lectures accompanying the book: Solid State Physics: An ...philiphofmann.net/book_material/lectures/lecture10_gen.pdf · •lectures accompanying the book: Solid State Physics: An

• lectures accompanying the book: Solid State Physics: An Introduction, by Philip Hofmann (2nd edition 2015, ISBN-10: 3527412824, ISBN-13: 978-3527412822, Wiley-VCH Berlin.

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Superconductivity

The discovery of superconductivity

• Resistance of Mercury at low temperatures (Heike Kamerlingh Onnes in 1911, Nobel price in 1913).• At 4.2 K the resistance drops to an unmeasurably small value.

This seems to be a phase transition with a transition range of around 10-3 K.

What about the low-temperature conductivity of metals?

• Metals should have a finite resistance at low temperatures.

resi

stiv

itty l

(arb

. uni

ts)

l

0

T (arb. units)

0

normal metal

superconductor

0 TC

Periodic table of superconducting elements

Fr Ra Ac Rf

La Hf

Cr Mn Fe Co Ni Cu

Ag

B C O F Ne

l Xe

Po At

Br Kr

Rn

YbTmErHoDyTbGdEuSmPmNdPr

Np Pu Am Cm Bk Cf Es Fm Md No Lr

S ArCl

N

Pd

Pt Au

Rh

Db Sg Bh Hs Mt Ds Rg Uub

Sc

Sr

Ca

Mg

ZrTi V

MoW

Zn GaAl

Th Pa U

SnCd InHg Tl Pb

RuOs Ir

TcRe

NbTa

BeSiGe As Se

Te

LuCe

P

BiSbY

BaCsRb

K

NaLiH Hesuperconducting under high pressure

superconducting under normal conditions

Is the resistivity of a superconductor really zero or just very small?

• A two-point probe measurement will not work because the the contact resistance and the resistance of the wires dominates everything.• A four point probe measurement can be tried but it does not

work either: one is limited by the smallest voltage one can measure.

V I I

R1 R2 R3 R4

Rsample Rsample Rsample

V

R1 R2

I I

Rsample

Is the resistivity of a superconductor really zero or just very small?

• A super-current can be induced in a ring.• The decay of the current is given by the relaxation time

(10-14 s for a normal metal). For a superconductor it should be infinite. Experiments suggest that it is not less than 100.000 years.

Chapter 9

Superconductivity

BT = Bc

1

T

Tc

2(9.1)

I(t) = I0et/ (9.2)

Eg = 3.53kBTc (9.3)

Vc =Eg

2e(9.4)

Bint = Bextez/ (9.5)

> /

2 (9.6)

Bc2 =

2

Bc (9.7)

B = µ0nI (9.8)

26

I

FB

Destruction by magnetic fields

Fr Ra Ac Rf

La Hf

Cr Mn Fe Co Ni Cu

Ag

B C O F Ne

l Xe

Po At

Br Kr

Rn

YbTmErHoDyTbGdEuSmPmNdPr

Np Pu Am Cm Bk Cf Es Fm Md No Lr

S ArCl

N

Pd

Pt Au

Rh

Db Sg Bh Hs Mt UunUuuUub Uuq

Sc

Sr

Ca

Mg

Zr

Ti V

Mo

W

Zn Ga

Al

Th Pa U

SnCd In

Hg Tl Pb

Ru

Os Ir

Tc

Re

Nb

Ta

Be

Si

Ge As Se

Te

LuCe

P

Bi

SbY

BaCs

Rb

K

Na

Li

H

Period

1

2

1

1

2

Group

1716151413 18

18

Group

109876543 11 12Group3

4

5

6

7

He

superconducting under high pressuresuperconducting under normal conditions

Figure 11.2: Periodic table of the elements with the superconducting elementsin bold large letters. Light grey letters indicates that the elements do onlybecome superconducting in a high pressure modification.

small? In fact, one does not know this and it is not possible to answer thisquestion from any experiment. One can, however, try to find and upper limitfor the resistivity. The experimental approach to this is to induce a currentin a superconducting loop using a magnetic field. One should then expectthis current to go on forever without any decay. Observing the decay (orrather the lack of it) over a long time (years!) gives the possibility to give anupper limit for the resistivity. Currently, this upper limit is thought to be10−25 Ωm or so.

Apart from the temperature, there are two other important parameterswhich can destroy the superconducting state. These are a magnetic field anda current through the sample. The combined effect of a magnetic field andthe temperature is shown in Fig. 11.3(a). For low temperatures and smallmagnetic fields, in the area below the curve, the solid is in its supercon-ducting state. For too high a magnetic field or too high a temperature, thesuperconductivity is lost and the solid is in the area above the curve. Fora given temperature T < TC we can assign a critical magnetic field B0C(T )which turns out to be

B0C(T ) = B0C(0)

!

1 −"

T

TC

#2$

.

178

normal

superconducting ext

erna

l fie

ld B

0

temperature T TC

BC(0)

00

BC(T)

Destruction by magnetic fields / current density

TT

j

C

jC(B0,T)

B0

BC(0)

Example: medical MRI scanner

The Meissner effect

• Superconductors (type I) are perfect diamagnets (with χm=-1) which always expel the external field.

• This so-called Meissner effect goes beyond mere perfect conductivity.

The Meissner effect

The Meissner effect perfect conductor

T>TC

B=0 B=Ba


T>TC

B=0 B=Ba

A

IEdl = dB

dt


T>TC

B=0 B=Ba

cool below TC

B=BaB Ba

A

IEdl = dB

dt


T>TC

B=0 B=Ba

cool below TC

B=BaB Ba

B 0 B 0

A

IEdl = dB

dt

The Meissner effect

• For a perfect conductor, the magnetic field inside the conductor cannot change. An enclosed field is possible.

perfect conductor

T>TC

B=0 B=Ba

cool below TC

B=BaB Ba

B 0 B 0

A

IEdl = dB

dt

The Meissner effect movement without friction

perfect conductor

T>TC

B=0 B=Ba

cool below TC

B=BaB Ba

B 0 B 0

m@v

@t= qE

@j

@t=

nsq2

mE


perfect conductor

T>TC

B=0 B=Ba

cool below TC

B=BaB Ba

B 0 B 0

m@v

@t= qE

with Maxwell equation

curlE = @B

@t@

@t

m

nsq2curlj+B

= 0

@j

@t=

nsq2

mE


perfect conductor

T>TC

B=0 B=Ba

cool below TC

B=BaB Ba

B 0 B 0

A

@

@t

Zm

nsq2curljdA+

ZBdA

=

@

@t

Im

nsq2jdl +

ZBdA

= 0.

m@v

@t= qE


curlE = @B

@t@

@t

m

nsq2curlj+B

= 0

@j

@t=

nsq2

mE


perfect conductor

T>TC

B=0 B=Ba

cool below TC

B=BaB Ba

B 0 B 0

mag. flux by j mag. flux by ext. fieldA

@

@t

Zm

nsq2curljdA+

ZBdA

=

@

@t

Im

nsq2jdl +

ZBdA

= 0.

m@v

@t= qE


curlE = @B

@t@

@t

m

nsq2curlj+B

= 0

@j

@t=

nsq2

mE

The Meissner effect

• The essence of the Meissner effect is that the superconductor ALWAYS EXPELS THE MAGNETIC FIELD (ideal diamagnet).

perfect conductor superconductor

T>TC T>TC

B=0

B=0

B=Ba

B=Ba

B=Ba

cool below TC

B=BaB Ba B Ba

B 0 B 0 B 0 B 0

cool below TC

Why levitation?

• The levitation stems from the same reason as ordinary diamagnetic levitation: a combination of gravitational force and a magnetic force due to the inhomogeneous field.

• The only difference is the different strength because of the much higher (negative) susceptibility.

Isotope effect

• Changing the isotope of a material should have a very minor influence on the electronic states.• The isotope effect suggests that the lattice vibrations are

involved in the superconductivity.

remember: energy for an harmonic oscillator

with the frequency1

ω =

!

γ

M.

The total energy for one harmonic oscillator is

E =1

2Mv2

x +1

2γx2.

Assuming classical motion, we can estimate the amplitude of the vibrationby using the equipartition theorem of statistical mechanics. This theoremstates that every generalised momentum or position coordinate which ap-pears squared in the Hamilton function (or in the total energy in our case),should contribute with a mean energy kBT/2 to the system. Here we havetwo squared coordinates and therefore the the mean energy of such an oscil-lator in contact with a heat bath is kBT . At the highest displacement theoscillator has only potential energy and so

1

2γx2

max = kBT

and thus

xmax =

"

2kBT

γ

#1/2

(5.2)

This is usually a few percent of the lattice spacing.

5.1.2 An Infinite Chain of Atoms

A more realistic model for crystal vibrations is a chain of atoms. If werestrict ourselves to longitudinal vibrations in this chain, the model is onlyone dimensional. However, we can learn a lot about vibrations in threedimensional solids already from this. We start with infinite chains of oneand two different atoms per unit cell before passing on to chains of finitelength.

One Atom Per Unit Cell

An atomic chain with one atom per unit cell is shown in Fig. 5.1. From aformal point of view, this is a one dimensional atomic lattice with one atomper unit cell and a lattice vector of length a. This means that the reciprocallattice is also one-dimensional and the reciprocal lattice vector has a length

1For simplicity, we will often speak merely of ’frequency’ when ’angular frequency’would be the correct term.

44

2.05 2.10

0.565

0.575

0.585

~M-1/2

Sn

log (M / 1 amu)

log

(TC

/ 1K)

London equations (1935)

zero resistance 1. London equation@j

@t=

nsq2

msE


zero resistance 1. London equation

mag. flux by j mag. flux by ext. field

@

@t

ms

nsq2curlj+B

= 0

@

@t

Zms

nsq2curljdA+

ZBdA

=

@

@t

Ims

nsq2jdl +

ZBdA

= 0

@j

@t=

nsq2

msE




@

@t

ms

nsq2curlj+B

= 0

@

@t

Zms

nsq2curljdA+

ZBdA

=

@

@t

Ims

nsq2jdl +

ZBdA

= 0

2. London equationMeissner effect ms

nsq2curlj+B = 0

@j

@t=

nsq2

msE




@

@t

ms

nsq2curlj+B

= 0

@

@t

Zms

nsq2curljdA+

ZBdA

=

@

@t

Ims

nsq2jdl +

ZBdA

= 0

2. London equationMeissner effect ms

nsq2curlj+B = 0

Ims

nsq2jdl +

ZBdA = 0

@j

@t=

nsq2

msE

Field penetration

now we have and the Maxwell equation(inside the superconductor)

Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

mvt

= eE (9.3)

jt

=nse2

mE (9.4)

curlE = Bt

(9.5)

t

m

nse2curlj + B

= 0 (9.6)

t

m

nse2curlj + B

= 0 (9.7)

m

nse2curlj + B = 0 (9.8)

L =m

nse2(9.9)

curlj = 1L

B (9.10)

curlB = µ0j (9.11)

curlcurlB = µ0curlj = µ0

LB (9.12)

graddivBB = B = µ0

LB (9.13)

B µ0

LB = 0 (9.14)

E = iR =

Edl = dB

dt;

curlEdA =

BdA; curlE = B (9.15)

27

ms

nsq2curlj+B = 0

Field penetration


Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

mvt

= eE (9.3)

jt

=nse2

mE (9.4)

curlE = Bt

(9.5)

t

m

nse2curlj + B

= 0 (9.6)

t

m

nse2curlj + B

= 0 (9.7)

m

nse2curlj + B = 0 (9.8)

L =m

nse2(9.9)

curlj = 1L

B (9.10)

curlB = µ0j (9.11)


LB (9.12)

graddivBB = B = µ0

LB (9.13)

B µ0

LB = 0 (9.14)

E = iR =

Edl = dB

dt;

curlEdA =


27

ms

nsq2curlj+B = 0

combining this gives

curlcurlB = µ0curlj = µ0nsq2

mB

Field penetration


Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

mvt

= eE (9.3)

jt

=nse2

mE (9.4)

curlE = Bt

(9.5)

t

m

nse2curlj + B

= 0 (9.6)

t

m

nse2curlj + B

= 0 (9.7)

m

nse2curlj + B = 0 (9.8)

L =m

nse2(9.9)

curlj = 1L

B (9.10)

curlB = µ0j (9.11)


LB (9.12)

graddivBB = B = µ0

LB (9.13)

B µ0

LB = 0 (9.14)

E = iR =

Edl = dB

dt;

curlEdA =


27

ms

nsq2curlj+B = 0



mB

Field penetration


Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

mvt

= eE (9.3)

jt

=nse2

mE (9.4)

curlE = Bt

(9.5)

t

m

nse2curlj + B

= 0 (9.6)

t

m

nse2curlj + B

= 0 (9.7)

m

nse2curlj + B = 0 (9.8)

L =m

nse2(9.9)

curlj = 1L

B (9.10)

curlB = µ0j (9.11)


LB (9.12)

graddivBB = B = µ0

LB (9.13)

B µ0

LB = 0 (9.14)

E = iR =

Edl = dB

dt;

curlEdA =


27

ms

nsq2curlj+B = 0



mB

curlcurlB = graddivBB = µ0nsq2

mB

Field penetration


Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

mvt

= eE (9.3)

jt

=nse2

mE (9.4)

curlE = Bt

(9.5)

t

m

nse2curlj + B

= 0 (9.6)

t

m

nse2curlj + B

= 0 (9.7)

m

nse2curlj + B = 0 (9.8)

L =m

nse2(9.9)

curlj = 1L

B (9.10)

curlB = µ0j (9.11)


LB (9.12)

graddivBB = B = µ0

LB (9.13)

B µ0

LB = 0 (9.14)

E = iR =

Edl = dB

dt;

curlEdA =


27

ms

nsq2curlj+B = 0



mB


mB

B =µ0nsq2

mB

Field penetration


Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

mvt

= eE (9.3)

jt

=nse2

mE (9.4)

curlE = Bt

(9.5)

t

m

nse2curlj + B

= 0 (9.6)

t

m

nse2curlj + B

= 0 (9.7)

m

nse2curlj + B = 0 (9.8)

L =m

nse2(9.9)

curlj = 1L

B (9.10)

curlB = µ0j (9.11)


LB (9.12)

graddivBB = B = µ0

LB (9.13)

B µ0

LB = 0 (9.14)

E = iR =

Edl = dB

dt;

curlEdA =


27

This permits only an exponentially damped B field in the superconductor (exercise)

ms

nsq2curlj+B = 0



mB


mB

B =µ0nsq2

mB

Field penetration

The decay length is in the order of 1000 Å.

superconductorB0

λL

4B =µ0nsq2

mB

4j =µ0nsq2

mj

L =pm/µ0nsq2

Ginzburg-Landau theory (1950)

1

ns = |Ψ∗Ψ| = Ψ

2

0

1

order parameter

coherence length ξ(length over which

can change appreciably)

superconductorHext

λL

Figure 11.6: Exponential damping of an external magnetic field near thesurface of a superconductor.

so-called order parameter Ψ(r) which can be derived from an equation similarto the Schrodinger equation and which has a form which is very reminiscentof a free electron wave

Ψ(r) = Ψ0eiφ(r), (11.4)

in which φ(r) is a macroscopically changing phase and |Ψ∗Ψ| = Ψ20 is equal

to the density of superconducting carriers ns. From the spacial characterof Ψ(r) it emerges that ns cannot change instantaneously, for example fromzero to a high value at the surface of a superconductor. Appreciable changesof Ψ(r) and thereby ns have to happen over a length scale ξ, the so-calledcoherence length of the superconductor.

11.4.2 Microscopic BCS Theory

The microscopic theory of superconductivity was formulated in 1957 by J.Bardeen, L. N. Cooper and J. R. Schrieffer. It is commonly referred to asBCS theory, following the names of its inventors. As pointed out in theintroduction, a full appreciation of the theory is only possible using methodsfrom quantum field theory. Here we will restrict ourselves merely to the verybasic ideas and not go into any detail.

Even before the theory was formulated, some basic requirements wereclear. Any theory should of course be able to explain the vanishing resistivityand the Meissner effect. It should probably involve lattice vibrations in someway in order to account for the isotope effect. Finally, it should not bespecific to any particular material since we have seen that superconductivityis a wide spread phenomenon.

185

(r) = 0(r)ei(r)

Key-ideas of the BCS theory of 1957(Bardeen, Cooper, Schrieffer)

• The interaction of the electrons with lattice vibrations (phonons) must be important (isotope effect, high transition temperature for some metals which are poor conductors at room temperature).

• The electronic ground state of a metal at 0 K is unstable if one permits a net attractive interaction between the electrons, no matter how small.

• The electron-phonon interaction leads to a new ground state of Bosonic electron pairs (Cooper pairs) which shows all the desired properties.

The electron-phonon interaction/ Cooper pairs

• Polarization of the lattice by one electron leads to an attractive potential for another electron.

The electron-phonon interaction / Cooper pairs


k

The electron-phonon interaction / Cooper pairs


k k'

The electron-phonon interaction/ Cooper pairs


k k'

What is the “distance” between the electrons forming Cooper pairs?

The lattice atoms move much slowerthan the electrons. The polarization isretarded

Establishing the polarization takes

the electrons move with

so distance between theelectrons forming the Cooperpair is around

Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

me v t

= eE (9.3)

j t

=ne2

meE (9.4)

rotE = dBdt

(9.5)

t

me

ne2rotj + B

= 0 (9.6)

E = iR =

Edl = dB

dt;

rotEdA =

BdA; rotE = B (9.7)

BT = Bc

1

T

Tc

2(9.8)

I(t) = I0et/ (9.9)

Eg = 3.53kBTc (9.10)

Vc =Eg

2e(9.11)


> /

2 (9.13)

Bc2 =

2

Bc (9.14)

B = µ0nI (9.15)2D

= 1013s (9.16)

vF = 106ms1 (9.17)

107m (9.18)

27

BT = Bc

1

T

Tc

2(9.16)

I(t) = I0et/ (9.17)

Eg = 3.53kBTc (9.18)

Vc =Eg

2e(9.19)


> /

2 (9.21)

Bc2 =

2

Bc (9.22)

B = µ0nI (9.23)2

D 1013s (9.24)

vF 106ms1 (9.25)107m (9.26)

TC = 1.13D exp1

g(EF )V(9.27)

coherence and flux quantization

E(r)E(r)/4 = n(r)h (9.28)

E(r) =

4h

nei(r) (9.29)

E(r) =

4h

nei(r) (9.30)(r) =

nei(r) (9.31)

(r) =

nei(r) (9.32)

pds = rh (9.33)

p = mv + qA (9.34)j = nqv (9.35)

p =m

nqj + qA (9.36)

rh =m

nq

jds + q

Ads (9.37)

rh

q=

m

nq2

jds +

Bdf (9.38)

rh

q=

Bdf (9.39)

h

2e(9.40)

m

nq2

jds +

Bdf =

m

nq2

curljdf +

Bdf (9.41)

m

nq2curlj + B = 0 (9.42)

28

BT = Bc

1

T

Tc

2(9.16)

I(t) = I0et/ (9.17)

Eg = 3.53kBTc (9.18)

Vc =Eg

2e(9.19)


> /

2 (9.21)

Bc2 =

2

Bc (9.22)

B = µ0nI (9.23)2

D 1013s (9.24)

vF 106ms1 (9.25)107m (9.26)

TC = 1.13D exp1

g(EF )V(9.27)


E(r)E(r)/4 = n(r)h (9.28)

E(r) =

4h

nei(r) (9.29)

E(r) =

4h

nei(r) (9.30)(r) =

nei(r) (9.31)

(r) =

nei(r) (9.32)

pds = rh (9.33)

p = mv + qA (9.34)j = nqv (9.35)

p =m

nqj + qA (9.36)

rh =m

nq

jds + q

Ads (9.37)

rh

q=

m

nq2

jds +

Bdf (9.38)

rh

q=

Bdf (9.39)

h

2e(9.40)

m

nq2

jds +

Bdf =

m

nq2

curljdf +

Bdf (9.41)

m

nq2curlj + B = 0 (9.42)

28

k

Coherence of the superconducting state• The electrons paired in this way are called Cooper pairs. They

have opposite k vectors (a total k=0) and (in most cases) opposite spins, such that S=0.

• Cooper pairs behave like Bosons and can condense all in a single ground state with a macroscopic wave function. This leads to an energy gain with a size depending on how many Cooper pairs already are in the ground state.

• Cooper pairs are example of “quasiparticles” appearing in solids (like holes or electrons with an effective mass or phonons)

k k'

Ginzburg-Landau theory (1950)

1

ns = |Ψ∗Ψ| = Ψ

2

0

1

order parameter

coherence length ξ(length over which

can change appreciably)

superconductorHext

λL

Figure 11.6: Exponential damping of an external magnetic field near thesurface of a superconductor.

so-called order parameter Ψ(r) which can be derived from an equation similarto the Schrodinger equation and which has a form which is very reminiscentof a free electron wave

Ψ(r) = Ψ0eiφ(r), (11.4)

in which φ(r) is a macroscopically changing phase and |Ψ∗Ψ| = Ψ20 is equal

to the density of superconducting carriers ns. From the spacial characterof Ψ(r) it emerges that ns cannot change instantaneously, for example fromzero to a high value at the surface of a superconductor. Appreciable changesof Ψ(r) and thereby ns have to happen over a length scale ξ, the so-calledcoherence length of the superconductor.

11.4.2 Microscopic BCS Theory

The microscopic theory of superconductivity was formulated in 1957 by J.Bardeen, L. N. Cooper and J. R. Schrieffer. It is commonly referred to asBCS theory, following the names of its inventors. As pointed out in theintroduction, a full appreciation of the theory is only possible using methodsfrom quantum field theory. Here we will restrict ourselves merely to the verybasic ideas and not go into any detail.

Even before the theory was formulated, some basic requirements wereclear. Any theory should of course be able to explain the vanishing resistivityand the Meissner effect. It should probably involve lattice vibrations in someway in order to account for the isotope effect. Finally, it should not bespecific to any particular material since we have seen that superconductivityis a wide spread phenomenon.

185

(r) = 0(r)ei(r)

The BCS ground state

occupied single electron states en

ergy

(ar

b. u

nits

)

excited unpaired electrons

normal metal

unoccupied single electron states

EF



ergy

(ar

b. u

nits

)


normal metal


EFAs the Cooper pairs are formed due to the electron-phonon interaction,

one can expect the size of the gap to be in the order of a typical phononenergy hωD. This is indeed the case. Typically the gap size is given inrelation to the critical temperature TC. BCS theory predicts the gap atzero temperature to be ∆ = 3.53kBTC which amounts to just a few meVfor most superconductors. TC, in turn, can be predicted from the Debyetemperature, the electronic density of states at the Fermi level and V , aparameter measuring the electron-phonon coupling strength. The BCS resultfor the transition temperature, which is not derived here, is

TC = 1.13ΘD exp−1

g(EF )V. (11.5)

This equation tells us a lot about BCS type superconductivity. First ofall, the transition temperature is proportional to the Debye temperature.This readily explains the isotope effect since ΘD ∝ ωD ∝ M−1/2. It alsoshows that the Debye temperature basically sets the temperature scale ofthe phenomenon. The exponential cannot become larger than one so thatone cannot hope to get a transition temperature higher than 1.13ΘD. Inreality, the exponential is much smaller than one. Both the density of statesand the interaction strength enter the expression in the same way, meaningthat both increase the critical temperature.

To make matters even more complicated, the energy gain 2∆ for theformation of one Cooper pair depends on the number of Cooper pairs alreadythere, so that also the size of the gap in the excitation spectrum depends onthis number. As the temperature is raised, the number of Cooper pairsdecreases until it reaches zero at TC. Therefore also the gap size decreases.The gap size as a function of temperature is given in Fig. 11.9.

The existence of Cooper pairs and the gap in the excitation spectrum canbe used to explain all observed phenomena discussed above. Here we merelydiscuss the resistance free transport and the existence of a critical currentdensity and a critical magnetic field. We do not show how the Meissner effectcan be explain but the basic idea is that an equation similar to the secondLondon equation (11.2) can be derived in the BCS model.

In order to explain the resistance free flow of current, it is useful to lookback at Fig. 7.13 and (7.21), (7.22) describing the conduction of normalelectrons in the quantum picture. The basic idea of the process was thatthe external electric field merely shifts the distribution of the electrons in kspace. For bands crossing the Fermi level this would lead to an increasinglyskewed distribution, i.e. to an increasing current density j. This processis limited by the fact that the electrons are scattered by lattice vibrations,defects and impurities.

189

superconductor at T=0

occupied single electron stats


occupied single electron states

2Δ

Hydrogen molecule

54321interatomic distance (a0)

∆E

∆E

bonding

anti-bonding


As the Cooper pairs are formed due to the electron-phonon interaction,one can expect the size of the gap to be in the order of a typical phononenergy hωD. This is indeed the case. Typically the gap size is given inrelation to the critical temperature TC. BCS theory predicts the gap atzero temperature to be ∆ = 3.53kBTC which amounts to just a few meVfor most superconductors. TC, in turn, can be predicted from the Debyetemperature, the electronic density of states at the Fermi level and V , aparameter measuring the electron-phonon coupling strength. The BCS resultfor the transition temperature, which is not derived here, is


g(EF )V. (11.5)





189


ergy

(ar

b. u

nits

)


normal metal superconductor at T=0


EF




26


transition temperature

densityof states

Debye temp.

el-phinteractionstrength

Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

me v t

= eE (9.3)

j t

=ne2

meE (9.4)

rotE = dBdt

(9.5)

t

me

ne2rotj + B

= 0 (9.6)

E = iR =

Edl = dB

dt;

rotEdA =

BdA; rotE = B (9.7)

BT = Bc

1

T

Tc

2(9.8)

I(t) = I0et/ (9.9)

Eg = 3.53kBTc (9.10)

Vc =Eg

2e(9.11)


> /

2 (9.13)

Bc2 =

2

Bc (9.14)

B = µ0nI (9.15)2D

= 1013s (9.16)

vF = 106ms1 (9.17)107m (9.18)

TC = 1.13D exp1

g(EF )V(9.19)

27





g(EF )V. (11.5)





189

isotope effect

densityof states

Debye temp.


Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

me v t

= eE (9.3)

j t

=ne2

meE (9.4)

rotE = dBdt

(9.5)

t

me

ne2rotj + B

= 0 (9.6)

E = iR =

Edl = dB

dt;

rotEdA =

BdA; rotE = B (9.7)

BT = Bc

1

T

Tc

2(9.8)

I(t) = I0et/ (9.9)

Eg = 3.53kBTc (9.10)

Vc =Eg

2e(9.11)


> /

2 (9.13)

Bc2 =

2

Bc (9.14)

B = µ0nI (9.15)2D

= 1013s (9.16)

vF = 106ms1 (9.17)107m (9.18)

TC = 1.13D exp1

g(EF )V(9.19)

27





g(EF )V. (11.5)





189

isotope effect

densityof states

Debye temp.


Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

me v t

= eE (9.3)

j t

=ne2

meE (9.4)

rotE = dBdt

(9.5)

t

me

ne2rotj + B

= 0 (9.6)

E = iR =

Edl = dB

dt;

rotEdA =

BdA; rotE = B (9.7)

BT = Bc

1

T

Tc

2(9.8)

I(t) = I0et/ (9.9)

Eg = 3.53kBTc (9.10)

Vc =Eg

2e(9.11)


> /

2 (9.13)

Bc2 =

2

Bc (9.14)

B = µ0nI (9.15)2D

= 1013s (9.16)

vF = 106ms1 (9.17)107m (9.18)

TC = 1.13D exp1

g(EF )V(9.19)

27

number of electronsin Cooper pairs g(EF ) g(EF )D





2Δ



g(EF )V. (11.5)





189





g(EF )V. (11.5)





189

isotope effect

densityof states

Debye temp.


Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

me v t

= eE (9.3)

j t

=ne2

meE (9.4)

rotE = dBdt

(9.5)

t

me

ne2rotj + B

= 0 (9.6)

E = iR =

Edl = dB

dt;

rotEdA =

BdA; rotE = B (9.7)

BT = Bc

1

T

Tc

2(9.8)

I(t) = I0et/ (9.9)

Eg = 3.53kBTc (9.10)

Vc =Eg

2e(9.11)


> /

2 (9.13)

Bc2 =

2

Bc (9.14)

B = µ0nI (9.15)2D

= 1013s (9.16)

vF = 106ms1 (9.17)107m (9.18)

TC = 1.13D exp1

g(EF )V(9.19)

27

number of electronsin Cooper pairs g(EF ) g(EF )D





2Δ



g(EF )V. (11.5)





189

total energy gain g(EF )2 g(EF )22D

The gap at higher temperatures

1.00.8Temperature T(TC)

Gap

siz

e 6

(6(T

=0))

0.60.40.20.00.0

1.0

0.8

0.6

0.4

0.2

In

PbSn

Detection of the gap: tunnellingSuperconductorNormal metal

Insu

lato

r


Insu

lato

r

2D




EF

insu

lato

r

U=0



occupied single electron states occupied single

electron states


Insu

lato

r

2D




EF

insu

lato

r

U=0




electron states

occupied single electron statesoccupied single electron states


normal metal


EF

insu

lato

r

0<U<D/e

2D






Insu

lato

r

2D




EF

insu

lato

r

U=0




electron states



normal metal


EF

insu

lato

r

0<U<D/e

2D






normal metal



EF

insu

lato

r

U>D/e

2D







Insu

lato

r

2D




EF

insu

lato

r

U=0




electron states



normal metal


EF

insu

lato

r

0<U<D/e

2D






normal metal



EF

insu

lato

r

U>D/e

2D






tunn

ellin

g cu

rren

t (ar

b. u

nits

)

D/etunnelling voltage U

Detection of the gap: optical

• For radiation below hν < 2Δ no absorption, light is transmitted through a thin sheet of superconductor.

• For radiation below hν > 2Δ absorption takes place.

• Relevant light wavelength in the infrared.

hν hν

Detection of the gap: heat capacity

0

Al

1.0 2.00

1

2

3

4

T (K)

heat

cap

acity

(arb

. uni

ts)

cn

cs

Zero Resistance

Zero Resistance

• All Cooper pairs are in the same state and have THE SAME total k. For k=0, no current. For k≠0 supercurrent.

Zero Resistance


• For a normal metal, single electrons can be scattered and be lost for the current. For a superconductor, one can only change the k of all Cooper pairs at the same time. This is VERY unlikely.

Zero Resistance


• For a normal metal, single electrons can be scattered and be lost for the current. For a superconductor, one can only change the k of all Cooper pairs at the same time. This is VERY unlikely.

• Cooper pairs can be split up but this costs 2Δ and they might condensate quickly again. At high current densities, there is enough energy available to break them up. Also at high magnetic field densities.

Superconducting current in a ring

I

FB


I

FB

=h

p


I

FB

=h

p

n = 0,1,2...

Macroscopic coherence: apply Bohr-Sommerfeld quantization

2r = n = nh

p

2rp = nh


I

FB

=h

p

n = 0,1,2...

Macroscopic coherence: apply Bohr-Sommerfeld quantization

2r = n = nh

p

2rp = nh

more generalIpdr = nh


I

FB

p qAdr = hn

Ipdr = nh


I

FB

p qAdr = hn

Ipdr = nh

p = msvj = nsqv

ms

nsq

Ijdr q

IAdr = nh


I

FB

p qAdr = hn

Stoke’s integral theoremAdr =

curlAda =

Bda = B

Ipdr = nh

p = msvj = nsqv

ms

nsq

Ijdr q

IAdr = nh


I

FB

p qAdr = hn

Stoke’s integral theoremAdr =

curlAda =

Bda = B

Ipdr = nh

p = msvj = nsqv

ms

nsq

Ijdr q

IAdr = nh

ms

nsq2

Ijdr B = n

h

q


I

FBms

nsq2

Ijdr B = n

h

q


I

FB

integration path inside the ring material

B = nh

q

ms

nsq2

Ijdr B = n

h

q


The magnetic flux through the ring is quantized. q turns out to have a value of -2e in the experiment. So one flux quantum is

BT = Bc

1

T

Tc

2(9.16)

I(t) = I0et/ (9.17)

Eg = 3.53kBTc (9.18)

Vc =Eg

2e(9.19)


> /

2 (9.21)

Bc2 =

2

Bc (9.22)

B = µ0nI (9.23)2

D= 1013s (9.24)

vF = 106ms1 (9.25)

107m (9.26)

TC = 1.13D exp1

g(EF )V(9.27)


E(r)E(r)/4 = n(r)h (9.28)

E(r) =

4h

nei(r) (9.29)

E(r) =

4h

nei(r) (9.30)

(r) =

nei(r) (9.31)

(r) =

nei(r) (9.32)

pds = rh (9.33)

p = mv + qA (9.34)

j = nqv (9.35)

p =m

nqj + qA (9.36)

rh =m

nq

jds + q

Ads (9.37)

rh

q=

m

nq2

jds +

Bdf (9.38)

rh

q=

Bdf (9.39)

h

2e(9.40)

28

subject to the Pauli exclusion principle and as the temperature is loweredthey can condense into a single quantum state, the BCS ground state, de-scribed by one single wave function. This means that all the electrons form-ing Cooper pairs participate in a single wave function which macroscopicallyextends over the sample. Therefore one speaks of a high coherence of thesuperconducting state.

The concept of the single wave function can be used to describe the re-sistance free transport. In a normal conductor the individual electrons areaccelerated by an electric field and they can be scattered, giving rise to resis-tance. In a superconductor the whole condensate is accelerated. Scatteringcannot occur unless there is enough energy involved to destroy a Cooper pair.Therefore the current can flow without resistance.

Macroscopic coherence effects are of particular importance when consid-ering the penetration of a magnetic field through a superconducting ring,as shown in Fig. 11.13(a). If we assume that the superconducting currentis carried by the entire BCS condensate of Cooper pairs, we can apply theBohr quantisation condition as in the hydrogen atom but here for the totalmomentum p of the whole condensate.

!

pds = rh,

where r is some integer number. If the total momentum is quantised, thisimplies that also the total current through the ring and the magnetic fluxthrough its centre are quantised. It turns out that the magnetic flux canonly penetrate the superconducting ring in flux quanta of h/q where q is thecharge of the supercurrent carriers. Precise measurements have given a valueof q = 2e which is consistent with the BCS theory in which the Cooper pairshave a charge of 2e.

The flux quantum h/2e is very small, only 2.067×10−15Tm2. To set thisinto context, consider the earth’s magnetic field which is of the order of10−5T. In an area of one square millimetre, there would still be 1013 fluxquanta or so.

The fact that the flux in a superconducting ring is quantised can be ex-ploited in so-called superconducting quantum interference devices (SQIDs),as shown in Fig. 11.13(b). We cannot describe how these devices work indetail but the key idea is as follows. The device consists of two superconduct-ing ’forks’ and thin oxide layers or point contacts between them (so-calledweak links). The Cooper pairs from one superconductor can tunnel throughthe weak links into the other superconductor and there is definite relationbetween the phase of the superconducting state in one fork with that of theother fork. A magnetic field entering perpendicular through the forks gives

194

=

I

FB

integration path inside the ring material

B = nh

q

ms

nsq2

Ijdr B = n

h

q

movie of superconducting maglev trainfrom IFW Dresden

Many open questions

Many open questions

• Why does the train follow the tracks?

Many open questions


• Why does the train not fall down when turned upside down?

Many open questions


• Why does the train not fall down when turned upside down?

• How does the train “remember” the distance it has to be above the tracks?

Type I and type II superconductors

External magnetic field B0

Type I

l

BC

l

B = 0B = B

B

0Bin

t

int

int

C

M

0

0

0

0

0

Type I and type II superconductors

External magnetic field B0

Type I

l

BC

l

B = 0B = B

B

0Bin

t

int

int

C

M

0

0

0

0

0

Type II

B B

l =0

C1 C2

B

= B 0

int

B

= 0

int

0<B <B0int

B BC1 C2

lB

int

M

0

0

0

0

0

normalphase

mixedphasepure

supc.phase

external magnetic field B0

type II

B B

l =0

C1 C2

B

= B 0

int

B

= 0

int

0<B <B0int

B BC1 C2

lB

int

M

0

0

0

0

0

Vortices in type II superconductors

• The field penetrates through filaments of normal material, the rest remains superconducting.• These filaments are enclosed by vortices of

superconducting current.

side view

ΦB

top viewvortex

j

=h/2e

How does the magnetic flux penetrate the type II superconductor

• The magnetic field enters in the form of flux quanta in normal-state regions. These regions are surrounded by vortices of supercurrent.

• This can be made visible by Bitter method, optical techniques, scanning tunnelling techniques....

25 x 25 μm (educated guess)


• The vortices repel each other.

• If they can move freely, they form a hexagonal lattice.

• But if there are defects which “pin” the vortices, they cannot change their position.

25 x 25 μm (educated guess)


• Here the vortices can move quite freely.

• Their number changes as the external field is changed.

• For a very high field, the vortices move so close together that the superconductivity in between breaks up.

25 x 25 μm

25 x 35 μm



• The superconductor used in the train movie is a type II superconductor.



• It “pins” the vortices very strongly. Therefore the magnetic field it experienced while cooling through the phase transition is “frozen in”.



• It “pins” the vortices very strongly. Therefore the magnetic field it experienced while cooling through the phase transition is “frozen in”.

• This can explain the effects we did not understand before (staying on the tracks, train upside down, train on walls...).

Current-induced vortex movement

• A current through the material causes the vortices to move perpendicular to the current and to the magnetic field.

• This is very reminiscent of the Lorentz force.

• Vortex movement leads to energy dissipation and therefore resistance at very small current densities! This can be avoided by pinning the vortices.

Chapter 9

Superconductivity

F = jB (9.1)

U = iR =

(9.2)

mevt

= eE (9.3)

jt

=ne2

meE (9.4)

rotE = dBdt

(9.5)

t

me

ne2rotj + B

= 0 (9.6)

E = iR =

Edl = dB

dt;

rotEdA =

BdA; rotE = B (9.7)

BT = Bc

1

T

Tc

2(9.8)

I(t) = I0et/ (9.9)

Eg = 3.53kBTc (9.10)

Vc =Eg

2e(9.11)


> /

2 (9.13)

Bc2 =

2

Bc (9.14)

B = µ0nI (9.15)

27

force on vortex

j

Characteristic length scalestype I or type II superconductor

The penetration depth λ The coherence length ξsuperconductor

B0

λL

High-temperature superconductors

• Discovery of ceramic-like high-temperature superconductors

1900 1920 1940 1960 1980 2000020406080

100120140

Year

Hg Pb NbNbNNb3Sn

Nb3Ge

La1.85Sr0.15CuO4

YBa2Cu3O7-x Bi2Sr2Ca2Cu3O10+x

HgBa2Ca2Cu3O8+x

high

est d

iscov

erd

T C(K

)

Cs2RbC60

MgB2

Structure

• Superconductivity in the Cu+O planes. Very anisotropic

Some properties

• Extreme type II behaviour (this is why the train worked).

• Low critical current densities (because of the anisotropy)

• Ceramic-like: difficult to turn into wires

What makes the high-temperature superconductors superconduct?

• One is reasonably sure that Cooper pairs are also responsible here.

• But there is a big dispute on how they are formed: interaction of electrons with phonons, magnetic excitations (magnons), collective excitations of the electron gas (plasmons) or something else?

High-temperature superconductors

BCS XYZ

1900 1920 1940 1960 1980 2000020406080

100120140

Year

Hg Pb NbNbNNb3Sn

Nb3Ge

La1.85Sr0.15CuO4

YBa2Cu3O7-x Bi2Sr2Ca2Cu3O10+x

HgBa2Ca2Cu3O8+x

high

est d

iscov

erd

T C(K

)

Cs2RbC60

MgB2

Conclusions

Conclusions

• Very broad field. An exception to almost every rule we have discussed.

Conclusions


• Ferromagnetism and superconductivity apparently mutually exclusive.

Conclusions



• But we have seem some similarities in the phase transition.

Conclusions




• Both are ways to lower the energy of the electrons at the Fermi level. The relative energy gain may seem small but this is only because the Fermi energy is so high.

Conclusions




• Both are ways to lower the energy of the electrons at the Fermi level. The relative energy gain may seem small but this is only because the Fermi energy is so high.

• There are also other mechanisms for this and interesting situations arise when several compete.

Documents

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