Transport and thermodynamic properties of high temperature superconductors (HTS) - an experimentalist's view (contd.)

Powerpoint slides shown on Thursday 23rd April 2015

Notes on whiteboard

Magnetic work and contribution to Free Energy/unit volume F.Sphere demagnetising factorType I s/c long cylinder. First order transition in field.Second order in zero fieldType II second order/continuous phase transition in all fields. (GL mean field case)Entropy conserving property, C/T vs T for s/cFirst Josephson Equation I =IJ sin (f1 – f2)

Second derived in slides

Relaxation method for heat capacity measurements

2eVt

θ

2

ThermodynamicsSeveral different ways of treating magnetic work. Many texts use Gibbs free energy. Follow convention of Waldram (Chapter 3) use Helmholtz free energy.Microscopic local field B(r) = BE(r) + BM(r) (BE from external currents and BM from s/c currents). Energy density:

0

2

M

0

ME

0

2

E

0

2

2μ

B

μ

B.B

2μ

B

2μ

B

(3.3)

With this convention only include last term in internal energy U of system. When change external field by dBE, work done by system is m.dBE (consider back emf from BE/t in current loop representing m) . So work done on system is:

dW = - m.dBE (3.4)

If consider unit volume dU = dQ + dW = TdS - M.dBE (-pdV) (3.5)

Helmholtz free energy is F=U-TS so:

dF = -SdT – M.dBE (-pdV) (3.6)

3

ThermodynamicsSeveral different ways of treating magnetic work. Many texts use Gibbs free energy. Follow convention of Waldram (Chapter 3) use Helmholtz free energy.Microscopic local field B(r) = BE(r) + BM(r) (BE from external currents and BM from s/c currents). Energy density:

0

2

M

0

ME

0

2

E

0

2

2μ

B

μ

B.B

2μ

B

2μ

B

(3.3)

With this convention only include last term in internal energy U of system. When change external field by dBE, work done by system is m.dBE (consider back emf from BE/t in current loop representing m) . So work done on system is:

dW = - m.dBE (3.4)

If consider unit volume dU = dQ + dW = TdS - M.dBE (-pdV) (3.5)

Helmholtz free energy is F=U-TS so:

dF = -SdT – M.dBE (-pdV) (3.6)

4

Standard thermodynamic relations are then:

MB

F

TV,E

(3.8) S

T

F

EBV,

(3.9)

dQdTCV (3.10) dTT

C S

T

0

V

(3.11)

dTC U

T

0

V (3.12)

So by measuring the heat capacity Cv as a function of field and temperature, S, U, F and M can be found.Same thermodynamic convention used in quantum statistical mechanics:

TV,

2(2)

i

(1)

i

(0)

iiB

i

iB

F M :and ....BεBεεε lnZ, Tk - F , )βεexp(Z

Check e.g. for spin 1/2 in a magnetic field

Integrate Eqn. (3.8): EB

0

EE B.dMT)F(0,T),F(B (3.13)

5

So as for any diamagnetic material F rises as increase field. This is why the most paramagnetic (least diamagnetic) axis of a crystal tends to align with BE.

For a long cylinder of Type I s/c (radius >> l), then if no magnetic flux penetrates the cylinder, M = - H = -BE/0, and integrating Eqn. (3.13) gives:

) )0

2

E

SES2μ

BT0,FT,BF (3.14)

Experimentally it is found that at a certain thermodynamic critical field Bc(T) the superconductivity of a Type I material is destroyed abruptly, and that the T-dependence of Bc is approximately parabolic.Type I s/c cylinder first orderphase transition in a field.

)T

T(0)(1B(T)B

2

c

2

cc (3.15)

6

Conclude that at Bc(T), phase transition because F(B,T) of s/c rises with field and at Bc(T) it becomes equal to that of the normal state, that is:

(3.16)

But magnetic response of a normal metal (Pauli paramagnetism) is usually very small

{ (normal metal) ~ 10-5 cf. (superconductor) = –1} so to a good approximation:

) ) )T,BFT,BF 2μ

(T)B T0,F cncs

0

2

c

s

) )T,BFT0,F cnn (3.17)

giving ) )0

2

c

ns2μ

BT0,FT0,F (3.18)

and finally ) )0

2

c

0

2

EnEs

2μ

B

2μ

BT0,FT,BF (3.19)

The above thermodynamic relations between F (or more precisely the heat capacity) and the measured critical field Bc(T) have been verified experimentally for many Type I s/c. For example, as shown below for Gallium metal there is a sharp jump in heat capacity at Tc. (Another important feature for classical s/c is the limiting exponential behaviour at low T showing the presence of an energy gap).

7

Table: Examples of the good agreement between measured values of heat

capacity jumps and those calculated from experimental Bc(T) curves.

Cv(T) for Gallium

Metal Cs-Cn (meas.) Cs –Cn (calc.) (mJ mole-1K-1)In 9.75 9.62Sn 10.6 10.68Ta 41.5 41.6

So can use experimental Bc(T) law to understand physics, e.g. in zero field:

T

B

μ

BSS

T

)F(F c

0

c

ns

sn

(3.20)

8

So at Tc no entropy change, no latent heat. Discontinuity in Cv, second order phase transition. Bc/T < 0, so Ss < Sn, more ordered state as expected.

For Type II s/c, area under the reversible M-B curve is also the free energy difference between normal and s/c states, definition of Bc(T) – thermodynamic critical field. Found experimentally that Bc, Bc1 and Bc2 have approx same T- dependence all given by Eqn. 3.15. Also experimentally M (BE-Bc2) near Bc2 (and theoretically from G-L equations), so second order transition both in a field and in zero field.

Other general pointsCondensation energy Bc(0)2/20 - very small: Al (Bc(0) = .01T, Tc=1.2 K) 0.05 mK/Al atom, YBa2Cu3O7 (Bc(0)= 1T, Tc = 92 K), 5 K /unit cell.Entropy conserving property in plots of Cv/T versus T. (Sn=Ss at Tc and T0)

Cuprates: Cv jump less sharp and anomaly more symmetric (especially 2D ones). Thermodynamic fluctuations when free energy in a coherence volume of order kBT (number of Cooper pairs in a coherence volume smaller than in classical superconductors)

Useful in context of pseudogap in cuprates (later)

9

C/T vs T data for an YBa2Cu3O7 crystal showing the broadening and shift of the main peak by a magnetic field . Small sharp peaks mark first order phase transition at Bm(T). From A. Schilling et al. PRL 78, 4833 (1997). Compare Nb3Sn data below. Lortz et al. Phys. Rev. B 74, 104501 (2006)

Field broadening of in-plane resistivity curves. Again in classical s/c s the r(T) curves remain sharp in a field.

10

Irreversibility line : Important technical limitation ofcuprate superconductors.

Large region below the Bc2(T) line where M(BE) curves are completely reversible and therefore the electrical resistance is not zero, even in the presence of many pinning centres. There are dissipationless currents only at much lower fields and temperatures defined by the irreversibility line Birr(T).

Not completely understood e.g. may be a vortex liquid, 2D vortex "pancakes" or large thermodyamic fluctuations for Bc2(T) > BE > Birr(T). Empirically Birr(T) is much smaller for cuprates with weak interlayer coupling. Reason for predominance of YBCO. Some evidence that conducting charge reservoir layers could increase Birr(T) - possible research area.

d (nm)

J.L. Tallon et al. Phys. Rev. B53, 11972 (1996)

d

Cuprates

Optimally doped

CuO2 bi-layers

11

Qualitative description of superconducting state. (“Superconductivity of Metals and Cuprates” J.R. Waldram, IoP Publishing 1996)Macroscopic quantum state, wave-function:

(r,t) = np ei(r,t) (2.1)np is effective density of Cooper pairs (charge -2e mass 2me).

London equations and magnetic penetration depth l.

12

Pair Stateand are the amplitude and phase of the order parameter in the superconducting

state. See from Ginzburg-Landau theory that np varies as (Tc – T) just below Tc. For all T<Tc, assume is unique and therefore the pairs have no entropy. Imagine adding two electrons to the system to form an extra pair. Using the thermodynamic relation:

dU = TdS + dN (2.2)

with dS = 0, see that energy (U) of a pair is 2. Also follows that entropy at finite T arises solely from the elementary excitations of ground state (quasi-particles - loosely speaking un-paired electrons or broken pairs).

Effective Field and Potential

pn

Two metals not in electrical contact, same vacuum levels.

13

So the usual equation relating the electric field strength E to the magnetic vector potential Aand the electrostatic potential(f) is replaced by:

Vt

Aμ

1

t

AE

e

In contact, V=0, equalise (need very little charge transfer). Applying voltage –V, raises electron energy RHS. A voltmeter connected to two points in metal measures difference in V or /e. So in a metal or a superconductor V is equivalent to the electrostatic potential (f) for a charged particle in a vacuum. Note that for this to apply, must be measured relative to the vaccum, and not the bottom of the conduction band as is often done in Solid State Physics.

(2.3)

Time dependence of phase By analogy with time dependent Schrodinger equation, with e = 2= -2eV , /t of Eq. 2.1 gives:

(2.4)

etΨ/HΨ i

2eVt

θ

14

This is the second Josephson equation. (To check signs of QM operators, recall that plane wave state ei(k.r - t) , has momentum k and energy ).

Variation of phase with positionFrom definition of QM operators for particle mass m charge Q, moving with velocity vs in a magnetic field, B = xA: the canonical momentum operator is :

-i = mvs + QA. Hence for Cooper pairs of mass 2me and charge –2e:

)A2eθ2m

1v

e

s (2.5)

The current density is given by:

A

2θ

m

env2enj

e

p

sps

e (2.6)

This equation can also be derived using the standard QM expression for the currentdensity in Waldram P.17. The London equations and magnetic penetration depth lDifferentiating Eqn. 2.5 w.r.t. time and using Eqn. 2.4 for /t gives:

Em

e

t

AV

m

e

t

ν

ee

s

(2.7)

15

This is simply Newton’s law ( in rewriting eqn. 2.7 we have also made use of Eqn. 2.3)Or equivalently:

)E

t

jΛ s

, where

p

2

e

n2e

m Λ (2.8)

This is the first London equation. It shows that no electric field is needed for a steady supercurrent (js = const corresponds to E = 0). This is in contrast to normal metals where the field only accelerates the electrons for a finite time in between two scattering events thus giving j = E.Taking the curl of Eqn. 2.5 gives the second London equation:

) BjΛ s (2.9)

Combining Eqn. 2.9 with the Maxwell equation, (Ampere’s rule), with B = 0H

and the vector identity, :

sjH

B)B.()B( 2

2

2

λ

BB (2.10)

(recall that div B =0) gives

16

p

2

0

e2

ne2μ

mλ Where (2.11)

Eqn. 2.10 is a screening equation, it shows that B and Js fall away exponentially towards the interior of a superconductor over a characteristic distance l, the London penetration depth. In 1 D:

x/λx/λ

22

2

eBeAB ,λ

B

dx

Bd

for infinite slab, B’ = 0, both j and B decay exponentially. For a sphere, radius a, can show that magnetic moment is :

aλfor , 15λ

am

,aλfor , a

3λ1m

a

λ3

λ

acoth

a

λ31

Ba 2m

2

2

max

max

2

0

E3

(2.12 a,b,c)

Note that Eqn. 2.12(b) corresponds to a “normal” shell of thickness l on the surface of the sphere, and Eqn. 2.12(c) always has the same l-2 dependence.

17

For ordinary metals, high electron density, low effective mass, l is typically of order 30 nm at low T. It increases with T because np falls to zero at Tc.

For the cuprates l is anisotropic. In Eq. 2.5 then have , the inverse effective mass tensor, which also detemines the electrical conductivity (ii, i = x,y,or z) in the normal state. Two principal directions of are in the CuO2 planes and the third is perpendicular to the planes. When js flows in the CuO2 planes the penetration depth is determined by the in-plane effective mass, and is typically 200 nm. When js flows across the planes the penetration depth can be very large if the interplane coupling is small, e.g. lc(T=0) ~ 100 m in Bi:2212 crystals.

-1m

-1m

18

BCS theory: electrons attract via positive ions. QMlly phonon exchange. Lower energy if electrons near FS form Cooper pairs.

0| compare

0|)(

kkkk

NS

kkk

i

k

kBCS

cc

ccveu

F

222

kkkE e )1(2

12

k

kk

Eu

e )1(

2

12

k

kk

Ev

e

'

' '

'

2kk

k k

kk V

E

BCS self-consistent gap equation (T=0):

If V is a constant then gap parameter

constant, s-wave superconductor

kk

i

kkk

kk

i

kkk

cec

cec

vu

vu

New quasi-particles in s/c state

k’

k’ + q

19

Penetration depth in BCS theorySupercurrent arises from pairing of k+s and –k+ s electron states. The “fuzzy” FS is displaced by s in k-space. At T=0 gives j = nevs where n is total electron density. But energy of quasi-particle excitations also shifted by mevs .vF, so get “back-flow” current when T>0. Total supercurrent is sum of these two terms and hence effective pair density np falls from n/2 at T = 0, and l(T) diverges as T Tc.

DBk

E

dE

VN

0

2/122 ))0(()0(

1

For s-wave and N(0)V small (weak coupling limit)

])0(/1exp[2)0( VNk DB

)])0(/1exp[14.1 VNT Dc

2

2

λ

BB

n(T)eμ

mλ(T)

2

0

e2

20

d-wave s-wave

22 and )()( kkkEdndEEN eee

N(E) obtained using the relations:

21

Initial evidence for d-wave pairing - T-dependence of London penetration depth l(T)

.

In BCS theory T-dependence of l comes from “back-flow” of thermally excited, unpaired electrons (more precisely the Bogoliubov quasi-particles with energy E). So for s-wave l(T) - l(0) ~l(0)e-/kT at low T, but for d-wave l(T) - l(0) ~l(0)kBT/ as observed by Hardy et al. for YBCO.

From J.W. Loram, J. Phys. C 19, 6113 (1986), now computerised version

From J.W. Loram, J. Phys. C 19, 6113 (1986), now computerised version

24

Top view of CuO2 plane

Typical Cu-O co-ordination in cuprate superconductors

Crystal structure of highest Tc cuprate

H. Alloul, T. Ohno and P. Mendels, PRL, 63,1700 (1989)

Early NMR evidence for pseudogap -at that time referred to as a “spin gap”

26

Specific heat capacity (J.W. Loram et al (1989- …..)

Total (including phonons)

1 YBa2Cu3O7,

2 YBa2(Cu0.93Zn0.07)3O7

From J.W. Loram et al. PRL 71 1740 (1993)

27

Several hundred samples measured. Typical results for Y0.8Ca0.2Ba2Cu3O6+x (after subtracting phonon part to give = Cel/T) shown below. Gm.atom units used because high T limit of phonon term given by number of atoms.

Over-dopedp > 0.16

Under-doped p < 0.16

J.W. Loram et al., J. Phys. Chem. Solids 59, 2091 (1998)

28

29

J.W. Loram et al. Physica C 235-240, 134 (1994)

[6] is N. Athanassopoulou, Ph.D thesis, University of Cambridge (1994)

30J.W. Loram et al. Physica C 235-240, 134 (1994)

31

J.W. Loram et al. Physica C 235-240, 134 (1994)[11] is J.M. Wade et al., J. Superconductivity, 7, 261 (1994)

32

Analysis, entropy-conserving construction

Use thermodynamic formulae from earlier, e.g

theoryBCS coupling-in weak )0()N(E 0.25 2μ

(0)B U(0)

T)dTS(S)0(F)0(F )0(U)0( UU(0)

T

F S dT,

T

C S

2

F

0

2

c

c

0

snsnsn

BV,

T

0

V

E

33

Tc, pseudogap energy Eg, superfluid density rs(0) and condensation energy U(0) versus p for Bi2Sr2CaCu2O8+x

Tc, pseudogap energy Eg, condensation energy U(0) and U(0)/nTc

2 versus x for Y0.8Ca0.2Ba2Cu3O6+x from J. W. Loram et al. J. Phys. Chem Solids, 62, 59 (2001).

Empirical Fermion density of states

etc. EEN(E)f(E)dTT

γand dEE

fN(E)μ χ not, if

)N(Eμ χ ),N(Ek 3

π γconstant, N(E) If

2

B

F

2

BF

2

B

2

34From J.W. Loram et al. IRC Research Review 1998

35

Magnetic susceptibility of Y1-yCayBa2Cu3O6+x, with various hole concentrations p.

S.H. Naqib, JRC and J.W. Loram, Phys. Rev., B79, 075906 (2009)

36

This figure shows that at a fixed temperature (T0) the entropy of several cuprates increases unusually quickly with p, and that the product of Tbehaves in a similar way to S. Note that for a normal metal we expect

F

0B

E

Tk

p

S

From J. W. Loram et al. J. Phys. Chem Solids, 62, 59 (2001).

37

Superfluid density of Bi-2212

AC susceptibility of same Bi-2212 (a) and Bi(15%Y)-2212 (b) powder samples (randomly oriented grains ~ 1 m diameter) after different annealing treatments, i.e. different hole content (p).

Bi-2212 anisotropy so large that ACS signal determined by l ab(T), London penetration depth when screening currents flow in CuO2 planes. No need to align crystallites magnetically in epoxy resin as in earlier work with C. Panagopoulos et al. Make periodic checks that signal/unit weight same for a given p. Can see main effect in raw data.

P(O2) = 0.1 - 1000

millibars

TA= 350 - 650 C

LN2 77 K

From W. Anukool et al. Phys. Rev. B 80, 024516 (2009)

38

Figure showing that the pseudogap strongly reduces Birr of

the same 4 mg YBCO7- d crystal. (Babic et al Phys. Rev. B60,

698, 1999). Data taken using a VSM. For each panel the

lower (more diamagnetic) curve corresponds to the

magnetic field B being increased while the upper (more

paramagnetic) curve corresponds to B being reduced to

zero. The two curves separate at the irreversibility field Birr.

For B < Birr, flux lines are pinned and there is a finite,

persistent current. For B > Birr there is still a substantial

super-conducting diamagnetic signal but zero critical

current (non-zero resistivity). All curves were taken at the

same reduced temperature, T/Tc(B=0) = 0.8. The data also

show that oxygen vacancies pin the flux lines, since the

hysteresis increases with d.

Complementary experiments: TEP scaling

39Adapted from J.R. Cooper et al. Physica C 341-348, 855 (2000)

40

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5

S/S

*

T/T*

0.63

0.1

0

200

400

0.2 0.4 0.6 0.8 1

EG(K

)

x

-10

0

10

20

30

40

0.3 0.5 0.7 0.9

S(290) V/K

x

J.R. Cooper et al. Physica C 341-348, 855 (2000)

YBa2Cu3O7-d

p = pmin

p = popt

p = pmax

“Systematics in the thermoelectric power of high Tc oxides”, S D Obertelli, J R Cooper and J L Tallon, Phys. Rev., Rapid Comm. B46, 14928 (1992).

p from BVS 123Y and 1212 Tl

Others from:

Examples of usefulness of OCT plot when studying substitution effects

YBa2(Cu1-zMz)3O7-d

Black solid lines OCT formulae for S(290, p) plus

YBCO JRC and J W Loram, J. de Physique, 6, 2237 (1996) plus Pr data, Hui Chang and Literature. BSCCO T M Benseman, J R Cooper, C L Zentile, L Lemberger and G BalakrishnanPhys. Rev. B84, 144503 (2011).