies W À λs, the current is zero in the bulk.
Then, the fluxoid quantization reduces to that of the flux. Little and Parks [53]
demonstrated in their experiment that it is indeed Φ′ and not Φ which is quantized.
12 2.4
2.4 Pair breaking
The attractive interaction in a superconductor pairs electrons of time reversed states.
In a clean superconductor these are states of opposite momenta and opposite spins.
Anderson theorem [50, 62] states that if a static external perturbation does not break
the time-reversal symmetry, and does not cause a long-range spatial variation of
the order parameter (namely, ∆(r) is constant), the transition temperature and other
thermodynamic properties of the superconductor remain unchanged in the presence of
this perturbation. Non-magnetic impurities do not break the time-reversal symmetry
of the Hamiltonian. Including weak disorder affects the expression for the order
parameter only by replacing the density of states of the clean system around the
Fermi energy, N (0), in Eq. (A.8) by the one of the disordered system, N dis(0). In
three-dimensional weakly disordered systems, kF ` À 1, the density of states is almost
unchanged, N dis(0) ' N (0). Thus, disorder essentially does not modify Eqs. (A.8),
(A.9), and (2.40) for ∆ and Tc. However, when a pair-breaking perturbation (namely,
a perturbation that breaks the time reversal symmetry) is introduced, the disorder
may modify the thermodynamics. The PC of a superconducting ring discussed in
Section 2.3 is an example for that. This current is generated due to the AB flux which
corresponds to a term in the Hamiltonian that breaks the time-reversal symmetry.
Indeed, the PC depends on the amount of disorder in the ring, see Eq. (2.26).
Magnetic impurities, e.g., manganese ions (Mn) or gadolinium ions (Gd), may
suppress the transition temperature, since they break the time-reversal symmetry.
The reduction of Tc due to magnetic impurities was calculated by A. A. Abrikosov
and L. D. Gorkov in 1961 [63]. They showed that the relation between Tc and the
spin-scattering time due to magnetic impurities, τs, is given by
ln
(Tc
T 0c
)+ Ψ
(1
2+
1
2πTcτs
)−Ψ
(1
2
)= 0 . (2.33)
The digamma function is defined by
Ψ(z) = Γ′(z)/Γ(z) = − ln(γE)− 1
z+
∞∑n=1
(1
n− 1
n + z
), (2.34)
where γE ' 1.78 is the Euler constant. A derivation of Eq. (2.33) is given in Chapters
3 and 4. A full suppression of Tc is obtained when
~/τs ≥ πT 0c /2γE ' 1.76T 0
c . (2.35)
We can write Eq. (2.35) in terms of length scales: the magnetic-impurities scattering
length Ls =√
Dτs, and the superconducting coherence length in the absence of pair
breaking ξ(0) =√~D/Tc. When Eq. (2.35) is obeyed pair breaking occurs on a scale
smaller than the the characteristic distance between two paired electrons, Ls < ξ(0).
Thus, the bulk material cannot superconduct.
2.5 13
While the attractive interaction creates Cooper-pair fluctuations above Tc, the pair
breakers may flip the spin of one of the electrons, and thus reduce these fluctuations.
The effect of pair breakers must be taken into consideration for temperatures lower
than or of the order of the pair-breaking energy scale. This energy scale is given by
~/τs when the pair-breaking is caused by magnetic impurities. Magnetic fields can
also suppress Tc [62, 64]. The suppression of Tc in a ring geometry due to an AB
flux is discussed around Eq. (2.31). A contact between the superconductor and a
normal conductor (proximity effect) induces a reduction of Tc [62, 65, 66]. We study
the effect of proximity on the PC in Ref. [40] and also discuss the additional effect of
magnetic impurities.
Abrikosov and Gorkov [63] have shown that for temperatures very close to, but
smaller than, Tc(τs), the system is in a gapless superconducting phase. In this phase
the order parameter, ∆, defined in Eq. (A.2) is finite. However, the excitation spec-
trum is gapless. Namely, E = 0 is an excitation energy of the system [64]. The phase
diagram of systems with other pair-breaking mechanisms also includes the gapless
phase [67].
In the limit of strong disorder there are localized states in the system. Even before
the system is in the localization regime, the effective Coulomb repulsion increases due
to a weakened renormalization of the interaction because of disorder. Thus, ∆ and Tc
are reduced [68]-[70]. Resistivity measurements of homogenous Bi layers, deposited
onto Ge, of varying thickness were carried out in Ref. [71]. These measurements
demonstrate the relation between disorder and superconductivity. The normal-phase
sheet resistivity increases with decreasing thickness of the layer. In Ref. [71], layers of
low normal-phase sheet resistance of ∼ 0.1kΩ are of thickness of 7nm, and layers of
high normal sheet resistance of ∼ 100kΩ are of thickness of 0.5nm. Low (respectively,
high) normal sheet resistance are found [71] to correspond to a decrease (respectively,
increase) of the resistance with decreasing temperature.
2.5 Renormalization of the electron-electron inter-
action and its sensitivity to pair breaking
At low temperatures, electrons in a solid may have a net attraction among them-
selves. This is somewhat surprising since the bare Coulomb repulsion is larger than
the bare phonon-mediated attraction. The net attraction occurs due to the decrease
(respectively, increase) of the strength of the renormalized repulsive (respectively, at-
tractive) interaction as the temperature decreases. Both interactions are renormalized
by second-order processes with intermediate states of higher energy, which we term
intermediate scattering. In this section we summarize the renormalization procedure.
The obtained renormalized interaction, Eq. (2.39), is then related to the coefficient of
14 2.5
|∆|2 in the Ginzburg-Landau free energy, and to the renormalization procedure used
in Sections 3 and 4 for an AB ring with magnetic impurities.
We consider an energy-band of width D0 and a bare interaction between the
electrons V0 = V (D0). When T ¿ D0, direct scattering into or from states whose
energy is much larger than T is forbidden. (The energy is measured from EF .)
However, scattering between two low-energy states has a contribution coming from
processes that involve second-order processes with intermediate states of high energy.
These contributions are taken into account in the renormalized interaction between
low-energy particles. The renormalization of the interaction is done in a series of
steps. In each step an energy-segment of width δD is taken out from the edges of the
effective energy-band. Thus, at each step the band is effectively reduced from [−D, D]
to [−D + δD, D− δD]. In each step the interaction must be modified, to incorporate
the contribution of intermediate scattering into states whose energies were removed
from the energy band,
|Ek − EF | ∈ [D − δD, D] . (2.36)
Here Ek = ~2k2/2m, and for now D = D0. In each step the contribution of these
intermediate scattering is small. Thus, we use second-order perturbation theory in
each step in the calculations of the renormalized interaction, see Fig. 2.1.
!
Figure 2.1: A step in the renormalization of the interaction. The wavy left line stands
for the dressed interaction, after reducing the width of the effective energy-band from
D to D−δD. To second order in perturbation theory, the dressed interaction is given
by a single scattering process - the bare interaction (circle), plus a double scattering
process. The latter is composed of two bare interactions and two Green functions (the
solid lines), where the respective energy terms in these intermediate Green functions
are given by Eq. (2.36).
We introduce the dimensionless variable of the interaction λ = −N (0)V . Thus, λ
is positive for an attractive interaction, and negative for a repulsive one. Each step
in the renormalization changes the interaction by
λ(D − δD) = λ(D) + λ2(D)δD/D . (2.37)
In Appendix C we give a detailed derivation of Eq. (2.37), and specify the approx-
imations that we made in order to obtain it. We assume δD in Eq. (2.37) to be
infinitesimal, and obtaindλ
λ2= −dD
D. (2.38)
2.5 15
The renormalized interaction is given by the solution of Eq. (2.38)
λ(D) =[λ−1
0 − ln(D0/D)]−1
, (2.39)
where λ0 = −N (0)V0. One can see from Eq. (2.39) that a repulsive interaction is
renormalized downwards. Namely, its strength is reduced as D is reduced. On the
other hand, an attractive interaction is renormalized upwards. When the temperature
is much larger than the other energy scales in the system (apart from D0), one may
replace D by T . Therefore, even though the bare Coulomb repulsion can have a
larger magnitude than the bare phonon-mediated attraction, the net interaction at
low temperatures / energies may be attractive.
The initial energy band associated with the Coulomb repulsion is approximately
EF . In low Tc superconductors, the attractive interaction is mediated by phonons,
and thus the effective bandwidth is given by the Debye energy, ~ωD. We notice from
Eq. (2.39) that for the attractive interaction, λ diverges at low enough temperatures.
This temperature is identified as the critical temperature for superconductivity, Tc.
It is given by
Tc = ~ωD exp(−1/λ0) . (2.40)
Apart from a numerical correction (a factor of 1.14) in the prefactor, this result is in
agreement with the result of the BCS theory, Eq. (A.9). Here
λ0 = λatt,0 + λrep(~ωD) , (2.41)
where the bare phonon-mediated attraction coefficient, λatt,0, is positive, and the
negative screened Coulomb interaction coefficient is renormalized from EF down to
~ωD using Eq. (2.39). Substituting D = T in Eq. (2.39) and using Eq. (2.40) one
obtains
λ(T ) =
[ln
(T
Tc
)]−1
. (2.42)
At D = ~ωD the magnitudes of the coupling coefficients of the attractive and the
repulsive interactions are similar, see Fig. 2.2.
One way to relate λ(T ) to the coefficient of |∆|2 in the Ginzburg-Landau free
energy is the following. One can use the effective interaction λ(T ), given by Eq. (2.42),
in the partition function, see Eqs. (C.1)-(C.5), in which only very low fermionic
degrees of freedom should be left, |Ek−µ| . T . Employing the Hubbard-Stratonovich
transformation will give a coefficient of βN (0)/λ(T ) that multiplies |∆(q = ν = 0)|2in the action. [In this section we consider only the renormalization of the q = ν = 0
term, see the discussion around Eq. (C.12).] Note that
λ(T )−1 −−−→T→Tc
T − Tc
Tc
, (2.43)
see Eq. (2.42), in agreement with the Ginzburg-Landau theory.
16 2.5
0
ln(EF/T)
λ
rep
λatt
λ
ln(EF/ω
D) ln(E
F/T
c)ln(E
F/T
c,att)
Figure 2.2: A schematic plot of the renormalization of the repulsive, attractive, and
total interactions, for a bulk system without pair breaking. The renormalized coupling
constants of the interactions are plotted using Eqs. (2.39) and (2.41) with D = T , as
a function of ln(EF /T ). For the Coulomb repulsion D0 = EF , and for the phonon-
mediated attractive interaction and the total interaction D0 = ~ωD. Note that λ
equals the sum of λrep and λatt only at D = ~ωD. The renormalized coupling constant
for the attractive interaction diverges at D = Tc,att, where Tc,att > Tc.
An alternative way to obtain the coefficients of |∆|2 is used in Sections 3 and 4. In
these sections we first introduce the field ∆(q, ν) in the partition function, and then
we integrate over all the fermionic degrees of freedom. In this analysis we consider an
AB ring with magnetic impurities. The lowest wave-number for the variation of ∆
is qmin = ϕ4πφ/L. Identifying the coefficient of |∆(qmin, ν = 0)|2 with βN (0)/λ(T )
results in [63, 73], see Eq. (4.46),
λ(T )−1 = λ−10 + Π(qmin, ν) = ln
( T
T 0c
)+ Ψ
(1
2+
8π2Ecφ2 + ~/τs
2πT
)−Ψ
(1
2
). (2.44)
Here, we change the notation and φ denotes φ modulo 1, whose values are in the
range (−1/2, 1/2]. In Eq. (2.44), the limit φ = 1/τs = 0 is consistent with Eq. (2.42).
The renormalized interaction, see Eq. (2.44), does not diverge when
|φ| >(
T 0c
16πEcγE
)1/2
or when ~/τs > πT 0c /2γE . (2.45)
Then, the system cannot become superconducting. Note that using φ = 0 and
λ(T )−1 = 0 in Eq. (2.44) yields Eq. (2.33). The length L/φ plays a similar role
to the one of Ls in destroying the Cooper pairs, see the discussion below Eq. (2.35).
From Eq. (2.45) we see that when L/φ < ξ(0) the AB ring cannot superconduct.
In Fig. 2.3 the renormalization of the coupling constant of the attractive interaction
is shown for systems with and without pair breakers. The renormalization of the
effective interaction is further discussed in Section 4.6.
2.5 17
00
λ
ln(ωD
τs) ln(ω
D/T)ln(ω
D/T
c0)
Figure 2.3: The coupling constants of the interaction are schematically plotted as a
function of ln(ωD/T ) using Eq. (2.44). For a system without pair breakers (solid line)
λ diverges at T = T 0c . The dashed line corresponds to the situation where Tc vanishes
due to strong enough pair-breaking rate, see Eq. (2.45).
In Sections 3 and 4 [33, 34] we consider the effect of magnetic impurities in a nor-
mal metal with attractive interactions. We show that the PC is roughly determined
by the interaction on the energy scale D ∼ maxEc, ~/τs, T, see also Section 4.6.
Thus, there is a large window of pair-breaking rate given by Eq. (1.1), in which the
bulk transition temperature is suppressed but the PC is not.
18 2.5
19
Chapter 3
Effect of pair breaking on
mesoscopic PCs well above Tc
Published in: Phys. Rev. Lett. 101, 057001 (2008).
3.1 Abstract
We consider the mesoscopic normal persistent current (PC) in a very low-temperature
superconductor with a bare transition temperature T 0c much smaller than the Thouless
energy Ec. We show that in a rather broad range of pair-breaking strength, T 0c .
~/τs . Ec, the transition temperature is renormalized to zero, but the PC is hardly
affected. This may provide an explanation for the magnitude of the average PCs in
the noble metals, as well as a way to determine their T 0c ’s.
3.2 Introduction
The magnitude of the equilibrium averaged persistent currents (PCs) [7, 57] in normal
metals has been a long-standing puzzle. Experiments [24]-[26] produce a current larger
by at least 2 orders of magnitude than the theoretical prediction for non-interacting
electrons [10, 14, 17] and seem to indicate that the low-flux response is diamagnetic.
The average PC of a diffusive system with interactions was calculated first in this
connection [74] in Refs. [21] and [20]. The Resulting PC was found to be much larger
than that of a non-interacting system, but nevertheless not large enough to explain
the experiments.
Repulsive electron-electron interactions [20] result in a paramagnetic response (at
small magnetic fluxes) whose magnitude is smaller than the experiment by about
a factor of 5. This disagreement is due to the downward renormalization of the
interaction [64, 72]. Attractive interactions [21] result in a diamagnetic response,
20 3.2
0.1 1 100
0.5
1
s
I/I(s=0)
Tc/T
c0
Figure 3.1: The first flux harmonic [m = 1; see Eq. (3.2)], in units of I(s = 0), of the
PC at T = Ec (full line) and Tc/T0c (dashed line) as functions of the pair-breaking
strength, s = 1/(πT 0c τs), displayed on a logarithmic scale.
whose magnitude (due to the very low superconducting transition temperature), is
again smaller by a factor of order five than the measured one. This is in spite of the
renormalization upward of the attractive interaction. Attractive interactions, at low
energies, imply (with no pair breaking) a transition into a superconducting state, and
the PC of such an interacting system depends on its transition temperature. These
temperatures are very low [36]-[38] for the noble metals used in the PC experiments
– hence the too small predicted values for the PC.
Here we consider attractive interactions. We show that the presence of a very
small amount of pair breakers, e.g., magnetic impurities (which seem to be very
difficult to avoid in these metals [35]), may change the picture profoundly. Obviously
one may consider other pair breakers, such as a two-level systems [75] or simply a
magnetic field [59]. In this Letter we treat specifically the case of magnetic impurities.
We find that within a significant range of the pair-breaking strength, the magnetic
impurities suppress the transition temperature down to immeasurable values, leaving
concomitantly the PC almost unchanged. The physical reason for this remarkable
observation is that the PC is determined by the interaction on the scale of the Thouless
energy Ec = ~D/L2 (∼ 20mK for a typical experimental system), while the bare
transition temperature, T 0c , is much smaller. (The circumference of the ring is denoted
by L and D is the diffusion coefficient.) This gives rise to a rather wide range of pair-
breaking strengths, presented here by the spin-scattering time τs,
T 0c . ~/τs . Ec, (3.1)
in which the actual transition temperature Tc will drop to zero [63], but the PC will be
hardly affected. As a result, it is the bare transition temperature of the system without
the magnetic impurities, T 0c , as opposed to Tc, which dominates the expression for
PC, see Fig. 3.1. We concentrate here [34] on the experimental results of Ref. [24]. In
order to explain them, it is necessary to assume a T 0c in the 1mK range for copper.
Our basic assertion is that this may indeed be the correct order of magnitude of T 0c
for ideally clean copper, but that it is knocked down to zero or to a very low value by
3.3 21
a minute, . ppm, amount of unwanted [35] pair breakers. We emphasize, however,
that our result concerning the fundamentally different sensitivities of Tc and PC to
pair breaking in the range given by Eq. (3.1), remains valid regardless of the situation
in specific materials. The Kondo screening of the spins is not considered here. Other
effects of magnetic impurities have previously been considered in Ref. [23].
3.3 Results
The expression we obtained for the PC in a diffusive ring with magnetic impurities
can be expressed as a sum over the harmonics of the magnetic flux through the ring
φ, in units of the flux quantum h/e,
I = −8eEc
∞∑m=1
sin(4πmφ)
m2
∑ν
∫ ∞
0
dxx sin(2πx)Ψ′(F (x, ν))
ln(T/T 0c ) + Ψ(F (x, ν))−Ψ(1
2)
,
F (x, ν) =1
2+|ν|+ 2/τs
4πT+
πEcx2
m2T, (3.2)
(using ~ = 1). Here ν denotes the bosonic Matsubara frequency [76], Ψ and Ψ′ are the
digamma function and its derivative, and T is the temperature. Our expression (3.2)
generalizes the result of Ref. [21] for the case where spin-scattering is present: the
Matsubara frequency |ν| is shifted by 2/τs. However, the superconducting transition
temperature (which appears formally in the denominator of the integrand) is not the
one modified by the pair breakers, but retains its bare (magnetic impurities free)
value. Interestingly enough, it follows that by measuring the PC one may determine
T 0c (which would be directly measurable only if all low-temperature pair breaking
could be eliminated).
In Fig. 3.2 the PC is plotted using Eq. (3.2). At the critical pair-breaking time
1/τs ' T 0c , corresponding to s = 1/πτsT
0c ' 1/π, the transition temperature vanishes
[63], while the PC is hardly affected. The measured PC in the copper samples of
Ref. [24] is I(T . Ec) ' −eEc. The curve with s = 1 in the upper panel, taken with
T 0c = 1.5mK and Ec = 15mK (the value for the samples of Ref. [24]) gives a PC lower
by only 25%. A better fit is possible by changing the parameters somewhat, but we do
not regard this as crucial at the present stage. Likewise, we can qualitatively explain
the result of Ref. [25]. The high frequency results of Ref. [26] require a separate
discussion [26]. The PC is reduced significantly once 1/τs ≥ Ec, or Ls ≡√
Dτs ≤ L.
For T 0c /Ec = 0.1 (0.01), the condition for Ecτs ∼ 1 is s = 10 (100).
3.4 Derivation
For completeness, we outline below the derivation of the PC in the presence of mag-
netic scattering [34]. The PC, Eq. (3.2), is obtained by differentiating the free energy
22 3.4
0
0.5
1
I/ I*
s=0s=1s=10
0 2 4 6 80
0.5
1
T/Ec
I/ I*
s=0s=10s=100
Tc0/E
c=0.1
Tc0/E
c=0.01
Figure 3.2: The first flux harmonic of the PC in units of I∗ = −eEc as a function of
the temperature, for two values of T 0c /Ec and several values of s. Keeping, at T . Ec,
up to the 100 lowest values of |ν|, was necessary for convergence. Note that the s = 0
curve in the upper panel is valid only for T/Tc ≥ 1 + Gi, where Gi is the Ginzburg
parameter (Gi ∼ 0.1 for the samples of Ref. [24]).
with respect to the flux. Our system is described by the Hamiltonian [63]
H =
∫dr
(ψ†α(r)
[(H0 + u1(r))δαγ + u2(r)S · σαγ
]ψγ(r)
− g
2ψ†α(r)ψ†γ(r)ψγ(r)ψα(r)
), (3.3)
in which the last term is the attractive interaction, of coupling g. The spin components
are α and γ, σ is the vector of the Pauli matrices, and H0 = (−i∇ − eA)2/2m − µ
(µ is the chemical potential and A is the vector potential describing the flux through
the ring). The scattering, both nonmagnetic and magnetic, is assumed to result from
Ni point-like impurities, such that
u1(r) + u2(r)S · σ ≡Ni∑i=1
(δ(r−Ri)− 1
V
)(u1 + u2SRi
· σ) , (3.4)
where V is the system volume. In averaging over the impurity disorder one assumes
that the impurity locations, Ri, are random, and so are their classical spins, such
that 〈SRi〉 = 0, and 〈SRi
· SRj〉 = δijS(S + 1).
The partition function, Z, is calculated by the method of Feynman path inte-
grals [77], combined with the Grassman algebra of many-body fermionic coherent
states in terms of the variables ψα(r, τ) (ψα(r, τ)). Introducing the bosonic fields
3.4 23
∆(r, τ) via the Hubbard-Stratonovich transformation leads to the partition function
Z =∫
D(ψ(r, τ), ψ(r, τ))D(∆(r, τ), ∆∗(r, τ))e−S with
S =
∫dr
∫ β
0
dτ( |∆(r, τ)|2
g− 1
2Ψ(r, τ)G−1
r,r;τ,τΨ(r, τ))
, (3.5)
where Ψ = (ψ↑, ψ↓, ψ↑, ψ↓). The inverse Green function G−1 (at equal positions r and
equal imaginary times τ) is given by
G−1r=r′;τ=τ ′ =
−∂τ − hφ↑ −2u2S− 0 ∆
−2u2S+ −∂τ − hφ↓ −∆ 0
0 −∆∗ −∂τ + h−φ↑ 2u2S+
∆∗ 0 2u2S− −∂τ + h−φ↓
≡
[G−1
p ∆
∆† G−1h
].
(3.6)
where h±φα = H0(±A) + u1 + sgn(α)Szu2, and S± = (Sx ± iSy)/2.
The integration over the fermionic part of the action (3.5) yields
Z =
∫D(∆(r, τ), ∆∗(r, τ))
× exp(1
2Tr ln(βG−1)−
∫dr
∫ β
0
dτ|∆(r, τ)|2
g
). (3.7)
In order to treat the boson fields ∆, we expand Tr ln(βG−1) up to second order in ∆.
This expansion is valid for temperatures well above the transition temperature, and,
strictly speaking, above the Ginzburg critical region. The zeroth order is omitted as
it leads to the tiny magnitude PC of non-interacting, grand-canonical, normal metal
rings [10]. The result in Fourier space reads (the dependence on the magnetic flux is
specified below)
Tr ln(βG−1)∣∣∣2nd
= −∑
q1,q2,ν
∑
k1,k2,ω
Tr[Gp(k1 + q1,k2 + q2, ω + ν)
× ∆(q2, ν)Gh(k2,k1,−ω)∆†(q1, ν)]
. (3.8)
The resulting expression for the partition function may be simplified considerably.
First, the terms that survive the disorder-average in the sum of Eq. (3.8) are those
for which [78] q1 = q2. Secondly, the particle and the hole Green functions, Gp and
Gh, [see Eq. (3.6)] are related,
Gh(k,k′; ω) = −Gtp(−k,−k′, ω) , (3.9)
where the superscript t denotes the transposed. Carrying out the integration in
Eq. (3.7),
Z =∏q,ν
N (0)(V
g− T Π(q, ν)
)−1
, (3.10)
24 3.4
where N (0) denotes the extensive density of states at the Fermi level. The polariza-
tion is
Π(q, ν) =1
2
∑ω
εαγKωαγ(q, ν) (3.11)
with
Kωαγ(q, ν) =∑
k1,k2
〈Gαα′(k1 + q,k2 + q, ω + ν)εα′γ′Gγγ′(−k2,−k1,−ω)〉 . (3.12)
Here ε is the anti-symmetric tensor, εαα = 0, and ε↑↓ = −ε↓↑ = 1, and G denotes the
particle Green function.
In Ref. [63] K(0, 0) was calculated using a Dyson equation. We generalize their
calculation to obtain K(q, ν) and, consequently, the polarization becomes [34]
T
N (0)Π(q, ν) = Ψ
(1
2+
ωD
2πT+|ν|+ Dq2
4πT
)−Ψ
(1
2+
Dq2 + |ν|+ 2/τs
4πT
). (3.13)
Here ωD is the cutoff frequency on the attractive interaction, and the pair-breaking
time τs is given by
1
τs
= 2πN (0)NiS(S + 1)u22 . (3.14)
The transition temperature of the system in the absence of pair breakers, T 0c , is
obtained from the q = 0, ν = 0 pole of Z, upon setting 1/τs = 0,
V
gN (0)= Ψ
(1
2+
ωD
2πT 0c
)−Ψ
(1
2
). (3.15)
(Note that the same procedure in the presence of the pair breaking reproduces the
decrease in the transition temperature Tc, as found in Ref. [63].) Since ωD À T 0c , T
we may use the asymptotic expansion of the digamma function. In this way we obtain
Z =∏q,ν
[ln
( T
T 0c
)+ Ψ
(1
2+
Dq2 + |ν|+ 2/τs
4πT
)−Ψ
(1
2
)]−1
. (3.16)
Finally, the PC is given by I = (e/h) ∂T lnZ/∂φ. In our ring geometry, the flux
enters the longitudinal component, q‖, of the vector q as
q‖ =2π
L(n + 2φ) , (3.17)
where n is an integer. Only the zero transverse momentum contributes significantly to
the current. Our result (3.2) is obtained upon inserting Eq. (3.16) into the definition
of the current and employing the Poisson summation formula. It then follows from
3.4 25
Eq. (3.2) that values of τs which are detrimental to Tc, may hardly affect the PC (see
Fig. 3.1).
We conclude by further explaining the physical argument behind our result. Very
roughly, the renormalization of the dimensionless attractive interaction λ (> 0) from
a higher frequency scale ω> to a lower one, ω<, is given by λ(ω<) = λ(ω>)
1−λ(ω>)ln(ω>ω<
).
At T 0c and 1/τs = 0, the attractive interaction should diverge. Using this to eliminate
λ(ωD) (≡ gN(0)/V ), we obtain that for T 0c . ω << ωD, λ(ω) v 1/ln(ω/T 0
c ), which
around the Thouless scale, is close to the value found in Ref. [21]. The pair breaking
stops the renormalization at 1/τs, but does not significantly change the interaction
on the much larger scale of Ec. Our prediction can also be tested with very small
rings made of known low Tc superconductors.
We point out that the mechanism suggested by Kravtsov and Altshuler [22], re-
lating extrinsic dephasing to an enhanced PC, is different than ours, since it relies on
the rectification of the noise.
26 3.4
27
Chapter 4
Pair-breaking effect on mesoscopic
persistent currents
Published in: Phys. Rev. B 80, 024509 (2009).
4.1 Abstract
We consider the contribution of superconducting fluctuations in the mesoscopic persis-
tent current (PC) of an ensemble of normal metallic rings, made of a superconducting
material whose low bare transition temperature T 0c is much smaller than the Thou-
less energy Ec. The effect of pair breaking is introduced via the example of magnetic
impurities. We find that over a rather broad range of pair-breaking strength ~/τs,
such that T 0c . ~/τs . Ec, the superconducting transition temperature is normalized
down to minute values or zero while the PC is hardly affected. This may provide an
explanation for the magnitude of the average PCs in copper and gold, as well as a
way to determine their T 0c ’s. The dependence of the current and the dominant super-
conducting fluctuations on Ecτs and on the ratio between Ec and the temperature is
analyzed. The measured PCs in copper (gold) correspond to T 0c of a few (a fraction
of) mK.
4.2 Introduction
Equilibrium persistent currents (PCs), flowing in normal mesoscopic metallic rings,
have been a challenge for both experimentalists and theorists. The persistent current
is a manifestation of the Aharonov-Bohm effect–it appears when the ring is threaded
by a magnetic flux, and it is periodic in the flux enclosed in the ring [7, 57]. Due to
energy-averaging and phase-coherence limitations, one expects to monitor in experi-
ment only the lowest harmonics in the flux quantum h/e.
28 4.2
Surprisingly enough, the magnitudes of the PCs measured on huge collections of
rings (107 copper rings [24] and 105 silver rings [26]) turned out to be larger than
those expected theoretically. The periodicity observed in these large ensembles is
h/2e, i.e., half of the magnetic flux quantum. On the other hand, measurements on a
single ring [31, 79] or on a small number [25] of gold rings showed the h/e periodicity.
In the collection of 30 gold rings [25], both the h/2e harmonic and the h/e harmonic
were observed. Overall, the sign of the amplitude of the h/2e harmonic measured on
metallic rings seems to indicate that the low-flux response is diamagnetic [25, 26].
In the experiments on ensembles of rings [24]-[26], the average PC was found by
measuring the magnetic moment produced by all rings, which was then divided by
the number of rings, N , to yield the net average current of a single ring. In most of
the experiments [24]-[26], [31], the magnitude of the average PC, at low temperatures,
is roughly on the order of eEc/~. Here Ec = ~D/L2 is the Thouless energy, L is the
circumference of the ring, and D = vF lel/3 is the diffusion coefficient, where lel is
the elastic mean-free path, and vF is the Fermi velocity. (We consider the diffusive,
L À lel, case.)
The first theoretical studies of the PC have been carried out on grand-canonical
systems of non-interacting electrons [10, 12, 14, 39, 57]. In these theories, the current
in each ring is h/e periodic. The sign and magnitude of the PC of the individual
rings vary randomly due to their high sensitivity to the disorder and to the system’s
size. This results in a very small average PC, which is dominated by the exponential
factor exp(−L/2lel). Hence, the typical magnitude of the current is predicted to be√N times the standard deviation of the PC of non-interacting electrons, which at low
temperatures is on the order of eEc/~. Consequently, the persistent current carried
by non-interacting electrons is too small to explain the large-ensemble experiments.
Similarly, the PC predicted for non-interacting electrons in the canonical ensemble
[17] is substantially too small to explain the observed amplitude of the h/2e harmonic.
The theory for interacting electron systems [20, 21] predicts h/2e periodicity of
the interaction-dependent part of the PC. According to this theory, the average mag-
nitude of the PC per ring due to interactions is independent of the number of rings.
The total measured PC, divided by N , is thus expected to have an N -independent
contribution due to interactions, and an interaction-independent contribution propor-
tional to N−1/2. The presence of the h/e harmonic in the measurements performed
on a single ring [31, 79] and on a few [25] rings, and its absence in large ensembles
[24, 26], are in agreement with these theoretical predictions. Experiments on a single
ring [30] and on a large ensemble [27, 80] of semiconducting rings show the h/e and
the h/2e periodicities, respectively, consistent with the arguments given above.
Notwithstanding the order of the harmonics, their amplitudes, in particular, that
of the h/2e one, remained unexplained for the large-ensemble measurements. On the
other hand, the magnitudes of the h/e harmonic measured in Refs. [25] and [31] agree
4.2 29
roughly with the prediction for non-interacting electrons, while the PC measured by
Chandrasekhar et al. [79] turns out to be much larger.
Here we study the PC of large ensembles, focusing on the role of electronic in-
teractions. These, attractive and repulsive interactions of reasonable strengths, give
rise to comparable magnitudes of the averaged PC (within an order of magnitude),
but predict opposite signs. While repulsive electron-electron interactions [20] result
in a paramagnetic response at small magnetic fluxes, attractive interactions yield a
diamagnetic response [21], as indeed seems to be indicated in the experiments. The
magnitude of the PC predicted for electrons which interact repulsively is smaller [81]
by a factor of about 5 than, e.g., the magnitude of the PC measured in copper [24].
The effective coupling strength of repulsive interactions decreases as the temperature
decreases, due to interactions mediated by states whose energies are large compared
with the temperature [64, 72]. This “downwards” renormalization is the reason for
the disagreement between the theory for electrons interacting repulsively and the ex-
periments [20]. On the other hand, the attractive interaction is normalized “upwards”
at low temperatures [72], and eventually leads to a superconducting state. One ex-
pects the magnitude of the averaged PC due to attractive interactions, i.e., due to
superconducting fluctuations [74], to increase with the strength of the interaction, or
alternatively, to decrease as the (superconducting) transition temperature is reduced.
Since the transition temperatures of metals such as copper, gold, and silver – on
which the PC has been measured – are expected [36]-[38] to be extremely small or
zero, Ambegaokar and Eckern [21] have employed in their estimates small values of
the attractive coupling. Consequently, they came up with a magnitude for the PC
which is again smaller by a factor of order 5 than the measured one [24].
In order to reconcile the relatively large interaction required to fit the experiments
with the apparent absence of a superconducting transition, we propose that the rings
(of, e.g., copper) contain a tiny amount of magnetic impurities. We show that a
small concentration of these pair-breakers may suffice to hinder the appearance of
superconductivity, while hardly affecting the magnitude of the PC. Indeed, it seems
that a small amount of magnetic impurities is almost unavoidable in metals such as
copper. This is suggested by recent experiments [35], aimed to measure the temper-
ature dependence of the dephasing time in noble metal samples. Theoretically, one
expects [57, 82] this rate to vanish as the temperature goes to zero. However, it was
found that the dephasing time may cease to increase below a certain temperature.
This finding was attributed [35] to the presence of a small concentration of magnetic
impurities, which was reported to exist in these samples.
As is well-known, magnetic impurities act as pair breakers, leading to the vanishing
of the transition temperature Tc once the spin-scattering rate 1/τs is larger than the
bare transition temperature of the material without the magnetic impurities, T 0c [63].
At the same time, superconducting fluctuations can result in a significant PC provided
30 4.2
that the lifetime of a Cooper-pair (∼ τs at low temperatures) is larger than the time
it takes it to encircle the ring, ∼ ~/Ec. (In the experiments [24]-[26] Ec ∼ 10 mK.)
Therefore, the observation that the PC is almost unaffected by magnetic impurities
while Tc vanishes holds in the range
T 0c . 1/τs . Ec , (4.1)
(from now on we use units in which ~ = 1).
It is instructive to write the above condition in terms of lengths, for which Eq. (4.1)
reads
L . Ls . ξ(0) , (4.2)
where
Ls = (Dτs)1/2, and ξ(0) = (D/T 0
c )1/2 . (4.3)
Here the magnetic-impurities scattering length Ls is the distance a diffusing electron
covers during the time interval τs. The bulk superconducting coherence length, in
the absence of magnetic impurities ξ(0), is the characteristic distance between two
electrons forming a Cooper-pair. At low temperatures, a Cooper-pair fluctuation can
propagate a distance on the order of Ls until it is destroyed due to the scattering by
magnetic impurities. When L . Ls the pairs are sensitive to the Aharonov-Bohm
flux and consequently contribute significantly to the PC. When pair breaking occurs
on scales smaller than the characteristics distance between two paired electrons, i.e.,
when ξ(0) > Ls, then the bulk material would not become a superconductor. There-
fore, rings made of alloys which are not superconducting in the bulk due to pair
breakers, will have PCs due to Cooper-pair fluctuations provided that Eq. (4.2) is
satisfied. We show that the measured amplitude of the h/2e harmonic in copper [24]
and gold [25] rings can be understood theoretically, assuming a minute, less than one
part per million, concentration of pair breakers. Similar amounts of magnetic impu-
rities were obtained for the most purified copper and gold samples in Ref. [35]. We
point out that according to our considerations, the measurement of the PC provides
a way to estimate T 0c , which may well be unreachable by direct experiments.
This paper is organized as follows. In Section 4.3 and in Appendix D we derive
the expression for the PC due to superconducting fluctuations, taking into account
the effect of pair breakers. In Section 4.4 we characterize the dominant Matsubara
frequencies and wave numbers that contribute to the PC, and discuss the significant
harmonics. In Section 4.5 we expand the expression for the PC in the limits of high
and low temperatures. The effect of pair breaking on the renormalization of the
attractive interaction is discussed in Section 4.6. In Section 4.7 we present a detailed
comparison of our results with the experimental data, and estimate T 0c for copper
and gold. Finally, the results are summarized below.
4.3 31
In our analysis, the effect of pair breaking is brought about by the presence of
magnetic impurities disregarding the Kondo screening of the spins. Obviously one
may consider other pair breakers such as two-level systems [75], inelastic scattering
[83], or magnetic fields [59]. Other effects of magnetic impurities have previously been
considered in Ref. [23].
It was suggested by Kravtsov and Altshuler [22] that the measured currents have
a different source than the equilibrium PC discussed so far. A non-equilibrium noise,
for example, a stray ac electric field, can cause a dc by a rectification effect. In
Ref. [22] it was shown that the measured signal [24] may be explained provided that
there exists such a non-equilibrium noise. This mechanism is different from the one
suggested by us.
4.3 Derivation of the persistent current
The PC is obtained by differentiating the free energy of electrons residing in a ring
with respect to the magnetic flux enclosed in that ring. In this section, we derive the
term in the free energy which results from superconducting fluctuations. The system
consists of diffusing electrons which interact with each other attractively and are also
scattered by magnetic impurities that couple to their spin degrees of freedom. We
use the Hamiltonian [63]
H =
∫dr
(ψ†α(r)
[(H0 + u1(r))δαγ + u2(r)S · σαγ
]ψγ(r)
− g
2ψ†α(r)ψ†γ(r)ψγ(r)ψα(r)
), (4.4)
in which the last term represents the attractive interaction, with coupling g (> 0).
The spin components are α and γ, and σ is the vector of the Pauli matrices. The
free, spin-independent, part of the Hamiltonian is
H0 = (−i∇− [2π/L]φx)2/2m− µ , (4.5)
where m is the electron mass, µ is the chemical potential, and φ is the magnetic flux
through the ring, in units of h/e. The unit vector x points along the circumference
of the ring in the anti-clockwise direction. The scattering, by both nonmagnetic and
magnetic ions, is assumed to result from Ni point-like randomly-located impurities,
such that
u1(r) + u2(r)S · σ ≡Ni∑i=1
(δ(r−Ri)− 1
V
)(u1 + u2SRi
· σ) , (4.6)
where V is the system volume.
32 4.3
We calculate the partition function Z using the method of Feynman path inte-
grals combined with the Grassmann algebra of many-body fermionic coherent states
[77], in which the superconducting order-parameter is introduced by the Hubbard-
Stratonovich transformation [33]. Details of this procedure are given in the Ap-
pendix D. As is shown in the Appendix D, the partition function is (the temperature
is denoted by T )
Z = Z0
∏q,ν
(1− gT
VΠ(q, ν)
)−1
, (4.7)
where the polarization [84],
Π(q, ν) =1
2
∑ω
εαγKωαγ(q, ν) , (4.8)
consists of the Cooperon-dominated contributions
Kωαγ(q, ν) =∑
k1,k2
〈Gαα′(k1 + q,k2 + q, ω + ν)εα′γ′Gγγ′(−k1,−k2,−ω)〉 . (4.9)
Here ε is the anti-symmetric tensor, εαα = 0, and ε↑↓ = −ε↓↑ = 1, and G denotes the
particle Green function.
In Ref. [63] the polarization Π(q = 0, ν = 0) was calculated from the Dyson
equation for the Cooperon. Their calculation can be extended to general q, ν
Kωαγ(q, ν) =∑
k
Gαα(k + q, ω + ν)Gγγ(−k,−ω)
× [εαγ + Ni(u1δαα′ + u2S · σαα′)(u1δγγ′ + u2S · σγγ′)Kωα′γ′(q, ν)
]. (4.10)
Here, Gαγ = δαγGαα is the Green function averaged over the impurity disorder and
spin components (which makes it diagonal in spin-space). Averaging over the impurity
spins,
(u1δαα′ + u2S · σαα′)(u1 δγγ′ + u2S · σγγ′)
= u21δαα′δγγ′ +
1
3S(S + 1)σαα′
j σγγ′j u2
2 , (4.11)
is carried out employing Si = 0 and SiSj = δij S(S + 1)/3 (where i, j = x, y, z).
Following Ref. [63] we assume that Kωαγ = εαγKω, and then using σαα′j σγγ′
j εα′γ′ =
−3εαγ we obtain
Π(q, ν) =∑
ω
Kω(q, ν) ,
Kω(q, ν) =[1 + (2πN (0)τ−)−1Kω(q, ν)
] ∑
k
G(k + q, ω + ν)G(−k,−ω) , (4.12)
4.3 33
where the averaged Green function is
G(p, ω) = [iω − (p2/2m− µ) + isgn(ω)/2τ+]−1 . (4.13)
(The spin indices are suppressed since G is independent of them.) In Eqs. (4.12) and
(4.13),
1
τ±= 2πN (0)Ni(u
21 ± S(S + 1)u2
2) , (4.14)
where N (0) is the extensive density of states at the Fermi level. (Note that τ+ is the
elastic mean-free time.) Using Eq. (4.13) to calculate the sum over k in Eq. (4.12)
yields
∑
k′G(k′ + q, ω + ν)G(−k′,−ω)
= 2πN (0)τ+θ[ω(ω + ν)](1− τ+|2ω + ν| −Dq2τ+) . (4.15)
Upon inserting this expression into Eq. (4.12) and solving it one finds
Kω(q, ν) = 2πN (0)θ[ω(ω + ν)](Dq2 + |2ω + ν|+ 2/τs)−1 , (4.16)
where 1/τs is the pair-breaking rate
1
τs
= 2πN (0)NiS(S + 1)u22 . (4.17)
When τ+ ' τ− most of the disorder is due to the non-magnetic part. This, together
with the assumption [78] |2ω + ν|, Dq2 ¿ 1/τ+ was used in obtaining Eq. (4.16).
The summation in Eq. (4.12) over the Matsubara frequencies can be written ex-
plicitly as
T
N (0)Π(q, ν) =
∞∑n=0
[n +
1
2+|ν|+ 2/τs + Dq2
4πT
]−1
. (4.18)
Note that Eq. (4.18) also includes the negative Matsubara frequencies. This sum does
not converge and therefore a cutoff is required. The cutoff frequency on the attractive
interaction is the Debye frequency ωD and consequently, the sum is terminated at
n = ωD/2πT . As a result, the polarization is given by
T
N (0)Π(q, ν) = Ψ
(1
2+
ωD
2πT+|ν|+ 2/τs + Dq2
4πT
)
−Ψ(1
2+|ν|+ 2/τs + Dq2
4πT
), (4.19)
where Ψ is the digamma function.
34 4.3
We next express the polarization in terms of the bare transition temperature of
the system. This is the temperature at which Z/Z0 diverges for |ν| = 0 and the
smallest possible |q| in the absence of the pair breakers and the magnetic flux,
V
gN (0)= Ψ
(1
2+
ωD
2πT 0c
)−Ψ
(1
2
). (4.20)
Since ωD À T 0c , T we may use the asymptotic expansion of the digamma function,
Ψ(x À 1) ' ln(x) , (4.21)
to obtain
Z = Z0
∏q,ν
( V
gN (0)
[ln
( T
T 0c
)+ Ψ
(1
2+|ν|+ 2/τs + Dq2
4πT
)−Ψ
(1
2
)]−1). (4.22)
The effect of the pair breakers is represented by the term 2/τs in the argument of the
digamma functions.
As is mentioned above, the persistent current is given by
I = (e/2π) ∂T lnZ/∂φ . (4.23)
The flux enters the expression for Z through the longitudinal components of the
momenta, see Eq. (D.14). In our ring geometry, only momenta of zero transverse
components contribute significantly to the current, since the contribution of momenta
of higher transverse components can be shown to decay exponentially, as a function
of the ratio of L and the transverse dimension (e.g., the height) of the ring.
As is seen in Eqs. (4.22) and (4.23), the PC consists of two parts. The first arises
from differentiating Z0 and is the ensemble averaged PC of non-interacting, grand-
canonical, normal metal rings [10]. This contribution is much too small to account
for the measured amplitude of the h/2e harmonic (see Section 4.2) and, therefore,
will be omitted in the following. The other part of the PC comes from the free energy
due to the superconducting fluctuations,
I = −2eEc
∑n,ν
(n + 2φ)Ψ′(F (n, ν))
ln(T/T 0c ) + Ψ(F (n, ν))−Ψ(1
2)
, (4.24)
where we have introduced the function
F (n, ν) =1
2+|ν|+ 2/τs
4πT+
πEc
T(n + 2φ)2 . (4.25)
In particular, one notes the h/2e periodicity in the flux. Indeed, upon employing the
Poisson summation formula
I = −8eEc
∞∑m=1
sin(4πmφ)
m2
∑ν
∫ ∞
0
dxx sin(2πx)Ψ′(F (x, ν))
ln(T/T 0c ) + Ψ(F (x, ν))−Ψ(1
2)
, (4.26)
4.3 35
where
F (x, ν) =1
2+|ν|+ 2/τs
4πT+
πEcx2
m2T. (4.27)
Clearly, the fluctuation-induced PC decreases as the pair-breaking strength increases.
Our central result is that this decrease may be far less than the one caused in the
transition temperature.
In order to compare the dependence of the PC and the transition temperature on
the pair-breaking strength, we use the expression [63] for the transition temperature
in the presence of both pair breakers and magnetic flux
ln( Tc
T 0c
)+ Ψ
(1
2+
4πEcφ2
Tc
+1
2πTcτs
)−Ψ
(1
2
)= 0 . (4.28)
Here φ is in the range −1/2, 1/2, modulo unity [85]. We plot in Fig. 4.1 the ampli-
tude of the h/2e harmonic of the PC, as well as the transition temperature (in the
absence of the flux) as functions of the pair-breaking strength, using the dimensionless
parameter
s = 1/πT 0c τs . (4.29)
The transition temperature is reduced due to pair breaking and vanishes at s = 1/2γE,
where γE is the Euler constant. In contrast, for Ec À 1/τs the PC is hardly affected
for these values of pair-breaking strengths.
0.1 1 100
0.5
1
s
I/I(s=0)
Tc/T
c0
Figure 4.1: The h/2e harmonic (full line) and Tc/T0c (dashed line) as functions of
the pair-breaking strength, displayed on a logarithmic scale. The current, in units
of I(s = 0), is plotted for T = Ec and T 0c = 0.1Ec. The PC reduction at s = 10
corresponds to 1/τs = πEc.
Figure 4.2 portrays the PC plotted by numerically evaluating Eq. (4.26). In each
of the panels the upper curve is drawn for s = 0, while the second curve corresponds
to a pair-breaking strength [see Eq. (4.28) and Fig. 4.1] which is large enough to
destroy Tc. Nonetheless, the PC is hardly affected as long as Ls & L [see Eqs. (4.2)
and (4.3)]. The considerably-reduced PC due to a small Ls is presented by the dash-
dotted curves, which correspond to Ls ' 0.5L. The effect of the temperature on the
36 4.4
0
0.5
1
I/ I*
s=0s=1s=10
0 2 4 6 80
0.5
1
T/Ec
I/ I*
s=0s=10s=100
Tc0=0.1E
c
Tc0=0.01E
c
Figure 4.2: The amplitude of the h/2e harmonic in units of I∗ = −eEc, as a function
of the temperature, for two values of T 0c /Ec and several values of s. Note that the
s = 0 curve in the upper panel is valid only for T/Tc ≥ 1 + Gi, where Gi is the
Ginzburg parameter.
magnitude of the PC is manifested by its dependence of the ratio L/LT , where LT is
the thermal length,
LT =√
D/T , (4.30)
or equivalently the ratio T/Ec, see Fig. 4.2.
4.4 The dominant fluctuations
Our result for the PC [see Eq. (4.24)] consists of infinite sums over the frequencies and
over the momenta. One naturally asks oneself whether the characteristic features of
the expression are not given by the first few members of each sum, notably the static,
ν = 0, regime. It turns out that this is not the case over most of the relevant range:
to obtain the correct magnitude of the fluctuation-induced PC, numerous frequencies
and momenta are required.
In order to study this aspect, it is convenient to express the PC in a form which
is more amenable to numerical computations. To this end, we write Eq. (4.26) as
I =2ieT
π
∞∑m=1
sin(4πmφ)∑
ν
∫ ∞
−∞dxe2πix d
dxln
[Ψ(F (x, ν))− ln(T 0
c /4γET )]
, (4.31)
where the function F is given in Eq. (4.27). The x integration is carried out by closing
the integral in the upper half of the complex plane. Two sets of simple poles can be
identified in the integrand of Eq. (4.31). These sets result from (a) the zeros and (b)
4.4 37
the poles of the argument of the logarithm [86]. The first set of poles, denoted by
x`zero, is given by
Ψ(F `zero) = ln(T 0
c /4γET ) . (4.32)
The second set consists of the poles of the digamma function. These are denoted by
x`pole, and are obtained from the relation
F `pole = −` , ` = 0, 1, 2, . . . . (4.33)
The index ` runs over the poles in each set. The two sets of F `pole/zero given by
Eqs. (4.32) and (4.33), are shown in Fig. 4.3.
−2 −1 0 1 2 3 4−5
0
5
10
ψ(x)
x
l=0l=2 l=1
l=2 l=1 l=0
Figure 4.3: The digamma function (solid line) and ln(T 0c /4γET ) for T 0
c /T = 0.6
(dashed line). The first three solutions F `zero of Eq. (4.32) are marked on the x axis
with their indices indicated below it. The first values of the set F `pole, Eq. (4.33), are
marked by arrows.
Performing the Cauchy integration, the current takes the form [87]
I =− 4eT∞∑
m=1
sin(4πmφ)∑
ν
∞∑
`=0
[exp(2πix`
zero)− exp(2πix`pole)
]. (4.34)
Here x`pole/zero depends on the Matsubara frequency and the harmonic index m,
x`pole/zero = im
√T
2πEc
[1 +
|ν|+ 2/τs
2πT− 2F `
pole/zero
]1/2
. (4.35)
Note that all the exponents (2πix) in the two series in Eq. (4.34) are negative, and
their absolute value increases with increasing ν, l, or m. As can be seen from Fig. 4.3,
for each pair of poles F `zero > F `
pole, and consequently |x`zero| < |x`
pole|. This ensures
that the term in the square brackets of Eq. (4.34) is positive, and hence the response
of the ring to a small flux is diamagnetic, as it should be.
38 4.4
4.4.1 The dominant imaginary time fluctuations
The dominant terms in Eq. (4.34) are those for which the absolute value of x is smaller
than unity, but if the absolute values of all x are larger than one only the smallest
[x`=0zero(ν = 0,m = 1)] is the dominant one. The absolute value of the exponents
(which are given by 2π|xzero/pole|) is at least (|ν|/Ec)1/2. Thus, for Ec & T , the lowest
. 10Ec/T frequencies have the dominant contribution to the PC. The proportionality
factor, of order 10, had been determined numerically and resulted from the square-
root structure of the exponents, see Eq. (4.35). At high temperatures T > Ec,
the system is dominated by the classical fluctuations - namely, by the first (lowest
energy), ν = 0, Matsubara frequency. The effect of the quantum fluctuations for
which ν 6= 0 increases as the temperature decreases. This tendency has an exception
in two cases. First, for very strong pair breaking 1/τs > T, Ec, T2/Ec the significant
quantum fluctuations that have a dominant contribution to the PC are bounded by
|ν| <√
Ec/τs. Second, in the case of small or zero pair breaking when T → Tc, only
ν = 0 is the dominant frequency [73].
When Tc is finite, the n = ν = 0 pole of the partition function, Eq. (4.7), is
the most dominant one as T → Tc. Consequently, in this low-temperature regime,
physical properties, including the PC, are determined only by the ν = 0 fluctuations,
pertaining to the static Ginzburg-Landau free energy. We find, however, that in the
case of a vanishing Tc, quantum fluctuations, for which ν 6= 0, have a significant
contribution to the PC at low temperatures. Indeed, the quantum fluctuations of a
system with no magnetic impurities and for which |φ|2 > T 0c /(16πγEEc) have been
recently invoked in the context of the “strong” Little-Parks oscillations, see Ref. [59].
4.4.2 The dominant spatial fluctuations
The contribution of high Matsubara frequencies to the PC involve many spatial fre-
quencies q. Thus, at low temperatures and for a vanishing Tc, many wave vectors
contribute to the PC. We have estimated numerically their number by comparing the
PC computed with a relatively small number of frequencies and wave vectors with
the exact result, Eq. (4.34) for T = T 0c = 0.1Ec and s = 1. In this case ∼ 100
Matsubara frequencies are required (see the parametric analysis in the previous sub-
section). The highest momenta, Eq. (D.14), that contribute significantly are given by
|n| ∼ (1, 5, 100, 1000) for the frequencies ν/(2πT ) = (0, 5, 10, 100), respectively [88].
Figure 4.4 shows the PC as computed from Eq. (4.24) for different maximal |q| values
and without limiting the range of ν. It is thus seen that in the whole range of φ the
persistent current is not mainly determined by the lowest momenta, even when the
size of the system L is smaller than the thermal length LT , Eq. (4.30). This is differ-
ent to the situation of calculations of other properties (for example, weak-localization
corrections [89] to the conductivity), in which L ¿ LT is taken as a sufficient con-
4.4 39
dition for using only q = 0. We point out however that for Ec & 1/τs À T, T 0c at
φ = 0 the susceptibility (∂I/∂φ)φ=0, appears to be describable, within a numerical
factor of order unity, using the smallest wave number only. Using the lowest three
wave numbers gives almost quantitatively correct results for the susceptibility.
−0.2 −0.1 0 0.1 0.2
−1
0
1 I/eE
c
φ
Figure 4.4: The PC as computed from Eq. (4.24), with the summation over n cut
at 1000, 3, 1 (solid, dashed and dash-dotted curves, respectively). The plots are for
T = T 0c = 0.1Ec and s = 1.
4.4.3 The dominant harmonics
Examining the series in Eq. (4.34) one can see that the maximal harmonic of the flux,
mmax, that still has a significant contribution to the current is given by min√
Ec/T ,√
Ecτsor by one if the first two values are smaller than unity. This condition can be expressed
in terms of lengths as
mmax = minLs/L, LT /L , or 1. (4.36)
The upper limit on the harmonics results from the fact that the mth harmonic is asso-
ciated with paths that encircle the ring (coherently) m times and hence their length is
at least mL [16]. The sinusoidal shape I ∝ sin(4πφ) at high temperatures is modified
due to higher harmonics as the temperature decreases. In the absence of magnetic
impurities (upper panel in Fig. 4.5) the low-temperature current as a function of the
flux attains a sawtooth shape. Such a behavior is predicted also for the equilibrium
PC in superconductors at zero temperature [57] and for the persistent current in a
clean system of non-interacting electrons [13]. In the presence of pair-breakers the
upper bound on the harmonics Eq. (4.36) prevents the current from reaching the
sharp sawtooth shape. This suggests, in principle, a way to experimentally confirm
the role of pair breaking for this problem. In the lower panel of Fig. 4.5 the current
of a system with L ' Ls is plotted for several temperatures. At temperatures below
0.1Ec the shape of the current does not change anymore.
40 4.5
−1
0
1
I/eEc
−0.2 −0.1 0 0.1 0.2−1
0
1
φ
I/eEc
1/τs=0
1/π Tc0τ
s=1
Figure 4.5: The current, in units of eEc, as a function of the flux φ, for T 0c /Ec = 0.1
and for several temperatures, T/Ec = 5, 1 and 0.15 in the solid, dashed and dash-
dotted curves respectively. In the lower panel the dash-dotted curve corresponds to
T/Ec = 0.1. For s = 0 the current attains the sawtooth form (upper panel) which is
lost for s = 1 (lower panel).
4.5 The temperature dependence
Here we study the PC in the low- and high-temperature regimes. In particular, we
find that the PC decays exponentially as the length of the ring exceeds the thermal
length LT or the magnetic-impurity scattering length Ls, whichever is shorter.
4.5.1 High-temperature regime, T À max1/τs, T0c , Ec
When the temperature is much higher than all relevant energy scales, i.e., T Àmax1/τs, T
0c , Ec, the leading contribution to the double sum in Eq. (4.34) comes
solely from the first pole x`=0zero of the lowest Matsubara frequency, ν = 0 [see Eq. (4.35)].
In this temperature range the h/2e harmonic, corresponding to m = 1, is the domi-
nant one.
As the temperature increases, the horizontal line in Fig. 4.3 representing ln(T 0c /4γET )
moves further down, so that F `=0zero approaches zero. We use the expansion of the
digamma function for small arguments in Eq. (4.32) and obtain
F `=0zero =
[ln(T 0
c /4γ2ET )
]−1. (4.37)
Upon substituting this result in the dominant term of Eq. (4.34), we obtain the
current in the form
I ' −4eT sin(4πφ) exp(− L
LT
[2π +
2L2T
L2s
− 4
π ln(4γ2ET/T 0
c )
]1/2). (4.38)
4.5 41
We compare the full result, Eq. (4.34), with the high-temperature approximation
Eq. (4.38) in Fig. 4.6. The difference between the contributions of the first x`=0zero(ν = 0)
and the second x`=0pole(ν = 0) poles to the PC is the absence of the third term, which
includes a logarithm [see Eq. (4.38)], in the exponent of the latter. Therefore, this
approximation improves as T 0c increases.
0 2 4 6 8 100
0.5
T/Ec
I/ I*
exacthigh temperature limit
Figure 4.6: The amplitude of the h/2e harmonic is plotted in units of I∗ = −eEc as
a function of the temperature, for T 0c /Ec = 0.1 and s = 1. The exact results can be
approximated by Eq. (4.38) for T À Ec.
4.5.2 Low-temperature regime, Tc ¿ T ¿ 1/τs, EcIn the low temperature regime the argument (F ) of the digamma function and its
derivative is much larger than unity [see Eq. (4.27)], so that we can use their asymp-
totic expansions ln(F ) and 1/F , respectively. Substituting these approximations in
Eq. (4.26) gives
I = − 8
πeT
∞∑m=1
sin(4πmφ)∑
ν
∫ ∞
0
x sin(2πx)dx
ln[
4πγEEc
T 0c
(x2 + am,ν)](x2 + am,ν)
, (4.39)
where am,ν = m2(|ν|+2/τs+2πT )/(4π2Ec). For T À Tc the denominator in Eq. (4.39)
does not vanish. Then the term x2+am,ν in the logarithm in Eq. (4.39) can be replaced
by αam,ν , with, say, 1 < α < 3. Consequently,
I '− 4eT∑m
sin(4πmφ)∑
ν
e−m
√2πTEc
√1+
|ν|+2/τs2πT
/ ln
[2γEαT
T 0c
(1 +
|ν|+ 2/τs
2πT
)]. (4.40)
42 4.6
Since T ¿ Ec the summation over ν can be replaced by an integration. Approximat-
ing again the logarithm by its value at the dominant ν of the integration, yields
I '− 8
πeEc
∑m
sin(4πmφ)
m2
[1 + m
√2πL2
L2T
+2L2
L2s
]
× e−m
√2πL2
L2T
+ 2L2
L2s / ln
[γEαEc
πT 0c m2
z
], (4.41)
where z = max1, 2m2/τsEc.We compare in Fig. 4.7 the low-temperature approximation, Eq. (4.41), with the
full result, Eq. (4.34). As one can see from this comparison, the flux dependence of
the PC as well as its amplitude are well approximated by Eq. (4.41).
−0.2 −0.1 0 0.1 0.2
−0.5
0
0.5 I/eE
c
φ
exact
approx
Figure 4.7: The current in units of eEc as a function of the magnetic flux φ, plotted
for T = 0.1T 0c = 0.01Ec and s = 1. The low-temperature approximation Eq. (4.41)
is compared with the exact result Eq. (4.34). We take α = 3 in the logarithm of
Eq. (4.41).
4.6 Renormalization of the effective interaction
In this section we calculate the PC to first order in the interaction, in order to see
whether it suffices to explain our full result. To first order in the interaction, the
contribution of superconducting fluctuations to the free energy [see Eq. (4.7)] is
∆Ω = −(gT 2/V )∑q,ν
Π(q, ν) . (4.42)
The PC resulting from Eq. (4.42) has the same form as Eq. (4.24), except that
the denominator in the latter is replaced by the bare interaction gN (0)/V . Had
we tried to to fit the experimental data of Refs. [24] and [25] using Eq. (4.42), we
would have taken the implausible ratio Ec ∼ 0.1ωD [see Eq. (4.20)]. This first-order
approximation fails because of screening effects, which increase the magnitude of
the effective attractive interaction as the temperature decreases. Very roughly, the
4.7 43
renormalization of a dimensionless interaction λ, from a higher frequency scale ω> to
a lower frequency scale ω<, is given by [72]
λ(ω<) =
[λ−1(ω>)− ln
(ω>
ω<
)]−1
. (4.43)
For attractive interactions λ is positive and the high-frequency scale is ωD. At T = T 0c
and 1/τs = 0, the attractive interaction should diverge. Using this to eliminate λ(ωD)
(≡ gN (0)/V ), we obtain that for T 0c . ω ¿ ωD,
λ(ω) v 1/ ln(ω/T 0c ) . (4.44)
Replacing in the first-order approximation for the current the bare interaction by the
effective interaction, Eq. (4.44), gives
I1st = − 8eEc
ln(ω/T 0c )
∞∑m=1
sin(4πmφ)
m2
∑ν
∫ ∞
0
dxx sin(2πx)Ψ′(F (x, ν)) . (4.45)
The effective interaction is renormalized upwards with decreasing energy and, for
the bulk and no pair breaking, it blows up at T 0c . For 1/τs > T 0
c , this renormalization
stops at 1/τs and Tc disappears. In the mesoscopic range, the Thouless energy, Ec,
becomes a relevant scale and it may be expected (as is borne out by our results) that
the PC at low temperatures is determined by the interaction on that scale, as long as
Ec & 1/τs. Once 1/τs & Ec, we expect the renormalization to “stop at 1/τs” and the
PC to be depressed. Thus, the relevant range for our considerations is T 0c . 1/τs . Ec.
Using these bounds on the energy scale of the renormalized interaction in the first
order calculation Eq. (4.45), gives a good agreement with our result Eq. (4.34). In
Fig. 4.8 we plot the amplitude of the h/2e harmonic as a function of T/Ec, calculated
from the full expression (4.26) (thin curves) and from the first-order approximation
Eq. (4.45) (bold curves). The plotted curves are for T 0c = 0.1Ec.
A more precise expression for the renormalized attractive interaction depends
on q, ν on the order-parameter fluctuation. The renormalized attractive interaction
λ(q, ν), obtained from an infinite series of diagrams containing Cooperon contribu-
tions, is given by [84]
λ(q, ν) =[λ−1(ωD)− T
N (0)Π(q, ν)
]−1
. (4.46)
Upon substituting Eq. (4.19) in Eq. (4.46) one can identify λ(q, ν) from our result,
e.g., by comparing Eq. (4.45) with Eq. (4.24).
4.7 Comparison with experiments
Theoretically, only static magnetic fields have been considered here. However, exper-
iments have been carried out with an ac magnetic field. In the experiments on copper
44 4.7
0 1 2 3 4 50
0.5
1
T/Ec
I/ I*
s=0s=1s=10
Figure 4.8: The first-order approximation for the h/2e harmonic of the current
Eq. (4.45) (bold lines) is compared with the exact result (thin lines). Here T 0c =
0.1Ec. In drawing the former, we have used the simplest expression for the cutoff
ω = T + Ec + 1/τs .
[24] and gold [25], the sweeping frequencies of the magnetic field were very low (0.3
and 2Hz, respectively). Thus, one expects that the measured PC could be explained
using a theory for a static magnetic field. In the experiment on silver, on the other
hand, a very high sweeping frequency of the magnetic field was used (217 MHz). It is
plausible that in order to explain the results of Ref. [26] one may not confine oneself
to a static magnetic field. We therefore do not attempt to explain the experiment of
Ref. [26].
Here we explain the h/2e signal observed in copper [24] and gold [25] using our
result Eq. (4.34). In the left six columns of Table 4.1 we summarize the experimental
parameters for the (h/2e)-periodic signal [90].
Table 4.1: Experimental parameters in the left six columns. The magnitude of the
h/2e periodic current (column 4) is given for the lowest temperature (column 3)
reached in the experiment. The dephasing length Lϕ is given together with the
temperature at which it was measured. The last column is our estimate for a lower
bound on T 0c according to Eq. (4.34), see also Fig. 4.9.
Ec T L Lϕ
(mK) (mK) I/eEc (µm) (µm) Minimum T 0c
Copper [24] 15 7 1 2.2 2 (1.5 K) A few mK
Gold [25] 4.9 5.5 0.65 8.0 16 (0.5 K) A fraction of a mK
The metals used in the experiments are not superconductors at any measured
temperature in their bulk form. Therefore, it is not possible to obtain theoretically
4.7 45
0 0.1 0.2 0.3 0.4 0.5 0.60
0.2
0.4
Tc0/E
c
L/Ls
Figure 4.9: The bare transition temperatures corresponding to the measured PC
as a function of L/Ls. The dashed and dash-dotted curves correspond to the PC
measured in copper and gold, respectively. The solid curve gives the maximal possible
T 0c satisfying Tc = 0.
a large enough PC (to match the measurements [24, 25]) due to the attractive inter-
action without pair breaking; the required Tc ∼ 1 mK is too high to be considered as
realistic. We suggest that the bare transition temperature may indeed be on the order
of a mK, but the transition temperature of the real material is considerably reduced
due to pair breakers. Together with this assumption, the necessary condition to fit
the experiments is 1/τs & πT 0c /2γE so that Tc vanishes or is very strongly depressed
[63], see Eq. (4.28). This condition can also be written as
T 0c
Ec
. 2γE
π
(L
Ls
)2
. (4.47)
Note that we need Ls & L in order not to depress the PC [Eq. (4.34)]. The upper
limit on T 0c , corresponding to the equality in Eq. (4.47), is given by the solid line in
Fig. 4.9. The values for T 0c /Ec that correspond to a vanishing Tc are in the region
below this line. In the dashed and dash-dotted curves in Fig. 4.9 different values of
L/Ls are matched with an appropriate T 0c /Ec so that the measured values in columns
2− 4 of Table 4.1 remain the same.
The monotonically increasing shape of the curves in Fig. 4.9 results from the
fact that higher values of T 0c /Ec are required to describe the experiments as L/Ls
increases. The minimal T 0c ’s correspond to the points where the dashed and the dash-
dotted lines cross the solid line. In this way we obtain estimates of the lower bounds
on the value of T 0c for copper and gold. These lower bounds are given in the seventh
column of Table 4.1.
These estimates of T 0c are very sensitive [see Eqs. (4.41) and (4.45)] to the exper-
imental parameters. For example, Ref. [24] points out that the measured values of
the PC are correct up to a factor of 2. The exact minimal value of T 0c that satisfies
Eq. (4.47) for copper, based on the values quoted in Table 4.1, is 4mK. However,
assuming half of the value reported in Ref. [24] for the PC, results in a minimal T 0c of
about 0.3mK. The curves in Fig. 4.9 ignore the error bars in the experiments. Thus,
46 4.8
the values of T 0c /Ec in this figure should be considered only as rough estimates.
Besides spin-flip scattering from magnetic impurities, decoherence of the electrons
is also caused by other processes, e.g., electron-phonon inelastic interactions. Hence
Ls is always larger or on the order of the dephasing length. A lower bound on Ls is
given by equating it to the measured Lϕ. Those values (see Table 4.1) are small enough
to fulfill the condition in Eq. (4.47). In other words, we could account for the data
of copper and gold since the measured Lϕ was small enough. This is not the case for
silver [26], where L/Lϕ(= 0.3) is too small to explain the result I(T = 4.6Ec) = 1.6eEc
using Eq. (4.34). Our theory is not applicable to that experiment. We believe that
the reason for that, as explained above, is the high frequency used in that experiment.
4.8 Discussion
In our result for the PC, Eq. (4.34), there appears the bare transition temperature
and not the one reduced by the pair-breaking mechanism. Therefore we propose the
scenario, in which the bulk transition temperature vanishes due to the pair-breaking
mechanism, while the PC is dominated by a relatively high attractive interaction.
The bulk Tc vanishes due to pair breaking for Ls < ξ(0). However, we find that
the PC may still be hardly affected by pair breaking. The physical reason for that is
that as long as Ls > L the Cooper pair fluctuations can complete a circle around the
ring before being magnetically scattered, and hence respond to the Aharonov-Bohm
flux. The PC is immune to pair breaking in the regime given by Eq. (4.2) where the
bulk form is normal. This is demonstrated in Figs. 4.1 and 4.2.
In the pair-breaking regime given by Eq. (4.2), the upper bound on the dominant
quantum fluctuations (ν 6= 0) is determined by the Thouless energy. Dominant
fluctuations of high Matsubara frequencies necessitate high wave numbers. Therefore,
at low temperature T ¿ Ec high wave numbers are involved too in the dominant
fluctuations [see Fig. 4.4], in contrast to the effective dimensional reduction occurring
in other phenomena when L ¿ LT , notably weak-localization corrections [89]. The
maximal number of flux harmonics that contribute to the PC, Eq. (4.36), is bounded
due to thermal fluctuations and due to spin-flip scattering. Consequently, in a system
with magnetic impurities, even at zero temperature the PC may not have the sawtooth
shape, which appears for the PCs without pair breaking, see Fig. 4.5.
The effective interaction is renormalized upwards with decreasing energy; for the
bulk it stops at ∼ maxT 0c , 1/τs (which explains why Tc disappears for 1/τs & T 0
c ).
In the mesoscopic range, Ec & T 0c , the Thouless energy sets another bound for the
energy scale at which the renormalization stops. In Section 4.6 it is shown that these
considerations agree with our result for the PC, Eq. (4.34), see Fig. 4.8.
We found that in the high-temperature regime, the PC decreases exponentially
with L/Ls or with L/LT , whichever is larger. The explicit exponential decay of the
4.8 47
PC with L/Ls in both the high- and the low-temperature regimes [Eqs. (4.38) and
(4.41) respectively] for L & Ls is in agreement with the qualitative argument of
Eq. (4.2). Note that Eq. (4.41) is applicable only at very low temperatures, such that
T ¿ T 0c , Ec. The experiments on copper [24] and gold [25] rings correspond to
T 0c ∼ 1mK, thus Eq. (4.41) can be used only at very low temperatures T ¿ 1mK. In
the experiments the lowest temperature was ∼ 10 mK, and therefore the measured
PC cannot be precisely fitted by the approximate expression (4.41). In the low-
temperature regime the dependence of the PC on T 0c is logarithmically weak [see
Eq. (4.41)]. This weak dependence explains why in Ref. [21], where the transition
temperature was taken as 10µK (in the absence of pair breaking), the result was
smaller only by a factor of ∼ 5 compared with the experiment [24].
Interestingly enough, it follows from our work that by measuring the PC and the
pair-breaking strength, one may determine T 0c which would be directly measurable
only if enough low-temperature pair breaking could be eliminated. This elimination
is very hard to achieve in some materials. Our result Eq. (4.26) can explain the large
PC of Refs. [24] and [25], with Ls value larger than (or on the order of) the measured
Lϕ value (see Table 4.1 and Fig. 4.9). Even though Ls was not measured in the
PC experiments, we obtain a lower bound on the bare transition temperatures for
copper and gold. These minimal T 0c ’s correspond to minimal pair-breaking strength
given by Ls ∼ 5 µm in the copper sample [24] and Ls ∼ 25 µm in the gold sample
[25]. The fitted maximal Ls’s can be caused by a very low (less than one part per
million) concentration of magnetic impurities. These concentrations seem appropriate
for the purest copper and gold samples available experimentally [35]. Although, a full
consideration of the effect of the magnetic impurities, including Kondo physics, is still
necessary.
Our result concerning the fundamentally different sensitivities of Tc and PCs to
pair breaking is valid regardless of the situation in specific materials. Our idea can
be tested, for example, by measuring the persistent currents in very small rings made
of a superconducting material whose transition temperature is known, as functions
of possible pair-breaking mechanisms. For Ec & 100 mK, say, and a material with T 0c
of a few 10 mK, the range of pair breaking which satisfies Eq. (4.1) becomes easier
to control experimentally.
48 4.8
49
Chapter 5
Persistent currents of
non-interacting electrons in one-,
two-, and three-dimensional thin
rings
Published in: Phys. Rev. B 82, 144202 (2010).
5.1 Abstract
We thoroughly study the persistent current of non-interacting electrons in one, two,
and three dimensional thin rings. We find that the results for non-interacting electrons
are more relevant for individual mesoscopic rings than hitherto appreciated. The
current is averaged over all configurations of the disorder, whose amount is varied
from zero up to the diffusive limit, keeping the product of the Fermi wave number
and the ring’s circumference constant. Results are given as functions of disorder and
aspect ratios of the ring. The magnitude of the disorder-averaged current may be
larger than the root-mean-square fluctuations of the current from sample to sample
even when the mean-free path is smaller, but not too small, than the circumference
of the ring. Then a measurement of the persistent current of a typical sample will be
dominated by the magnitude of the disorder-averaged current.
5.2 Introduction
One of the consequences of the Aharonov-Bohm (AB) effect [2] is that a finite normal
(i.e. non-superconducting) mesoscopic ring exhibits a persistent current (PC) when
the AB magnetic flux through its opening is non zero [3]-[6]. The PC does not
50 5.2
decay with time when the dephasing and the thermal lengths are larger than the ring
circumference. This results from the fact that the PC reflects an equilibrium state
even when the ring has a finite resistance due to defect scattering [4, 7, 57]. The PC is
periodic in the flux Φ with a period given by the magnetic flux quantum Φ0 ≡ 2π~c/e.Measurements of the PC (Refs. [24]-[30]) often stimulated the theoretical studies [9]-
[18], [20], [21]. Today, this fundamental phenomenon of quantum mechanics still
challenges both theoreticians and experimentalists of mesoscopic physics [19], [28],
[31]-[34]. Persistent currents are also relevant for the orbital response of semimetals
and aromatic molecules [8], and for the ongoing interest in nanotubes [91].
At zero disorder, the azimuthal component of the velocity associated with each
single-particle eigenstate of the Hamiltonian of non-interacting particles is shifted due
to the AB flux Φ < Φ0/2, by ∆v = 2π~φ/ML. Here M is the electron mass, L is the
circumference of the ring and φ ≡ Φ/Φ0. One may naively assume that the current
density is −ne∆v, where n is the density of the electrons. In a normal ring, because
of level crossing, the occupation of the levels changes with the flux. As a result,
once level-crossing occurs, the PC density of the normal ring is much smaller than
−ne∆v. In a superconducting ring −ne∆v gives the value of the PC density at zero
temperature and zero disorder. It might be argued that in a perfect superconductor at
zero temperature, the above occupation switching is suppressed. Thus, the attractive
interaction in a superconductor, which enforces the pairing correlations, strongly
enhances the PC compared to the normal-state value. Note that the current of a
superconducting ring is an intensive quantity –it does not depend on the size of the
system. In the normal state, the current is only a mesoscopic effect–proportional to
an inverse power [-1 in the ballistic one-dimensional (1D) case [4]] of the system’s
length.
The current of non-interacting electrons in two-dimensional (2D) cylinders in the
grand-canonical ensemble was studied analytically in the limit of zero disorder and
in the diffusive limit [9]-[12] In these works the PC was calculated in two geometries:
“short” cylinders, H ¿ L, where H is the height of the cylinder, and “long” cylinders,
H À L. Cheung et al. [10] studied the case of a three-dimensional (3D) short and
thin diffusive cylinder as well. In the zero-disorder limit, the PC was calculated by
summing the velocities, with appropriate factors, of all the states that, after the
energy shift due to the flux, are below the Fermi energy [9]. In the diffusive limit the
PC may be averaged over the configurations of the impurities. It can be calculated
as a function of the magnetic flux from the density of states in the diffusive limit
[10, 11]. Entin-Wohlman and Gefen [12] calculated the impurity-ensemble-averaged
current of long cylinders using the linear-response theory in φ, which is valid only for
φ ¿ 1/2.
Our work extends the above research [9]-[12] in two ways. First, we describe the
current for any degree of disorder between the previously studied limits of perfectly
5.2 51
clean systems and diffusive systems. Second, we consider 3D thin rings with a finite
width W for which W ¿ L (in contrast to W . a, where a is the smallest microscopic
length of the system) [9, 11, 12]. We also correct, and generalize for any given value
of the flux, the expression for the PC as calculated by Entin-Wohlman and Gefen
for “long” 2D cylinders [12]. In the latter, a calculation error [92] gave a result of
incorrect sign and magnitude for the prefactor of the dominant (for L À `, where `
is the elastic mean-free path) exponential dependence.
The expression [10] for the disorder-averaged PC in the grand-canonical ensemble
at zero temperature is given in Section 5.3. This expression can be simplified in two
regimes, defined in Section 5.4, which we name the uncorrelated- and the correlated-
channel regimes [4]. In Secs. 5.5 and 5.6 we perform the simplifying steps that are
allowed in each regime, and then obtain the leading-order expressions of the PC in
the zero-disorder and the diffusive limits. The specific conditions for which these two
limits hold in both the uncorrelated- and the correlated-channel regimes are given in
Table 5.1. In Refs. [9]-[12] the same simplifying assumptions had been used but were
referred to as “short” and “long” cylinders. We find that these pictorial definitions do
not agree with the regimes in which the corresponding results hold. Our results for
PC of 2D cylinders in the zero-disorder and the diffusive limits for the uncorrelated-
channel regime, and in the zero-disorder limit for the correlated-channel regime, agree
with the ones obtained in Refs. [9]-[11]. For 2D cylinders, our result for the PC in
the correlated-channel regime in the diffusive limit is new.
The disorder-averaged PC is highly sensitive to the exact value of kF L, as it
contains a factor of sin(kF L), where kF is the Fermi wave number. In Section 5.7 we
discuss the way to compare the measured average PC in an ensemble of rings to the
theoretical results depending on the variance of the value of kF L among the rings.
In this section the disorder-averaged PC is also compared with the root-mean-square
(rms) fluctuations [10, 14] of the PC with respect to the disorder. We find that as long
as the system is not too diffusive, the magnitude of the disorder-averaged current may
be larger than the current rms fluctuations. As discussed in Section 5.8, our result for
the disorder-averaged PC of non-interacting electrons agrees with the PC measured
in a 2D clean annulus by Mailly et al. [30], but has a larger magnitude than the one
measured by Rabaud et al. [29]. The results of our study are discussed in Section 5.9.
In contrast with the Green’s function technique used in the main body of this
paper, we give in the Appendix E an approximation for the PC of a 3D ring in the
zero-disorder limit. This approximation is based on the canonical ensemble results
for a 1D ring, and on the probabilities that, at a given flux, the number of electrons
in a given transverse channel is odd or even.
52 5.3
5.3 Expression for the persistent current
In this section we obtain an expression [10] for the impurity-ensemble-average zero-
temperature PC of non-interacting electrons. We consider spinless electrons in a ring
of a mean circumference L, a width W , and a height H. In the absence of disorder,
the Hamiltonian is given by
H =1
2M(−i~∇+
e
cA)2 . (5.1)
The AB flux, which does not penetrate the ring itself, is given by the magnetic vector
potential A = ϕΦ/2πr, where r is the radial coordinate and ϕ is a unit vector oriented
along the ring. The eigenstates of H, in cylindrical coordinates, are
ψ(r, ϕ, z) = einϕ sin(πqz
H
)[C1Jn+φ(kr) + C2Yn+φ(kr)] , (5.2)
where n = 0,±1,±2, .., q = 1, 2, .., and
k =√
2Mε/~2 − (πq/H)2 . (5.3)
Here J and Y are the Bessel functions of the first and second kind. The boundary
conditions ψ[r = (L/2π −W/2)] = ψ[r = (L/2π + W/2)] = 0 set the ratio between
the prefactors C1 and C2 and the eigenenergies. For W ¿ L, the eigenenergies are
given by [93]
εq,s,n =~2π2
2M
[q2 − 1
H2+
s2 − 1
W 2+
[2(n + φ)]2
L2
]+
1
L2O
[(W/L)2
], (5.4)
where s is a positive integer. In this work, all energies are shifted so that the single-
particle ground state energy, for which q = s = 1, n = φ = 0, is zero. We henceforth
neglect the term of order (W/L)2 in Eq. (5.4).
We now introduce disorder, induced by impurities having point-like potentials.
The PC, averaged over a grand-canonical ensemble of disordered systems having the
same mean-free path but different impurity configurations, is given by [10]
〈I〉 =∑q,s,n
∫ ∞
−∞
dE
2πif(E)
[G+([q, s, n], E)−G−([q, s, n], E)
]I(0)n . (5.5)
Here the Fermi distribution function, f(E), sets the chemical potential as an upper
bound on the integration at zero temperature. The current associated with a single-
electron wave function is given by
I(0)n = −2π~e
ML2(n + φ) . (5.6)
5.3 53
In Eq. (5.5), the disorder-averaged retarded and advanced Green’s function are de-
noted by G+ and G−, respectively. The expressions for the disorder-averaged Green’s
function, for kF ` À 1 and within the Born approximation, are [94]
G±([q, s, n], E) =
[E − εq,s,n ± i~
2τ
]−1
, (5.7)
where τ is the elastic mean-free time. Equation (5.5) for the disorder-averaged PC
is given as a sum over channels (q,s). However, in the corresponding expression
for the non-averaged current, one should use the non-averaged Green’s function and
consequently for a specific configuration, the channels are mixed in the expression for
the PC [95].
We note that the (q, s) term in Eq. (5.5) is given by the averaged PC in a 1D ring
[11] with a shifted chemical potential
µ → µ(q, s) = µ− ε(q, s, n = 0, φ = 0) , (5.8)
namely,
〈I〉 =∑q,s
⟨I1D[µ(q, s)]
⟩. (5.9)
The current of a 1D ring, calculated in Ref. [11], is
⟨I1D
⟩= 2I0
∞∑m=1
sin(2πmφ)
πmcos (mkF L) e−
mL2` . (5.10)
Here I0 ≡ evF /L , where vF is the Fermi velocity [96]. In Eq. (5.9) each (q, s) term
has its Fermi wave number determined by Eq. (5.8)
kF (q, s) = kF
√µ(q, s)/µ . (5.11)
Equation (5.10) is valid for µ À ~/2τ, a, where a = 2π2~2/ML2 is the prefactor of
(n + φ)2 in the expression for the eigenenergies, see Eq. (5.4).
Substituting the 1D result, Eq. (5.10), in Eq. (5.9), we obtain that at zero tem-
perature
〈I〉 =∞∑
m=1
〈Im〉 sin(2πmφ) , (5.12)
where the disorder-averaged harmonics are given by
〈Im〉 =2I0
πm
Nz∑q=1
Nr
√1−(q2−1)/N2
z∑s=1
kF (q, s)
kF
× cos [mkF (q, s)L] exp
(− mL
2`kF (q, s)/kF
). (5.13)
54 5.4
The approximate numbers of the occupied channels corresponding to momenta in the
radial and the z directions are
Nr = kF W/π , Nz = kF H/π , (5.14)
respectively. In the upper bounds on the summations over q and s, one needs to take
the closest integer values for Nr and Nz from below (but not less than 1).
In Eq. (5.13) we sum over the contributions of the occupied channels, which obey
(s/Nr)2 + (q/Nz)
2 ≤ 1, so that µ(q, s) > 0. In a diffusive system, one might worry
about the contribution to 〈Im〉 of channels with high transverse momentum which
satisfy
`[kF (q, s)/kF ] < 1/kF (q, s) , (5.15)
and are therefore not diffusive. Their contribution is given by an expression simi-
lar to Eq. (5.10), where a term of√
4kF L multiplies the exponent and divides I0.
In Eq. (5.13) we ignore this extra reduction since only a few channels may satisfy
Eq. (5.15) and their contribution to the PC is anyhow small.
5.4 Approximations for the PC harmonics
In this section we identify different regimes in which the expression for the disorder-
averaged harmonics, see Eq. (5.13), can be simplified.
5.4.1 Dimensionality of the system
The ring is considered to have a significant thickness along the radial direction when
Nr À 1 [see Eq. (5.14)] and when the ratio between the exponential in Eq. (5.13)
with a small index s to the following s + 1 term is much smaller than, say, 10. Thus,
for the calculation of 〈Im〉 many s values give significant contributions when
kF W
πÀ
1 and
√mL
8`
. (5.16)
When the “much larger” sign in Eq. (5.16) is replaced by a “smaller” or “comparable”
one, the ring is considered to be of zero dimension along the radial direction, and we
use only s = 1.
Note that condition (5.16) depends on L/`. This can be understood by the fol-
lowing argument: the phase of the Green’s function of a particle that encircles the
ring depends on the specific disorder configuration. Averaging the PC over all config-
urations of disorder results in the exponential decay of⟨I1D
⟩, see Eq. (5.10) [16]. In
a multichannel ring, the overall path, and correspondingly the variance of the phase
5.4 55
shifts, increase as the transverse momentum increases. This results in the increase in
the exponential decay rate in Eq. (5.13) for increasing channel index. Indeed, as we
see in Eq. (5.16), increasing the disorder may decrease the effective dimensionality of
the system. The condition for considering the ring to have a finite height is given by
Eq. (5.16) upon replacing W with H. In this way the system is classified as one of
the following: 1D, 2D annulus, 2D hollow cylinder, or a 3D ring. In the 2D annulus
case one sums over s taking q = 1, and in the 2D cylinder case the summation is over
q keeping s = 1.
5.4.2 Contributions of consecutive channels to 〈Im〉The discrete summation over the channel indices in Eq. (5.13) makes the expression
for 〈Im〉 hard to handle analytically. In this section we define two regimes where one
can overcome this difficulty. The contributions to the mth harmonic of consecutive
transverse channels (s and s+1, or q and q +1) are uncorrelated when the change in
the arguments of the corresponding cosine terms, see Eq. (5.13), is larger than, say,
π/4. This difference between the arguments of the cosines increases with increasing
channel index. Hence, if the lowest two transverse indices obey this condition, then
higher indices will fulfill it as well, so that all channels are uncorrelated. Thus, the
channels associated with the z direction are uncorrelated when
H
mL<
2π
kF H. (5.17)
The same rule applies to channels of consecutive s indices upon replacing H with W .
The regime defined by Eqs. (5.16) and (5.17) will be referred to as the uncorrelated-
channel regime.
In the uncorrelated-channel regime the dependence of the PC on the parameters
kF L, Nz, and Nr, which appear in the arguments of the cosines in Eq. (5.13), is
non-trivial. This is demonstrated in Fig. 5.1. We thus turn to calculate the typical
magnitude of the disorder-averaged harmonics ( 〈Im〉2 )1/2. The overline denotes
averaging over kF L within a segment δ(kF L) ¿ kF L of a width of & 2π. Note
the different notations of averaging over kF L and averaging over disorder. In the
calculation of ( 〈Im〉2 )1/2 we use the approximation
cos [mkF (q, s)L] cos [mkF (q′, s)L] =δqq′
2, (5.18)
and obtain
(〈Im〉2
)1/2
=
√2
πmI0
√√√√∑q,s
(kF (q, s)
kF
)2
exp
(− mL
`kF (q, s)/kF
). (5.19)
56 5.4
1000 1020 1040 1060−0.5
0
0.5
kFL
<I m
=1
2D>
/I 0
Figure 5.1: The disorder-averaged PC depends on kF L, Nz, and Nr in a non-trivial
fashion. Here we plot⟨I2Dm=1
⟩, see Eq. (5.13), for L/` = 5 and for Nz = 70 (solid line)
and Nz = 100 (dashed line). The typical magnitude of the disorder-averaged current,
given by Eq. (5.19), for the above two values of Nz is 0.25I0 and 0.30I0, respectively.
We have confirmed numerically that the standard deviation of 〈Im〉 obtained from
Eq. (5.13) gives the same value for ( 〈Im〉2 )1/2 as given by Eq. (5.19). For the calcu-
lation of the standard deviation of 〈Im〉 we have inserted in Eq. (5.13) the parameters
of the ring used by Mailly et al. [30], see Section 5.8, and considered many values of
kF L in a segment of a width of 10π.
When the first harmonic is in the uncorrelated-channel regime, the harmonics
with m up to m ∼ 8k2F W 2`/π2L are also in that regime, see Eq. (5.16). In this case,
the contribution of higher harmonics is negligible. Therefore, in the approximate
expression
(〈I〉2
)1/2
=
√√√√∞∑
m=1
〈Im〉2 sin2(2πmφ) , (5.20)
we can use the expression given in Eq. (5.19) for 〈Im〉2 for all the relevant harmonics.
For a 2D cylinder, the maximal q whose contribution to 〈Im〉 is not negligible, see
Eq. (5.13), is
qmmax = minNz
√8`
mL, Nz . (5.21)
When Eq. (5.16) is satisfied and the cosines of sequential indices with q ≤ qmmax are
correlated, then the sum in Eq. (5.13) can be replaced by an integral. Since the
difference between the arguments associated with sequential channels increases as the
index of the channel increases, the condition for the channels to be correlated is
mL [kF (qmmax − 1, 1)− kF (qm
max, 1)] <π
4. (5.22)
When qmmax = Nz, condition (5.22) has the form H/L > 10m2kF L. The correlated-
channel regime for a 2D annulus is defined in the same way, but the limitation W ¿ L
5.5 57
Conditions associated Results
with the z direction
Uncorrelated: zero disorder 1 ¿ kF H/π < 2L/H (5.24), (5.25) [3D]
Uncorrelated: diffusive√
L/8` ¿ kF H/π < 2L/H (5.26) [3D]
Correlated: zero disorder H/L > 10m2kF L (5.29) [2D cylinder]
Correlated: diffusive H/L > max100`/H, π/kF L (5.30) [2D cylinder]
Table 5.1: The results for the PC in the zero-disorder (`/L → ∞) and the diffusive
(` ¿ L) limits. The conditions defining the uncorrelated- and the correlated-channel
regimes are given in the second column for a 2D cylinder. For a 2D annulus the
conditions are the same with H replaced by W . For a 3D ring the conditions should be
satisfied for both azimuthal directions. In the third column we refer to the appropriate
expressions for the PC.
of our analysis makes this regime irrelevant for that geometry. We refer to this point
in more detail at the end of Section 5.6. The expressions for the conditions for
the uncorrelated- and the correlated-channel regimes, in the zero-disorder and the
diffusive limits are summarized in Table 5.1.
5.5 Uncorrelated-channel regime
Consider a 3D ring in the uncorrelated-channel regime, defined by Eqs. (5.16) and
(5.17). To estimate ( 〈Im〉2 )1/2 we replace the sum in Eq. (5.19) by an integral over
x =√
(q/Nz)2 + (s/Nr)2, and add the factor 2xNtot, where Ntot = π4NrNz is the total
number of occupied channels
(〈I3D
m 〉2)1/2
=2√
2
πmI0
√Ntot
√∫ 1
0
x(1− x2) exp
(− mL
`√
1− x2
)dx . (5.23)
In Fig. 5.2 the magnitudes of the first and second harmonics are plotted as a function
of L/` using Eq. (5.23). Here one can see that with increasing disorder, the first
harmonic becomes more dominant.
Equation (5.23) can be further approximated in the zero-disorder and in the dif-
fusive limits. In the first limit
((I3D
m )2)1/2
=1
πmI0
√Ntot . (5.24)
From Eqs. (5.20) and (5.24) we obtain [97]
((I3D)2
)1/2
= I0
√Ntot
√|φ|(1− 2|φ|) . (5.25)
58 5.5
0 1 2 3 4 50
0.2
0.4
L/l
−
−−
−−
sqrt
( <
I m>
2 ) /
Nto
t1/
2
4 6 8 10 120
0.5
1
L/l
m=1
m=2
Figure 5.2: The PC of a 3D ring in the uncorrelated-channel regime. The typical
magnitudes of the first harmonic (solid line) and the second harmonic (dashed-dotted
line) are plotted in units of I0
√Ntot, using Eq. (5.23). In the inset ( 〈I3D
m=1〉2 )1/2/I0
(solid line) is obtained by substituting Nz = Nr = 20 in Eq. (5.23). For a later
discussion, the rms fluctuations of the PC with respect to the disorder, δI/I0, (dashed
line) are plotted using Eq. (5.33). Here ( 〈Im=1〉2 )1/2 = δI when L/` = 8.5, in
agreement with Eq. (5.34). The horizontal axis of the inset begins at L/` = 3 since
Eq. (5.33) is valid only in the diffusive limit.
Note the enhancement of the PC magnitude by the square root of the channel number.
Deep enough in the diffusive limit, L/` ≥ 10 , the PC is dominated only by its first
harmonic. Here, the magnitude of the PC is given by the limit L/` À 1 of Eq. (5.23)
(〈I3D〉2
)1/2
=2
π
√`
LI0
√Ntot e−
L2` sin(2πφ) . (5.26)
This reproduces the result [52] of Ref. [10].
The PC harmonics of a 2D annulus are given by
(〈I2D
m 〉2)1/2
=
√2
πmI0
√Nr
√∫ 1
0
(1− x2) exp
(− mL
`√
1− x2
)dx . (5.27)
Results for a 2D annulus in the uncorrelated-channel regime and the zero-disorder
limit are given by Eqs. (5.24) and (5.25) with Ntot replaced by 4Nr/3. Here, replacing
Nr with Nz gives the expression for the PC in a 2D cylinder obtained [52] by Cheung
et al. [9]. In the diffusive limit, the PC of a 2D annulus or a 2D cylinder in the
uncorrelated-channel regime amounts to multiplying the expression in Eq. (5.26) by
the factor√
πL/8` and replacing Ntot by Nr or Nz, respectively. The latter yields
the results obtained in Refs. [10] and [11]. The difference between the powers of L/`
between the 2D and the 3D expressions is due to the difference of the densities of
states of the transverse channels in these cases.
The similarity between the PC of a 2D annulus and the PC of a 2D cylinder
is hardly surprising since these two cases of finite width and of finite height are
topologically equivalent for the AB flux, and the eigenenergies are the same as long
as W ¿ L.
5.6 59
5.6 Correlated-channel regime
For a 2D cylinder, the correlated-channel regime is defined by Eq. (5.16) (with H re-
placing W ) and Eq. (5.22). In this case we replace the summation over q in Eq. (5.13)
by an integration and obtain
⟨I2Dm
⟩=
2
πmI0Nz
∫ 1
0
√1− x2 cos
(mkF L
√1− x2
)exp
(− mL
2`√
1− x2
)dx . (5.28)
In the zero-disorder limit, Eq. (5.28) yields the result [52] of Ref. [9]
I2Dm =
√2
πm3I0Nz
1√kF L
cos (mkF L− π/4) . (5.29)
The diffusive limit of the PC of a 2D cylinder in the correlated-channel regime is
found here to be given by
⟨I2D
⟩=
√2 sin(2πφ)√
πkF LI0Nz e−
L2` cos(kF L− π/4) . (5.30)
(The higher harmonics are negligible.) The conditions for the correlated-channel
regime in the zero-disorder limit, see Table 5.1, cannot be satisfied for the radial
direction together with the restriction W ¿ L, for most reasonable values of kF L.
The limit of a diffusive annulus, see Table 5.1, is satisfied, for W ¿ L, only when
L/` > 130, but then the disorder-averaged PC is irrelevant.
In Fig. 5.3 the magnitude of the disorder-averaged PC is plotted using Eq. (5.30)
as a function of L/` in the diffusive regime. The results, Eqs. (5.29) and (5.30) are
reduced by 1/√
kF L compared to the results in the uncorrelated-channel regime in
the zero-disorder and the diffusive limits, see Section 5.5. However, these results are
enhanced by√
Nz and by√
Nz(L/`)1/4, respectively.
5.7 rms fluctuations versus 〈I〉The disorder-averaged PC is very sensitive to the exact value of kF L, see e.g., the
cosine factor in Eq. (5.29). In contrast, the rms fluctuations of the current in respect
to the disorder [10, 14]
δI = [⟨I2
⟩− 〈I〉2]1/2 (5.31)
are not sensitive to kF L. The common practice in PC measurements is to determine
the total current, Itot, from the measurement of the overall magnetic response of N
rings. This current is related to both the disorder-averaged current and to the current
rms fluctuations by
Itot =
N 〈I〉 ±√
NδI δ(kF L) ¿ π
±√
N
[(〈I〉2
)1/2
± δI
]δ(kF L) > π
. (5.32)
60 5.7
5 10 15 200
0.2
0.4
0.6
L/l
<Im=1 2D >
δ I
Figure 5.3: The disorder-averaged PC of a 2D diffusive cylinder in the correlated-
channel regime (solid line) is plotted as a function of `/L. We replace sin(kF L) by
1/√
2 in Eq. (5.30) to obtain the typical magnitude, and use Nz = 103 and kF L =
5×103. The rms fluctuations (dashed line), see Eqs. (5.31) and (5.33), equals⟨I2Dm=1
⟩,
for the above parameters, at L/` ' 10, in agreement with Eq. (5.35). Both⟨I2Dm=1
⟩
and δI are given in units of I0.
Here δ(kF L) is the variation in kF L in an ensemble of N rings. Equation (5.32) hold
also for the harmonics (replacing I by Im). If the ring is in the uncorrelated-channel
regime, one may replace 〈I〉 by ±( 〈I〉2 )1/2 in the top equality of Eqs. (5.32), while
if the ring is in the correlated-channel regime, one needs to replace the cosine factor
in Eq. (5.28) for 〈Im〉 by 1/√
2 in order to obtain ( 〈Im〉2 )1/2 in the bottom equality.
The rms fluctuation due to the disorder of the h/e harmonic of the current for a
thin-walled (L À W,H) ring in the diffusive limit is given by [10, 14]
δI =
√8
π√
3
`
LI0 sin(2πφ) [` ¿ L] . (5.33)
This result is independent of the number of channels, i.e., of W and H. These current
rms fluctuations do not exist for `/L À 1, see Eq. (5.31). Thus, the contribution to
Itot which is not related to interactions, is expected to be given by Eq. (5.13) in the
zero-disorder limit. Equation (5.33) for δI is strictly valid in the diffusive regime, but
is expected to give a correct order of magnitude for systems in which ` and L are
comparable.
In Figs. 5.2 and 5.3, the crossover from the dominance of the disorder-averaged
PC to the dominance of δI can be observed. In the uncorrelated-channel regime,
the typical magnitude of the disorder-averaged current of a 3D ring is equal to δI
at L/` = 5, 10, and 14 for Ntot = 20, 103, and 105, respectively. These values are
obtained, for L/` > 1, by comparing Eq. (5.26) with Eq. (5.33)
Ntot > 0.7`
LeL/` ⇔
(〈I3D
m=1〉2)1/2
> δI . (5.34)
The analogous result for a 2D cylinder in the correlated-channel regime is
Nz > 0.9`
LeL/2`
√kF L ⇔
(〈I2D
m=1〉2)1/2
> δI . (5.35)
5.8 61
For kF L = H/L = 100, the equality ( 〈I2Dm=1〉2 )1/2 = δI is satisfied, see Eq. (5.35),
for Ntot = 22, 135, and 700 at L/` = 5, 10, and 14, respectively.
5.8 Discussion of experimental data
Since the first harmonic is not expected to be affected by electron-electron interactions
[20, 21], we may compare its measurements [25], [29]-[30] with calculations of the
typical magnitude of 〈I〉 and δI.
Mailly et al. [30] studied the PC in an almost ballistic annulus of GaAlAs/GaAs,
characterized by L = 8.5µm, ` = 11µm, kF = 1.5 × 108m−1, vF = 2.6 × 105m/s,
and W = 0.16µm. These parameters, which yield I0 = 5nA and Nr = 8, satisfy
conditions (5.16) and (5.17) for the uncorrelated-channel regime. We insert these
parameters in our result Eq. (5.27) and in Eq. (5.33), adding a factor of 2 due to
spin degeneracy. This yields ( 〈I2Dm=1〉2 )1/2 = 1.4I0, and δI = 1.3I0 sin(2πφ). We see
that δI and ( 〈I2Dm=1〉2 )1/2 are comparable, and both are in fair agreement with the
measured PC of (0.8 ± 0.4)I0. Using the expression for the PC of a 2D cylinder in
the zero-disorder limit obtained in Ref. [9] (replacing H with W ) yields a value larger
by a factor of ∼ 2 compared to our result. When ` ∼ L, the ballistic, diffusive and
exact expressions should give the same order of magnitude for the PC. Indeed, using
the expression for the PC of a diffusive annulus in the uncorrelated-channel regime
[10, 11] gives a value that is very close to the one obtained from Eq. (5.23) for the
parameters of the annulus measured in Ref. [30].
Rabaud et al. [29] measured the PC of an array of 16 ballistic rings of GaAlAs/GaAs.
Those rings are in fact squares whose external total edge length is 16µm and the in-
ternal one is 8µm, yielding L = 12µm. The rings are also characterized by ` = 8µm,
kF = 2 × 108m−1, W = 0.8µ m, and vF = 3.2 × 105m/s, implying I0 = 4.2nA and
Nr = 50. The measured total PC obtained for disconnected rings, divided by the
square root of the number of rings [98], was (0.33±0.07)I0. Neither the uncorrelated-
channel regime nor the correlated-channel regime can be associated with these rings,
since both Eqs. (5.17) and (5.22) are not obeyed by the above parameters. Therefore,
we use our result Eq. (5.13), with q = 1 and a factor of 2 due to spin degeneracy,
and obtain values for 〈Im〉 in the regime (−3I0, 3I0), whose standard deviation is
( 〈I2Dm=1〉2 )1/2 = 1.1I0. From Eq. (5.33) we find that δI = 0.7I0 sin(2πφ). The dis-
crepancy between the measured value, the above ( 〈I2Dm=1〉2 )1/2, and δI may be due
to the geometry (squares instead of rings) as well as due to decoherence [29]. The
relative large W may also play a role.
One may compare our result for ( 〈I2Dm=1〉2 )1/2 for the parameters of Ref. [29] with
results of previous theoretical studies for these “short” annuli [9]-[11]. The latter
correspond to ( 〈I2Dm=1〉2 )1/2 = 7.5I0 in the zero-disorder limit, and ( 〈I2D
m=1〉2 )1/2 =
62 5.9
4.5I0 in the diffusive limit [as given by Eqs. (5.24) and (5.26), adapted to 2D and
including a factor of 2 due to the spin degree of freedom, see Section 5.5]. Hence,
our result is in a smaller disagreement, compared to results of former studies [9]-[11],
with the measured one. This is due to the fact shown above that the conditions for
Eqs. (5.24) and (5.26) to be valid are not satisfied by the parameters of the rings
measured in Ref. [29].
The first harmonic, measured for the diffusive rings used in the studies of Jariwala
et al. [25], and of Bluhm et al. [31], fairly agrees with the theoretical value for δI.
Here the rings are deep enough in the diffusive regime, and so ( 〈I〉2 )1/2 ¿ δI. In the
very recent work of Bleszynski-Jayich et al. [28], where aluminum rings were used,
the high magnetic fields utilized in the experiment cause 〈I〉 to be negligible, but
leave δI unaffected [19]. Indeed, the rms fluctuations, given by Eq. (5.33), agree with
the measured PC [28].
5.9 Discussion
In this work we have studied the disorder-averaged persistent current of non-interacting
electrons. We have extended earlier analytical studies, which considered only the zero-
disorder and the diffusive limits [9]-[12], and have given an expression, Eq. (5.13), for
a general [95] ratio of L/`, as long as kF ` À 1. We define the uncorrelated- and the
correlated-channel regimes in which Eq. (5.13) can be simplified [52] to expressions
(5.23) and (5.28), respectively. While previous works [9]-[13] dealt mostly with 1D
rings or 2D cylinders, we have considered here also rings of finite narrow width. In
particular we have obtained an expression for 3D rings. In addition, our expression
for the PC in a 2D cylinder in the correlated-channel regime in the diffusive limit is
new.
The inset of Figs. 5.2 and 5.3 demonstrate that the disorder-averaged PC may
be a relevant contribution, compared with the fluctuation δI, for slightly diffusive
systems, typically with L/` . 10. The relation between the parameters of a ring that
satisfy ( 〈Im〉2 )1/2 > δI, is given in Eqs. (5.34) and (5.35) for the uncorrelated- and
the correlated-channel regimes, respectively. We find that for the parameters of the
rings used in Refs. [29] and [30] the disorder-averaged PC is relevant compared to δI.
Interactions, repulsive [20] or attractive [21], can contribute to an h/2e flux-
periodic disorder-averaged PC. However, as long as the sample is not superconduct-
ing, the PC remains a mesoscopic effect. We have recently suggested [33, 34] that
if the effect of pair-breaking is taken into account, attractive interactions can ex-
plain the h/2e signal measured in ensembles of copper [24] and gold [25] rings. The
contribution of interactions to the PC is not sensitive to the exact value of kF L.
Therefore, the interaction-induced PC may be compared to measurements using the
top equality in Eqs. (5.32), for any value of δ(kF L). In contrast, since in reality
5.9 63
δ(kF L) > π, the interaction-independent contributions of both δI and ( 〈I〉2 )1/2 are
compared to measurements using the bottom equality in Eqs. (5.32). Thus, as N
increases the interaction-dependent contributions to the PC become dominant over
the contributions which do not depend on electronic interactions. This explains why
measurements on ensembles of 105 and 107 rings revealed only the h/2e harmonic
[24, 27]. It seems that the h/e harmonic can be accounted for only by the part of the
PC that is independent of interactions, which we study here. However, since the h/2e
periodicity of the interaction-dependent part of the PC was obtained from calcula-
tions of the disorder-averaged PC [20, 21], further study is needed to assure that the
h/e harmonic is not present in the interaction-dependent parts of δI. The special case
of a single-channel (pure 1D) interacting system [99] can be solved using bosoniza-
tion techniques. Qualitative differences exist then between repulsive and attractive
interactions. In 1D rings, interactions affect the first harmonic of the sample-specific
current.
Each harmonic has a different temperature dependence. Higher harmonics decay
faster with temperature since they necessitate multiple paths around the ring [9, 10].
For this reason we treated the different harmonics separately, though our calculations
are carried out at zero temperature.
We call attention to the appearance of positive powers of the channel number
(although the negative power of kF L in the correlated-channel regime may partially
compensate that) in the PC magnitude. This implies that once multichannel ballistic
systems would be manufactured, relatively large PCs should appear. Both molecular
and clean semiconducting systems come to mind in this connection, and perhaps
semimetals, such as Bi (see first reference of Ref. [8]). On the other hand, in all
regimes, the disorder-averaged PC in the diffusive limit is highly suppressed by a
factor of exp(−L/2`). Again, achieving ` not too small compared with L, will be
helpful.
64 5.9
65
Appendix A
Supplementary background on
superconductivity
For the convenience of the reader, we summarize in this section known results of the
microscopic theory of superconductivity and the Ginzburg-Landau theory [48, 64,
100]. These results are used in Chapter 2.
A.1 The first experimental observations of super-
conductivity
The ability to liquify helium enabled experimentalists to study the properties of mate-
rials in the low temperature regime of a few degrees Kelvin. This led to the discovery
of superconductivity in 1911 by H. Kamerlingh Onnes [101] in Leiden. Onnes observed
that the resistivity of metals, such as mercury, lead, and tin, disappears below a cer-
tain temperature, Tc, which depends on the material. W. Meissner and R. Ochsenfeld
found in 1933 in Berlin [45] a second important characteristic of the superconducting
state: a superconductor is a perfect diamagnet. A magnetic field cannot penetrate
the bulk of a superconductor [102]. The superconductor expels the magnetic field also
when it is cooled below Tc in the presence of the field. Namely, this is an equilibrium
effect. A magnetic field penetrates the superconductor only via a thin outer layer.
More precisely, the Meissner effect appears in type I superconductors, but in type II
superconductors it is exhibited only at very low fields. In the first few decades after
the discovery of superconductivity, a fundamental understanding of this phenomenon
was missing.
66 A.2
A.2 Microscopic description of the superconduct-
ing phase
In 1956 L. N. Cooper showed [103] that even a weak attraction causes instability
of the Fermi sea with respect to the formation of bound pairs of electrons. In low
temperature superconductors, the attraction between the electrons is mediated by
the interaction with the phonons. An electron polarizes the medium by attracting
positive ions, and the latter attract a second electron, resulting in an effective at-
tractive interaction between the electrons. The role of the electron-lattice interaction
in explaining superconductivity was first suggested by H. Frolich [104] in 1950, and
was confirmed in the same year by measuring the isotope effect [105]. J. Bardeen,
L. N. Cooper, and J. R. Schrieffer (BCS) constructed a mean-field theory for super-
conductivity [106]. The attractive interaction between the electrons in BCS theory is
general, and may be related also to interactions other than the electron-phonon one.
The microscopic BCS ground state of a clean system is given by the wave function
|BCS〉 = Πk(uk + vkc†k↑c
†−k↓)|φ0〉 , |uk|2 + |vk|2 = 1 , (A.1)
where |φ0〉 is the vacuum state with no particles. We define
∆k ≡ −∑
l
Vkl〈c−l↓cl↑〉 , (A.2)
where Vkl is the interaction coefficient in the pairing Hamiltonian,
H =∑
kσ
~2k2
2mc†kσckσ +
∑
klσ
Vklc†k↑c
†−k↓c−l↓cl↑ . (A.3)
The self-consistency equation for ∆ is
∆k = −∑
l
Vklu∗l v∗l [1− 2f(εl)] = −
∑
l
Vkl∆l
2εl[1− 2f(εl)] . (A.4)
Here
εl =√
ξ2l + ∆2 , (A.5)
are the eigenenergies of the fermionic excitations of the superconductor, and
ξl =~2l2
2m− µ. (A.6)
The Fermi distribution function is denoted in Eq. (A.4) by f .
The Fermi sea and the Pauli principle play a major role in the pairing correlations.
These correlations are not so sensitive to the exact dynamical attractive interactions.
This allowed Bardeen et al. [106] to use a simplified interaction
Vkl =
−V if |ξk| and |ξl| ≤ ~ωc
0 otherwise. (A.7)
A.2 67
In low-temperature superconductors, the cutoff frequency, ωc, is given by the Debye
frequency, ωD. Inserting the simplified interaction, Eq. (A.7), into Eq. (A.4) results
in
1 = N (0)V
∫ ~ωc
−~ωc
dξ1− 2f(
√ξ2 + ∆2)
2√
ξ2 + ∆2. (A.8)
At the critical temperature the excitation-gap vanishes. Substituting ∆(Tc) = 0 in
Eq. (A.8) yields
Tc = 1.13~ωce−1/N (0)V = 0.57∆(T = 0) . (A.9)
Note that Tc is proportional to the gap at zero temperature. Comparing the energy
expectation-values of the BCS and the Fermi sea states shows that the first indeed
corresponds to a lower energy
〈E〉BCS − 〈E〉Fermi = −1
2N (0)∆2 . (A.10)
This is the condensation energy.
Two particles are correlated if the distance between them is smaller than a given
coherence length. This length can be estimated in the case of a pure superconductor at
zero temperature by the following considerations [64]. The significant energy interval
for superconductivity is given by
EF −∆ < E < EF + ∆ . (A.11)
Substituting in Eq. (A.11) E = ~2(kF + δk)2/2m gives
δk = 2∆/~vF . (A.12)
Then, the coherence length ξ0 may be estimated by the uncertainty principle, ξ0 '1/δk. Hence,
ξ0 =~vF
π∆= 0.18
~vF
Tc
. (A.13)
It is a convention to define ξ0 with the above factor of π. In the last equality in
Eq. (A.13) we apply Eq. (A.9).
In the case of a dirty superconductor, ` ¿ ξ0, the coherence length decreases due
to the impurities and is given, at zero temperature, by
ξ(0) '√
ξ0` '√~D/Tc , (A.14)
where D = vF `/d is the diffusion coefficient, and d is the dimensionality of the system.
68 A.3
A.3 Ginzburg-Landau theory
In 1950 V. L. Ginzburg and L. D. Landau [107] established a phenomenological de-
scription of superconductivity based on Landau theory of second-order phase transi-
tions. In their theory a pseudo wave function, Ψ, acts as the order parameter which
describes the superconducting electrons. The free-energy density of the supercon-
ductor in the vicinity of the phase transition, can be expanded in a series of the
form
f = fN0 + α|Ψ|2 +β
2|Ψ|4 +
1
2m∗
∣∣∣∣(−i~∇− e∗
cA
)Ψ
∣∣∣∣2
+1
8πB2 . (A.15)
Here fN0 is the free-energy density of the normal state in the absence of field. Hence-
forth, we substitute −2e = e∗ and 2m = m∗. These values of e∗ and m∗ are found by
the microscopic pairing theory. The parameter α is very sensitive to the temperature
α(T ) = [(T − Tc)/Tc]α′ , α′ > 0 , (A.16)
while β is positive and roughly constant near Tc.
Minimizing the free energy with respect to the complex conjugate of the order
parameter, Ψ∗, and with respect to A results in
αΨ + β|Ψ|2Ψ +1
4m
(−i~∇+
2e
cA
)2
Ψ = 0 , (A.17)
and
J =
[~eim
(Ψ∗∇Ψ−Ψ∇Ψ∗)− 4e2
mc|Ψ|2A
]=
c
4π∇×B , (A.18)
respectively. In the absence of the gradient term and the field, Eq. (A.17) yields
|Ψ0|2 = −α/β . (A.19)
When the material undergoes the superconducting phase-transition in the presence
of a magnetic field, the field is expelled from the superconductor. Therefore, in the
absence of a magnetic field, the difference between the free-energy density of the
normal state and the superconducting state is
(fN − fS)H=0 = α2/2β = H2c /8π , (A.20)
see Eqs. (A.15) and (A.19). Here Hc = Hc(T ) is the critical magnetic field above
which the material does not become superconducting. The energy-density difference
given by Eq. (A.20) equals the condensation energy, Eq. (A.10), found from the BCS
theory, divided by the volume of the system.
The spatial variation of the order parameter occurs on a length scale
ξ2 =~2
4m|α| , (A.21)
A.3 69
which is the coherence length. Both ξ and λs are proportional to |α|−1/2 ∝ [(T −Tc)/Tc]
−1/2. Equations (A.10), (A.19), (A.20), and (A.21) yields
|Ψ0|2 =4m
~2n(0)ξ2∆2 . (A.22)
Inserting the expressions for ξ at zero temperature, Eqs. (A.13) and (A.14), gives
|Ψ0|2clean ∼ n , (A.23)
and
|Ψ0|2dirty ∼ |Ψ0|2clean
`
ξ0
∼ nτ∆/~ . (A.24)
Here n is the electron density, and we use the relation n ∼ n(0)EF .
70 A.3
71
Appendix B
Gauge invariance of second order
perturbation theory
A gauge transformation of the magnetic vector potential A → A +∇ϕ, where ϕ is a
single-valued function, does not affect the physical picture. Thus, the eigenenergies
of the Hamiltonian are not modified. In Section 2.2 we considered the case where
the unperturbed Hamiltonian (in which A = 0) is time-reversal invariant and its
ground state in non-degenerate. We observed that the paramagnetic term and the
diamagnetic term in the correction to the total energy calculated in second-order
perturbation theory are, separately, not gauge invariant. In this section we show that
their sum is gauge invariant.
The contribution to the diamagnetic term, Eq. (2.19), of a gauge field ϕ is given
by
∆E(2)i,dia =
e2
2m〈i|[2A · (∇ϕ) + (∇ϕ)2]|i〉 . (B.1)
The contribution to the paramagnetic term, Eq. (2.22), is
∆E(2)i,para =
e2
4m2
∑j
1
Ei − Ej
×[|〈i| [2(A +∇ϕ) · ∇+ (∇ · (A +∇ϕ))] |j〉|2 − |〈i| [2A · ∇+ (∇ ·A)] |j〉|2
]
= − e2
4m2
∑j
1
Ei − Ej
[(〈i| [2A · ∇+ (∇ ·A)] |j〉〈j| [2∇ϕ · ∇+∇2ϕ
] |i〉+ c.c.)
+ 〈i| [2∇ϕ · ∇+∇2ϕ] |j〉〈j| [2∇ϕ · ∇+∇2ϕ
] |i〉]
.
(B.2)
We note that
〈i| [2∇ϕ · ∇+∇2ϕ] |j〉 = 〈∇2i|ϕ|j〉 − 〈i|ϕ|∇2j〉 = −2m(Ei − Ej)〈i|ϕ|j〉 . (B.3)
72
Equation (B.3) allows us to eliminate the summation on j in Eq. (B.2). Thus,
∆E(2)i,para = − e2
2m
(〈i| [2A · ∇+ (∇ ·A)] ϕ|i〉 − 〈i|ϕ [2A · ∇+ (∇ ·A)] |i〉
− 〈i|ϕ (2∇ϕ · ∇+∇2ϕ
) |i〉)
= − e2
2m〈i|2A · (∇ϕ) + (∇ϕ)2|i〉 .
(B.4)
In the last equality in Eq. (B.4) we use the fact that the ground state, |i〉, of an
unperturbed time-invariant Hamiltonian is real. From Eqs. (B.1) and (B.4) we see
that
∆E(2)i,para + ∆E
(2)i,dia = 0 . (B.5)
Thus, we have explicitly shown that for an unperturbed time-invariant Hamiltonian,
the sum of the paramagnetic and the diamagnetic corrections to the energy is gauge
invariant.
73
Appendix C
Derivation of the renormalized
interaction
In this section we give a detailed derivation of Eq. (2.37) for the momentum- and
frequency-independent part of the renormalized interaction. We write the partition
function, using Grassman variables [77], as a product of two subpartition functions
Z = Z<Z> . (C.1)
In the first factor
Z< =
∫D(ψ<, ψ<)e−S< , (C.2)
we integrate over all states whose kinetic energies satisfy
|Ek| ∈ [0, D − δD) . (C.3)
In the abbreviated notation ψ< we omit the indices of momentum, frequency and
spin. The action S< is defined below. The factor Z> is given by the same formula
as Z< is defined by, but now the integration is over the states, ψ>, whose kinetic
energy is close to the edges of the energy-band, see Eq. (2.36). The action associated
with Z> is denoted by S>. We consider a system with a point-like electron-electron
interaction and no disorder [78]. Thus, the action in the expression for Z is given by
S = S0 + Sint , (C.4)
where
S0 = β∑
k,ω,α
ψk,ω,α(−iω + Ek)ψk,ω,α
Sint = βV∑
k1,k2,q
∑ω1,ω2,ν
ψk1+q,ω1+ν↑ψ−k1,−ω1↓ψ−k2,−ω2↓ψk2+q,ω2+ν↑ ,(C.5)
Here β = 1/T and α denotes the spin. The actions S< and S> are defined analogously
to S in Eqs. (C.4) and (C.5) with the following modifications. In S<, instead of
74
summing over all momenta, we restrict ourselves to those which satisfy Eq. (C.3). In
the action S0> the momenta is summed only over values for which Eq. (2.36) is satisfied,
and in Sint> at least one of the momenta [k1 or k2, see Eq. (C.5)] is associated with
a ψ> state. In the renormalization procedure we integrate over the states ψ>. Then
we show that the full partition function can be given by Z< with a renormalized
interaction coefficient.
We use the property
ψ2 = 0 (C.6)
of Grassman algebra and expand the interaction-free term in Z>
e−S0> = eβ
∑ψ>(iω−Ek)ψ> =
∏[1 + βψ>(iω − Ek)ψ>] , (C.7)
and the interaction-dependent term in Z>
e−Sint> = e−βV
∑ψψψψ =
∏(1− βV ψψψψ)
= 1− βV∑
ψk1+q↑ψ−k1↓ψ−k2↓ψk2+q↑
+ (βV )2∑
ψk1+q↑ψ−k1↓ψ−k2↓ψk2+q↑ψk3+q′↑ψ−k3↓ψ−k4↓ψk4+q′↑
+ O[(βV )3] .
(C.8)
The states with momenta ki + q (respectively, ki + q′) have frequencies of ωi + ν
(respectively, ωi + ν ′) and spin-up, for i = 1, 2, 3, 4. The states with momenta −ki
have frequencies of −ωi and spin-down. In each term in the right hand side of the
last equality of Eq. (C.8), we sum over repeating momenta and frequency indices.
Keeping terms only up to order (βV )2 in Eq. (C.8) corresponds to considering at
most one intermediate-scattering event into states near the edges of the energy-band,
see Fig. 2.1. Using the properties
∫dψ = 0,
∫dψψ = 1 , (C.9)
and Eq. (C.6), we reach the following conclusions about Z>:
• The terms of order (βV )1 that have a non-vanishing contribution are only those
for which k1 = k2 and ω1 = ω2. Therefore, in these terms, the energy associ-
ated with both momenta belongs to the edges of the energy-band, Eq. (2.36).
However, these contributions can be shown to be smaller by V/(βD2) than the
contribution of the term of order of (βV )0. Therefore, we henceforth ignore the
terms of order (βV )1.
• Without loss of generality, we assume that in the terms of order (βV )2, the
momenta k1 and k4 satisfy Eq. (C.3), and thus correspond to states ψ<, while
k2 and k3 satisfy Eq. (2.36), and corresponds to states ψ>. These terms give a
75
nonzero contribution when integrated over only if k2 = k3, q = q′, ω2 = ω3, and
ν = ν ′. [We ignore the terms for which all the four momenta indices correspond
to ψ> states, since their contribution is smaller by (V/D)2 than that of the
(βV )0 order term.]
Therefore, up to second order in βV , we have
Z> =∏
k2,q,ω2,ν
∫d(ψk2+q↑, ψk2+q↑)d(ψ−k2↓, ψ−k2↓)
[β2ψk2+q↑ [i(ω2 + ν)− Ek2+q] ψk2+q↑ ψ−k2↓ [−iω2 − E−k2 ] ψ−k2↓
+ (βV )2∑
k1,k4,ω1,ω4
ψk2+q↑ψ−k2↓ψ−k2↓ψk2+q↑ ψk1+q↑ψ−k1↓ψ−k4↓ψk4+q↑]
.
(C.10)
Preforming the integration in Eq. (C.10), and dividing Z> by the term of order (βV )0,
we obtain
Z> =∏q,ν
[1 +
(∑
k2,ω2
V 2
[i(ω2 + ν)−D][−iω2 −D]
) ∑
k1,k4,ω1,ω4
ψk1+qψ−k1ψ−k4ψk4+q
]
'∏q,ν
exp
[(∑
k2,ω2
V 2
[i(ω2 + ν)−D][−iω2 −D]
) ∑
k1,k4,ω1,ω4
ψk1+qψ−k1ψ−k4ψk4+q
].
(C.11)
Here we neglect the momentum dependence of the high energies. To make connection
with the usual renormalization picture, we now restrict ourselves to the contribution
of ν = 0. Thus, the sum over k2 and ω2 in the inner parentheses of Eq. (C.11) is
given by δDN (0)βV 2/D. Replacing the product by a sum over q in the exponent we
obtain
Zν=0> = exp
[δD
DN (0)βV 2
∑
k1,k4,q,ω1,ω4
ψk1+qψ−k1ψ−k4ψk4+q
]. (C.12)
We substitute the expressions given by Eqs.(C.2) and (C.12) for Zν=0< and Zν=0
> ,
respectively, in Eq. (C.1) for Z. [Here one needs to omit the summation over ν in
Eq. (C.2)]. One can now see that Z is equal to Zν=0< upon replacing the interaction
constant V by the renormalized value [108],
V → V − δD
DN (0)V 2 . (C.13)
By this we have proved Eq. (2.37).
A shorter derivation of Eq. (C.13), which is valid in the static ν = 0 and q = 0
case, can be obtained using elementary second-order perturbation theory
δV = −∑
f
| 〈i|V |f〉 |2Ei − Ef
= δDN (0)V 2/D . (C.14)
76
Here |i〉 stands for the ground state, and we sum over the 2δDN (0) energetically-
forbidden intermediate states. In those states, two electrons are scattered to the high
energy segment given by Eq. (2.36), and thus Ei − Ef = −2D.
77
Appendix D
Derivation of the partition function
of interacting electrons in the
presence of magnetic impurities
Here we derive, using the method of Feynman path integral, the partition function,
Eq. (4.7). In terms of the Grassmann variables ψα(r, τ) [ψα(r, τ)], the partition
function reads
Z =
∫D(ψ(r, τ), ψ(r, τ)) exp(−S) , (D.1)
where the action S is
S =
∫dr
∫ β
0
dτ[ψσ(r, τ)∂τψσ(r, τ) +H(r, τ)
]. (D.2)
Here β = 1/T and H is given by the integrand of the Hamiltonian, Eq. (4.4), with
Grassmann variables (of the same imaginary time) replacing the creation and annihi-
lation operators. Introducing the bosonic fields ∆(r, τ) via the Hubbard-Stratonovich
transformation, the partition function takes the form
Z =
∫D(ψ(r, τ), ψ(r, τ))D(∆(r, τ), ∆∗(r, τ)) exp(−S) , (D.3)
where the differential of the bosonic field ∆(r, τ), D(∆(r, τ), ∆∗(r, τ)), contains a
factor of βV/πg. The action S is given by
S =
∫dr
∫ β
0
dτ( |∆(r, τ)|2
g− 1
2Ψ(r, τ)G−1
r,r;τ,τΨ(r, τ))
, (D.4)
78
where Ψ = (ψ↑, ψ↓, ψ↑, ψ↓), and the inverse Green function G−1 (at equal positions r
and equal imaginary times τ) is
G−1r=r′;τ=τ ′ =
−∂τ − hφ↑ −2u2S− 0 ∆
−2u2S+ −∂τ − hφ↓ −∆ 0
0 −∆∗ −∂τ + h−φ↑ 2u2S+
∆∗ 0 2u2S− −∂τ + h−φ↓
≡
[G−1
p ∆
∆† G−1h
].
(D.5)
Here h±φα = H0(±φ) + u1 + sgn(α)Szu2, and S± = (Sx ± iSy)/2, where sgn(↑) = 1
and sgn(↓) = −1.
The integration over the fermionic part in Eq. (D.3) yields
Z =
∫D(∆(r, τ), ∆∗(r, τ)) exp
(1
2Tr ln(βG−1)−
∫dr
∫ β
0
dτ|∆(r, τ)|2
g
). (D.6)
We expand Tr ln(βG−1) up to second order [109] in ∆
Tr ln(βG−1) = Tr ln(βG−10 )− 1
(βV )2
∫∫∫∫drdr′dτdτ ′ Tr
[Gp(r
′, τ ′; r, τ)
× ∆(r, τ)Gh(r, τ ; r′, τ ′)∆†(r′, τ ′)]
. (D.7)
The inverse Green function for non-interacting electrons, G−10 , is given by Eq. (D.5)
for ∆ = 0. The first term on the right hand side of Eq. (D.7), which is of zeroth order
in ∆, gives rise to the partition function of non-interacting electrons, Z0 = det(βG−10 ).
In Eq. (D.7), Gp (Gh) is the particle (hole) Green function. These functions are
the solutions of
G−1p/h(r, τ)Gp/h(r, τ ; r′, τ ′) = δ(r− r′)δ(τ − τ ′) , (D.8)
where G−1p/h are defined in Eq. (D.5). As can be seen in that equation, the particle
and the hole inverse Green functions are related to one another by
G−1h (r, τ, φ, S+, S−, Sz) = −G−1
p (r,−τ,−φ, S−, S+, Sz) . (D.9)
Therefore,
Gh(r, τ ; r′, τ ′, φ, S+, S−, Sz) = −Gp(r,−τ ; r′,−τ ′,−φ, S−, S+, Sz)
= −Gp(r, τ′; r′, τ,−φ, S−, S+, Sz) , (D.10)
where in the last equality we have used time-translational invariance to shift τ and τ ′
by τ + τ ′. Reversing the sign of the flux φ together with interchanging r and r′ leads
to the relation (the superscript t denotes the transposed matrix)
Gp(r, τ′; r′, τ,−φ, S−, S+, Sz) = Gt
p(r′, τ ′; r, τ, φ, S+, S−, Sz) . (D.11)
79
We have used Eqs. (D.10) and (D.11) to replace the hole Green function in Eq. (D.7)
by a particle Green function. Then, in momentum representation, the second term
on the right-hand side of Eq. (D.7) reads [110]
Tr ln(βG−1)∣∣∣2nd
=∑
q1,q2,ν
∑
k1,k2,ω
Tr[Gp(k1 + q1,k2 + q2, ω + ν)
× ∆(q2, ν)Gtp(−k1,−k2,−ω)∆†(q1, ν)
]. (D.12)
The flux dependence is incorporated into the momenta p, where p2/2m − µ are the
eigenvalues of H0(φ). Thus, the longitudinal components of the momenta in the
Green function G have the form
2π(n + φ)/L , (D.13)
while those of the momenta in the boson field ∆ are
2π(n + 2φ)/L , (D.14)
where n is an integer. The Matsubara frequencies of the Green functions, ω + ν and
−ω, are fermionic [= π(2n+1)T ]. The order-parameter fluctuations are characterized
by the Matsubara bosonic frequencies ν = 2πnT .
The resulting expression for the partition function may be simplified since the
terms that survive the disorder average in the sum of Eq. (D.12) are those for which
[78] q1 = q2. Following Ref. [78], we disorder average over the exponent in Eq. (D.6),
rather than over the free energy, to obtain an answer which is correct to leading order
in (µτ+)−1. Finally we trace over the product of the 2× 2 matrices in Eq. (D.12) and
integrate over ∆ in Eq. (D.6). In this way we obtain the partition function, Eq. (4.7).
80
81
Appendix E
An alternative statistical approach
for the description of the current
In Chapter 5 we have used the Green’s function technique for our calculations of the
PC of non-interacting electrons. In this section we develop an alternative statistical
approach to approximate the current in the uncorrelated-channel regime and the zero-
disorder limit. The following approach leads to the magnitude of the PC, which is
given [9] by Eq. (5.25), in a more intuitive way. We study here the probabilities that
the channels are filled with an odd or an even number of electrons, and use the results
for PCs in canonical 1D rings, to obtain the PCs of 2D or 3D rings.
In the regime −1/2 ≤ φ ≤ 1/2, the PC of a 1D ring with an odd or with an even
number of electrons, see, for example, Ref. [57], is given by
Iodd = −2φevF
L, (E.1)
Ieven = [sgn(φ)− 2φ]evF
L. (E.2)
These currents have periodicity of unity in φ. Consider a ring of finite width in the
grand-canonical ensemble at zero temperature. The contribution of the (q, s) channel
to the PC is obtained by replacing vF in Eqs. (E.1) and (E.2) by an effective Fermi
velocity vF (q, s) = MkF (q, s), see Eqs. (5.8) and (5.11). Here, the exact position
where the chemical potential crosses the energy levels of each channel determines
whether the channel is occupied by an even or an odd number of electrons, see Fig. E.1.
In an ensemble of rings with similar but not identical parameters, the energy levels
of a given channel are shifted (among the rings) due to fluctuations in H and W , see
Eq. (5.4). Also, the variation in these levels with φ is changing due to fluctuations in
L. Therefore, the exact position of µ relative to the energy levels of a given channel is
distributed randomly in the ensemble. When the levels with E ≤ Eq,s,−n in Fig. E.1
are occupied the channel consists of an even number of electrons, and when the
82
E
-n
n
-n-1
Figure E.1: The energy levels of a single channel are plotted as a function of the flux.
The consecutive energy levels for a given positive flux and longitudinal indices −n,
n, and −n − 1, are marked by full circles. The bottom level corresponds to n = 0.
The random choice of µ in the interval [Eq,s,−n(φ), Eq,s,−n−1(φ)] yields an odd number
of occupied levels when µ > Eq,s,n(φ) and an even number of occupied levels when
µ < Eq,s,n(φ). The former regime is marked by the bold line in the figure. Here,
without loss of generality, we take n > 0.
levels with E ≤ Eq,s,n are occupied the channel consists of an odd number. Taking
Eq,s,n(φ) ' µ the probability that a channel consists of an odd number of electrons is
determined by
Podd =Eq,s,−n−1(φ)− Eq,s,n(φ)
Eq,s,−n−1(φ)− Eq,s,−n(φ). (E.3)
We assumed here φ > 0 and n > 0. The difference appearing in the numerator is
shown in Fig. E.1 as a vertical line. Inserting the eigenenergies, Eq. (5.4), in Eq. (E.3)
(considering n À 1), yields
Podd = 1− 2|φ| , Peven = 2|φ| . (E.4)
These probabilities are independent of the channel index.
We calculate the average current in an ensemble of similar rings using the currents
83
and the probabilities given in Eqs. (E.1), (E.2), and (E.4), and find
I(q, s) = PoddIodd(q, s) + PevenIeven(q, s) = 0 ,
I =∑q,s
I(q, s) = 0 . (E.5)
For |φ| ¿ 1, the probability to have an odd number of electrons in a channel is much
larger than the probability to have an even number, see Eq. (E.4). However, since
|Ieven| À |Iodd|, see Eqs.(E.1) and (E.2), the average PC is zero. This suggests very
large fluctuations of the current at small flux. The typical magnitude of I(q, s) is
given by
(I2(q, s)
)1/2
=√
PoddI2odd(q, s) + PevenI2
even(q, s)
=√
2|φ|(1− 2|φ|) evF (q, s)
L. (E.6)
We add the assumption that the contributions of different channels to the PC are
uncorrelated, which, together with Eq. (E.5), yields
I(q, s)I(q′, s′) = δqq′δss′I2(q, s) . (E.7)
Using Eqs. (E.6) and (E.7) we obtain the standard deviation of the current
(I2
)1/2
=[∑
q,s
I2(q, s)]1/2
=√
2|φ|(1− 2|φ|) evF
LCD . (E.8)
Here
CD =
[∑q,s
v2F (q, s)
v2F
]1/2
=
1 1D√2Nz/3 2D√Ntot/2 3D
(E.9)
depends on the dimensionality of the ring. The nonanalytic√
φ behavior at φ ¿ 1
at zero temperature is due to the paramagnetic contributions, since Peven ∝ φ, while
Ieven ∝ ±const at φ → 0. Thus, the slope of Eq. (E.8) at φ = 0 diverges [111].
Equation (E.8) reproduces Eq. (5.25) obtained for the uncorrelated-channel regime
in the zero-disorder limit for 3D rings. For one- and two-dimensional rings, Eq. (E.8)
reproduces the results of Refs. [9] and [13]. The reason for this equivalence is that
Eq. (5.18), which yields Eq. (5.25), is equivalent to Eq. (E.7).
For a finite ensemble of N clean rings, whose typical number of channels is Ntot,
the probability that all channels in all rings will be occupied by an odd number of
electrons is given for small φ by
(Podd)NNtot −−−−−−→
φNNtot¿11− 2φNNtot . (E.10)
84
This probability becomes arbitrarily close to unity for φNNtot ¿ 1. Therefore, such a
measurement will produce the diamagnetic linear response of a clean superconductor
(see Section 5.2). By increasing the flux in a given finite ensemble (or by increasing
NNtot), even channels will appear one by one, each giving a large paramagnetic
contribution, eventually causing the zero average and anomalously large fluctuations
of the current.
Note that an ensemble of 1D rings, with equal probability for an odd and for an
even number of electrons in a ring, should exhibit a very large paramagnetic response,
see Eqs. (E.1) and (E.2).
BIBLIOGRAPHY 85
Bibliography
[1] H. Bouchiat, Physics 1, 7 (2008).
[2] Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).
[3] I. O. Kulik, JETP Lett. 11, 275 (1970).
[4] L. Gunther and Y. Imry, Solid State Commun. 7, 1391 (1969).
[5] E. N. Bogachek and G. A. Gogadze, Sov. Phys. JETP 36, 973 (1973).
[6] N. B. Brandt, D. V. Gitsu, A. A. Nikolaeva, and Ya. G. Ponomarev, Sov. Phys.
JETP 45, 1226 (1977).
[7] M. Buttiker, Y. Imry, and R. Landauer, Phys. Lett. 96A, 365 (1983).
[8] P. Ehrenfest, Physica (The Hague) 5, 388 (1925); Z. Phys. 58, 719 (1929); C. V.
Raman and K. S. Krishnan, Proc. R. Soc. London A113, 511 (1927); L. Pauling,
J. Chem. Phys. 4, 673 (1936); K. Lonsdale, Proc. Roy. Soc. A159, 149 (1937);
F. London, J. Phys. Radium 8, 397 (1937); for an early reference on mesoscopic
orbital magnetism in normal metals, see: F. Hund, Ann. Phys. 32, 102 (1938).
[9] H. F. Cheung, Y. Gefen, and E. K. Riedel, IBM J. Res. Develop. 32, 359 (1988).
[10] H. F. Cheung, E. K. Riedel, and Y. Gefen, Phys. Rev. Lett. 62, 587 (1989).
[11] E. K. Riedel, H. F. Cheung, and Y. Gefen, Phys. Scr. T25, 357 (1989).
[12] O. Entin-Wohlman and Y. Gefen, Europhys. Lett. 8, 477 (1989).
[13] H. F. Cheung, Y. Gefen, E. K. Riedel and W. H. Shih, Phys. Rev. B 37, 6050
(1988).
[14] E. K. Riedel and F. von Oppen, Phys. Rev. B 47, 15449 (1993).
[15] L. Wendler, V. M. Fomin, and A. A. Krokhin, Phys. Rev. B 50, 4642 (1994).
[16] N. Argaman, Y. Imry, and U. Smilansky, Phys. Rev. B 47, 4440 (1993).
86 BIBLIOGRAPHY
[17] B. L. Altshuler, Y. Gefen, and Y. Imry, Phys. Rev. Lett. 66, 88 (1991).
[18] G. Montambaux, H. Bouchiat, D. Sigeti, and R. Friesner, Phys. Rev. B 42, 7647
(1990).
[19] E. Ginossar, L. I. Glazman, T. Ojanen, F. von Oppen, W. E. Shanks, A. C.
Bleszynski-Jayich, and J. G. E. Harris, Phys. Rev. B 81, 155448 (2010).
[20] V. Ambegaokar and U. Eckern, Phys. Rev. Lett. 65, 381 (1990).
[21] V. Ambegaokar and U. Eckern, Europhys. Lett. 13, 733 (1990).
[22] V. E. Kravtsov and B. L. Altshuler, Phys. Rev. Lett. 84, 3394 (2000).
[23] U. Eckern and P. Schwab, J. of Low Temp. Phys. 126, 1291 (2002).
[24] L. P. Levy, G. Dolan, J. Dunsmuir, and H. Bouchiat, Phys. Rev. Lett. 64, 2074
(1990).
[25] E. M. Q. Jariwala, P. Mohanty, M. B. Ketchen, and R. A. Webb, Phys. Rev.
Lett. 86, 1594 (2001).
[26] R. Deblock, R. Bel, B. Reulet, H. Bouchiat, and D. Mailly, Phys. Rev. B 89,
206803 (2002).
[27] R. Deblock, Y. Noat, B. Reulet, H. Bouchiat, and D. Mailly, Phys. Rev. B 65,
075301 (2002).
[28] A. C. Bleszynski-Jayich, W. E. Shanks, B. Peaudecerf, E. Ginossar, F. von Op-
pen, L. Glazman, and J. G. E. Harris, Science, 326, 272 (2009).
[29] W. Rabaud, L. Saminadayar, D. Mailly, K. Hasselbach, A. Benoit, and B. Eti-
enne, Phys. Rev. Lett. 86, 3124 (2001).
[30] D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. Lett. 70, 2020 (1993).
[31] H. Bluhm, N. C. Koshnick, J. A. Bert, M. E. Huber, and K. A. Moler, Phys.
Rev. Lett. 102, 136802 (2009).
[32] A. Schmid, Phys. Rev. Lett. 66, 80 (1991).
[33] H. Bary-Soroker, O. Entin-Wohlman, and Y. Imry, Phys. Rev. Lett. 101, 057001
(2008).
[34] H. Bary-Soroker, O. Entin-Wohlman, and Y. Imry, Phys. Rev. B 80, 024509
(2009).
BIBLIOGRAPHY 87
[35] F. Pierre, A. B. Gougam, A. Anthore, H. Pothier, D. Esteve, and N. O. Birge,
Phys. Rev. B 68, 085413 (2003). This work highlighted experimentally the role of
minute amounts of magnetic impurities in producing the extra low temperature
dephasing in the noble metal samples.
[36] Former suggestions that copper and gold have a finite Tc are based on extrapu-
lations of the Tc of varying amounts of alloying [37], and on the proximity effect
[38] between these noble metals and superconducting elements.
[37] R. F. Hoyt and A. C. Mota, Solid State Commun. 18, 139 (1976).
[38] G. Deutscher, S. Y. Hsieh, P. Lindenfeld, and S. Wolf, Phys. Rev. B, 8, 5055
(1973).
[39] H. Bary-Soroker, O. Entin-Wohlman, and Y. Imry, Phys. Rev. B 82, 144202
(2010).
[40] O. Entin-Wohlman et al. to be published.
[41] W. E. Shanks and J. G. E. Harris, private communication.
[42] F. Bloch, Z. Physik 52, 555 (1928).
[43] For a multichannel ring changes of the population of states with the flux appear
also for rings in the canonical ensemble.
[44] F. and H. London, Proc. Roy. Soc. (London) A149, 71 (1935).
[45] W. Meissner and R. Ochsenfeld, Naturwissenschaften 21, 787 (1933).
[46] A. B. Pippard, in Proceedings of the Seventh International Conference on Low-
Temperature Physics, 1960, edited by G. M. Graham and A. C. Hollis (University
of Toronto Press, Toronto, 1961), p. 320 ff.
[47] F. London, Phys. Rev. 74, 562 (1948). F. London, Superfluids, Vol. I., Wiley,
New York (1950).
[48] J. R. Schrieffer, Theory of superconductivity, revised edition (Benjamin, New
York, 1983).
[49] N. Byers and C. N. Yang, Phys. Rev. Lett. 7, 46 (1961).
[50] P. W. Anderson, J. Phys. Chem. Sol. 11, 26 (1959).
[51] The dependence of Ntot in the parameters of the ring is described in Ref. [39].
88 BIBLIOGRAPHY
[52] The definitions of long and short cylinders, used in Refs. [9]-[13] do not agree
with the regimes for which the results hold.
[53] W. A. Little and R. D. Parks, Phys. Rev. Lett. 9, 9 (1962); Phys. Rev. 133, A97
(1964).
[54] J. S. Langer and V. Ambegaokar, Phys. Rev. 164, 498 (1967).
[55] D. J. Scalapino, M. Sears, and R. A. Ferrell, Phys. Rev. B 6, 3409 (1972).
[56] D. E. McCumber and B. I. Halperin, Phys. Rev. B 1, 1054 (1970).
[57] Y. Imry, Introduction to Mesoscopic Physics, 2nd ed (Oxford University Press,
Oxford, 2002).
[58] F. von Oppen and K. Riedel, Phys. Rev. B 46, 3203 (1992).
[59] G. Schwiete and Y. Oreg, Phys. Rev. Lett. 103, 037001 (2009). These results are
relevant to recent experiments in Al rings, see N. C. Koshnick, H. Bluhm, M. E.
Huber, and K. A. Moler, Science 318, 1440 (2007).
[60] Y. Liu, Yu. Zadorozhny, M. M. Rosario, B. Y. Rock, P. T. Carrigan, and H.
Wang, Science 294, 2332 (2001).
[61] M. Tinkham, Phys. Rev. 129, 2413 (1963).
[62] K. Maki, in Superconductivity, Vol. II, edited by R. D. Parks, (Dekker, New York,
1969).
[63] A. A. Abrikosov and L. P. Gorkov, Soviet Physics JETP 12, 1243 (1961).
[64] P. G. de Gennes, Superconductivity of metals and alloys, (Benjamin, New York,
1966).
[65] P. Fulde and K. Maki, Phys. Rev. Lett. 18, 675 (1965).
[66] P. G. de Gennes and S. Mauro, Solid State Commun. 3, 381 (1965).
[67] The gapless phase due to proximity is obtained when the order parameter, ∆, is
small.
[68] P. W. Anderson, K. A. Muttalib, and T. V. Ramakrishnan, Phys. Rev. B 28,
117 (1983).
[69] A. M. Finkelstein, Physica B 197, 636 (1994).
[70] Y. Oreg and A. M. Finkelstein, Phys. Rev. Lett. 83, 191 (1999).
BIBLIOGRAPHY 89
[71] D. B. Haviland, Y. Liu and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989).
[72] P. Morel and P. W. Anderson, Phys. Rev. 125, 1263 (1962); N. N. Bogoliubov,
V. V. Tolmachev, and D. V. Shirkov, A New Method in the Theory of Supercon-
ductivity (Consultants Bureau, Inc., New York, 1959).
[73] A. Larkin and A. Varlamov, Theory of Fluctuation in Superconductors (Claren-
don, Oxford, 2005).
[74] The fluctuation correction to the orbital magnetic response above Tc was cal-
culated first by L. G. Aslamazov and A. I. Larkin, Sov. Phys. JETP 40, 321
(1975).
[75] Y. Imry, H. Fukuyama, and P. Schwab, Europhys. Lett. 47, 608 (1999).
[76] The classical limit, i.e., retaining only the lowest Matsubara frequency ν = 0,
holds only when the temperature T is larger than Ec (see Fig. 1 in Ref. [21]).
[77] A. Altland and B. Simons, Condensed Matter Field Theory (Cambridge Univer-
sity Press, Cambridge, 2006).
[78] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum
Field Theory in Statistical Physics (Prentice-Hall, Englewood Cliffs, NJ, 1963).
[79] V. Chandrasekhar, R. A. Webb, M. J. Brady, M. B. Ketchen, W. J. Gallagher,
and A. Kleinsasser, Phys. Rev. Lett. 67, 3578 (1991).
[80] B. Reulet, M. Ramin, H. Bouchiat, and D. Mailly, Phys. Rev. Lett. 75, 124
(1995).
[81] See note added in proof in Ref. [20].
[82] B. L. Altshuler, A. G. Aronov, and D. E. Khmelnitskii, J. Phys. C 15, 7367
(1982).
[83] P. A. Lee and N. Read, Phys. Rev. Lett. 58, 2691 (1987).
[84] B. L. Altshuler and A. G. Aronov, in Electron-Electron Interactions in Disordered
Systems, edited by A. L. Efros and M. Pollak (North-Holland, Amsterdam, 1985).
[85] Equation (4.28) generalizes the result of Ref. [63] to include the effect of a finite
flux.
[86] We use the fact that any meromorphic function f can be written locally as
(z − z0)ng(z), where g(z0) 6= 0,∞. Hence,
f ′
f=
d
dz(ln f) =
n
z − z0
+g′
g,
90 BIBLIOGRAPHY
where f ′/f has a simple pole at z0 with a residue n.
[87] The difference in sign between the zero and the pole terms in Eq. (4.34) is due to
the different sign of n in the equation in Ref. [86] for the poles and the zeros of the
function in the logarithmic derivative. [We checked numerically that Eqs. (4.26)
and (4.34) are identical.]
[88] The derivation of Eq. (4.15) is valid for the diffusive regime, taking Dq2 and
|2ω +ν| much smaller than 1/τ+. Thus, for |n| & L/lel [∼ 100 in the experiments
(Refs. [24]-[26])] Eq. (4.15) should be modified (see Ref. [73]).
[89] B. L. Altshuler, A. G. Aronov, D. E. Khmelnitskii, and A. I. Larkin, in Quantum
Theory of Solids, edited by I. M. Lifshits (MIR, Moscow, 1983), Sec. 3.2.4.
[90] Prefactors in the definition of the Thouless energy vary in the literature.
[91] F. Kuemmeth, S. Ilani, D. C. Ralph, and P. L. McEuen, Nature 452, 448 (2008).
[92] The error is in the expansion of Eq. (14) of Ref. [12].
[93] M. Abramovich and I. A. Stegun, Handbook of Mathematical Functions, (Dover
Publications, New York, 1972), see Eqs. 9.5.(27)-(29) therein.
[94] S. Doniach and E. H. Sondheimer, Green’s function for Solid State Physicists,
2nd ed (Imperial College Press, London 1998). Besides to kF ` À 1, the Born
approximation requires ν(0) ¿ niτ/~, where ν(0) is the density of states and
ni is the concentration of the impurities in the system. The latter condition is
equivalent to σ ¿ k−2F , where σ is the impurity scattering cross section.
[95] We use Eq. (5.7) for a finite system although it is strictly valid only in the
thermodynamic limit. Thus the case L ¿ ` is not rigorously covered by this
formulation. It may be hoped though that this is a reasonable approximation for
a system with periodic boundary conditions.
[96] Equation (5.10) is derived for the grand-canonical ensemble. However, a proper
choice of the chemical potential (which reflects on the value of kF L) will give
the sawtooth shape of the PC in a ring with a fixed number of electrons at zero
disorder, see Ref. [13] and references therein.
[97] The nonanalytic behavior at small flux of Eq. (5.25) [which follows from an
effective 1/φ cutoff of the summation over m in Eq. (5.20) in the zero-disorder
limit] exists only at the T → 0 limit. At any temperature smaller than the single
channel level-spacing, ∆1, there will be a small linear portion for φ ¿ T/∆1,
with a slope proportional to ∆1/T .
BIBLIOGRAPHY 91
[98] Due to a difference in the definition of the parameter Itot in Ref. [29] and in this
paper, we multiplied the value of Itot given in Ref. [29] by the square root of the
number of rings used in that experiment.
[99] D. Schmeltzer, Phys. Rev. B 47, 7591 (1993); A. O. Gogolin and N. V. Prokofev,
Phys. Rev. B 50, 4921 (1994); D. Schmeltzer and R. Berkovitz, Phys. Lett. A
253, 341 (1999); M. Kamal, Z. H. Musslimani, and A. Auerbach, J. Phys. I 5,
1487 (1995).
[100] M. Tinkham, Introduction to Supercoductivity, 2nd ed (McGraw-Hill, Inc., New
York, 1996).
[101] H. Kamerlingh Onnes, Leiden Comm. 120b, 122b, 124c (1911).
[102] In type 2 superconductors above a certain magnetic field, Hc1, the field can
penetrate the bulk via vortices.
[103] L. N. Cooper, Phys. Rev. 104, 1189 (1956).
[104] H. Frohlich, Phys. Rev. 79, 845 (1950).
[105] E. Maxwell, Phys. Rev. 78, 477 (1950); C. A. Reynolds, B. Serin, W. H. Wright,
and L. B. Nesbitt, Phys. Rev. 78, 487 (1950).
[106] J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).
[107] V. L. Ginzburg and L. D. Landau, Zh. Experim. i Teor. Fiz. 20, 1064 (1950).
[108] A non-zero ν adds an imaginary part to the renormalized interaction. To first
order in ν/D Eq. (2.39) changes to
V → V − δD
D
(1 +
iν
D
)N (0)V 2 .
[109] This expansion, up to second order in ∆, is valid for temperatures well above
the transition temperature, and, strictly speaking, above the Ginzburg critical
region. In the case of interest to us, in which Tc vanishes, it is not obvious that
a finite Ginzburg critical region exists at all.
[110] The order of the momenta of the second Green function has been interchanged
in Eqs. (9) and (12) of Ref. [33]. This typo was not carried on in the derivation
there.
[111] It is nontrivial to produce simple formulae for Podd and Peven at finite tem-
peratures using the statistical approximation. The singularities at φ = 0 and
φ = ±1/2 of, respectively, Ieven and Iodd are rounded at finite temperatures. We
92 BIBLIOGRAPHY
expect that Peven(φ = 0) and Podd(φ = ±1/2) will have a finite contribution,
which will keep I = 0 and eliminate (as in Ref. [97]) the square-root singularities
in ( I2(q, s) )1/2 for small φ and φ = ±1/2.
![[1-ג€˜ג€˜C] DLג€”VaIIne, A Potential Pa ncreasג€”lmagin g Agent](https://img.pdfslide.us/doc/110x75/58a178ff1a28abb24d8c056c/1-c-dlvaiine-a-potential-pa-ncreaslmagin-g-agent.jpg)
