Correlated density and

Bernoulli potentials in superconductivity

Klaus Morawetz (Chemnitz University of Technology)

Pavel Lipavsky, Jan Kolacek (Institute of Physics, Academy of Sciences, Prague)

Ernst Helmut Brandt (Max Planck Institute for Metals Research, Stuttgart)

1. Nonlocal kinetic theory, correlated density

2. Bernoulli potential at superconductor surfaces - history

3. Measurements by Brown and Morris, charged vortices in HTSC probed by NMR

4. Bulk, surface charge, and surface dipole within Ginzburg-Landau theory - experimental suggestion

5. Conclusion, change of critical temperature due to bias, interaction with lattice deformation, vortex mass

MPI for the Physicsof Complex Systems

LPC/ISMRA Michigan StateUniversity

RostockUniversity

Ludwig Boltzmann

Born: 20 Feb 1844 Vienna, Austria

Died: 5 Oct 1906 Duino, Austria (Italy) collision in collision out

P

k

p

k−q

p+q

Sitzungsberichte der Mathematisch-Naturwissenschaftlichen Classe der

Kaiserlichen Akademie der Wissenschaften Abteilung IIa, Mathematik,

Astronomie, Physik, Meteorologie und Technik, 10.10.1872:∂fk∂t

=∑

pq

P (fk−qfp+q − fkfp)

instantaneous in time and local in space

Non-local corrections necessary since virial corrections are missing (Enskog, Bogoliubov, Green, Ernst, Thirring..)

Nonlocal kinetic equation

∂f1

∂t+∂ε1

∂k

∂f1

∂r− ∂ε1

∂r

∂f1

∂k=∑

pq

Pδ(

ε1+ε−2 −ε−3 −ε−4 −2∆E

)

[

(1−f1)(1−f−2 )f−3 f−4 − f1f

−2 (1−f−3 )(1−f−4 )

]

3

∆k-q

p+q

p

k

out

∆

∆

2

4

where

f1 ≡ f (k, r, t)

f−2 ≡ f (p, r−∆2, t)

f−3 ≡ f (k−q−∆K, r−∆3, t−∆t)

f−4 ≡ f (p+q−∆K, r−∆4, t−∆t)

with T-matrix

eT = + T iφ= | T |

and shifts

∆t =∂φ

∂Ω

∣

∣

∣

∣

ε1+ε2

∆2 =

(

∂φ

∂p− ∂φ

∂q− ∂φ

∂k

)

ε1+ε2

∆E = −1

2

∂φ

∂t

∣

∣

∣

∣

ε1+ε2

∆3 = −∂φ∂k

∣

∣

∣

∣

ε1+ε2

∆K =1

2

∂φ

∂r

∣

∣

∣

∣

ε1+ε2

∆4 = −(

∂φ

∂k+∂φ

∂q

)

ε1+ε2

P. Lipavsky, K. M., and V. Spicka: Kinetic equation for strongly interacting dense Fermi systems

Annales de physique, 26,1 (2001) ISBN 2-86883-541-4

Summary of Correlated Observables

Quasiparticle parts (Landau theory – like)

nqp =∑

k

f Qqp =∑

k

k f jqp =∑

k

∂ε∂kf

Eqp =∑

k

( k2

2m + 12Σmf)fk J qp

ij =∑

k

(

kj∂ε∂ki

+ δijε)

f − δijEqp

Two-particle correlated parts

nmol =∫

dP ∆t jmol =∫

dP ∆3

Qmol =∫

dP k+p2 ∆t Emol =

∫

dP εk+εp2 ∆t

J molij =

1

2

∫

dP

kj∆3i + pj(∆4i −∆2i) + qj(∆4i −∆3i)

Conservation laws

∂(nqp + nmol)

∂t+∂(jqp + jmol)

∂r= 0

∂(Qqpj +Qmol

j )

∂t+∑

i

∂(J qpij + J mol

ij )

∂ri= 0

∂E∂t

=∂(Eqp + Emol)

∂t= 0

Internal energy- and momentum–gain Igain =∫

dP∆E Fgain =∫

dP∆K

Two concepts of quasiparticles

Landau theory

number of particles = number of quasiparticles

n =

∫

dk

(2π)3fk

Quasiparticle energy

εk =δEδfk

Beth-Uhlenbeck equation of state ’37

number of particles = number of free + bound particles

n = nf + 2n2fB(n, T )

correspond to spectral concept

Quasiparticle energy as pole ε = k2

2m + Σ(k, ε)

A =Γ

(ω − k2

2m − Σ)2 + 14Γ2≈(

1 +∂Σ

∂ω

)

2πδ(ω − ε)+ ℘′

ω − εΓ

In extended quasiparticle picture n =∫

dω2π AfFD =

∫

dk(2π)3fk +

∫

dP∆t

total density = quasiparticle density + correlated density

• Coincides with balance from nonlocal kinetic equation → consistency

• Explicit calculation of Wigner function not necessary, correlated observables directly from quasiparticle

kinetic equation

Relation to Landau theory

Landau theory works only if collisions I treated instant and local: from kinetic equation ∂fk∂t = Ik follows:

number of particles

dn

dt=

∫

dk

(2π)3

∂fk∂t

=

∫

dk

(2π)3 Ik ≡ 0

energy balance

dEdt

=

∫

dk

(2π)3

δEδfk

∂fk∂t

=

∫

dk

(2π)3 εkIk ≡ 0

Nonlocal kinetic theory

dn

dt=

d

dt

∫

dk

(2π)3fk +d

dt

∫

dP∆tdEdt

=

∫

dk

(2π)3εk∂fk∂t

+d

dt

∫

dP εk+εp2

∆t−∫

dP∆E

Instant approximation, last term can be rewritten ∆E = −12∂φ∂t

−∫

dP∆E =1

2

∫

dP∂φ∂t

=

∫

dk

(2π)3

∫

dP δφδfk

∂fk∂t≡∫

dk

(2π)3ε∆∂fk

∂t

with rearrangement energy follows variational expression of Landau theory εk = εk + ε∆k

Landau theory mimes for energy gain, but no correlated density !

Consequences of correlated density

Luttinger theorem: Fermi liquids in ground state should have no correlated matter, but

• Many systems turns not to ideal Fermi liquid at low temperatures

• Electrons in metals and nucleons develop coherent state – superconductivity

• There is a small fraction of correlated density in superconducting state

• Correlated density shifts Fermi momentum, i.e., shifts chemical potential µ

• Since electrochemical potential µ + eϕ=const, follows electrostatic potential

Calculate normal density from density of states h nn = 2∑

p

Θ(µ− εp) ≈ n0 − (eϕ +m

2v2)

h(µ)

2π

From the two-pole structure of the spectral function

one finds the correlated density [ωD Debye frequency]ncorr ≈

∂h

∂µ

∆2

4πln

(

2ωD√e∆

)

System stays neutral, n = n0, therefore the two contri-

butions cancel

→ follows electrostatic potential of Bernoulli typeeϕ = −1

2mv2 +

∂ lnh

∂µ

∆2

2ln

(

2ωD√e∆

)

• Shift of chemical potential causes internal electric fields

• Bernoulli potential currently best argument for our nonlocal kinetic theory

History of Bernoulli potential in sc - Theory

Equation of motion for condensate, London condition mv = −eA

mv = −e∂A

∂t− e(v∇)A = e(E + v ×B) +∇

(

eϕ +1

2mv2

)

Compare with Newton equation of motion mv = e(E + v ×B) + Fs

∇eϕ = Fs −∇1

2mv2

Hydrodynamics of charged ideal gas

F. Bopp, Z. f. Phys. 107 (1937) 623, F. London, Superfluids (1950)

V. S. Sorokin, JETP 19 (1949) 553 free energy responsible for sc contributeseϕ = −1

2mv2

Quasiparticle Screening:

Force resulting from interaction between electrons and condensate acting

on electrons to keep them at rest Fn + eE = 0

A.G. van Vijfeijken and F. S. Staas, Phys. Lett. 12 (1964) 175

Interaction between superfluid and normal electrons (fountain effect)

reduces Bernoulli potential

Fn = e∇ϕnnFn + nsFs = 0

Fs = −nnnse∇ϕ

eϕ = −nsn

1

2mv2

Thermodynamic correction:

Condensate kinetic energy fkin = ns12mv

2 determines chemical potential

G. Ryckaizen, J. Phys. C 2 (1969) 1334

eϕ = −µ = − ∂

∂nfkin

= −nsn

1

2mv2 + 4

nnn

∂ lnTc∂ lnn

1

2mv2

Thermodynamic corrections strong close to Tc: Idea to measure material parameters

History of Bernoulli potential in sc - Experiment

Ohmic contacts: Null results due to constant electrochemical potential

H. W. Levis, Phys. Rev 92 (1953) 1149, T. K. Hunt, Phys. Lett. 22 (1966) 42

Capacitive coupling: No thermodynamic corrections observed !

J. Bok, J. Klein, PRL 20 (1968) 660; T. D. Morris, J. B. Brown, Physica 55 (1971)

760

eϕ = −nsn

12mv

2 = −1nB2

2µ0

It should be δf = 12nsmv

2 and (BCS for Pb at T = 7K)

eϕ = − ∂

∂nδf = −

(

nsn−4

nnn

∂ lnTc∂ lnn

)

m

2v2 = − (0.1 + 3.2)

m

2v2

Why no signal of thermodynamic corrections?

Budd-Vannimenus theorem for superconductors

Modification of Budd-Vannimenus theorem (PRL 31 (1973) 1218)

e (ϕsurf − ϕ) = = n∂

∂n

(

fel

n

)

Potential step at surface due to surface dipole in terms of free energy with no regards of potential inside

Answer (after 30 years) due to surface dipoles: Budd-Vannimenus theorem

with eϕ = − ∂∂nfel and fel = ns

12mv

2

eϕsurf = eϕ + n∂

∂n

(

fel

n

)

= eϕ +∂

∂nfel −

fel

n= −ns

n

1

2mv2

Surface dipole compensates thermodynamic corrections exactly for homogeneous sc

P. Lipavsky and J. Kolacek and J.J. Mares, K. Morawetz, PRB 65 (2002) 2507

Hope: inhomogeneous superconductors, vortices

Extended Ginzburg-Landau approach

Free energy f [ψ,A, nn] = fs + fkin + fCoul + fmag

Condensation energy Gorter and Casimir

(Phys. Z. 35 (1934) 963)fs = U − εcon$ −

1

2γT 2√

1−$Equilibrium ∂fs/∂$ = 0, at critical Tc is $ = 0

εcon =γT 2

4√

1−$← εcon =

1

4γT 2

c and order parameter $ = 1− T 4

T 4c

≈ nsns + nn

Kinetic energy

Ginzburg and Landau proposed wave function

JETP 20 (1950) 1064

|ψ|2 =ns2→ $ =

2|ψ|2

n=

2|ψ|2

2|ψ|2 + nn

fkin =1

2m∗|(−i~∇− e∗A)ψ|2

1. Variation with respect to Ψ: GL-equation

Effective potential (Bardeen Phys. Rev. 94 (1954) 554)

extends GL towards lower temperatures

Close to Tc: χ→ α + β|ψ|2

1

2m∗(−i~∇− e∗A)2ψ + χψ = 0

χ = −2εcon

n+γT 2

2n

1√

1− 2|ψ|2n

Abrikosov vortex lattices at temperatures below Tc

Numerical solution of Ginzburg-Landau equation

extended to low temperatures (Bardeen)(−i~∇− e∗A)2

2m∗ψ − γT 2

c

2n

1− t2√

1− 2n|ψ|2

ψ = 0

−0.50

0.51

1.5

−0.5

0

0.5

1

0

0.2

0.4

0.6

0.8

xy

ω(x

,y)

t =0.5, b=0.5, κ=1.5

Condensate density

−0.50

0.51

1.5

−0.5

0

0.5

1

0

0.1

0.2

0.3

0.4

0.5

0.6

xy

B(x

,y)

t =0.5, b=0.5, κ=1.5

Magnetic field

−0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n s / n

= ω

⋅ ( 1

− t4 )

x, y

t=0.1, 0.2, ... 0.9, 0.99

b=0.5

κ0=1.5

0.1

0.99n

s(x,0)/n

ns(0,y)/n

Profiles of the condensate density

n(x, y) at various temperatures

reduced temperature T/Tc = 0.5, mean magnetic field B/Bc2 = 0.5, GL parameter κ0 = 1.5

• ns smaller at boarders than nonmagnetic value → nonlocal effects

• B higher than applied field in core → sc compresses magnetic field in vortices

P. Lipavsky, J. Kolacek, K. Morawetz, E. H. Brandt, PRB 65 (2001) 144511

Surface potential within the Ginzburg-Landau theory

Bardeen’s low temperature extension of GL

(free energy by Gorter Casimir two-fluid, subtraction of free energy of normal state)

fel =1

2γT 2 +

1

2m∗ψ (−i~∇− e∗A)2 ψ−εcon

2|ψ|2

n− 1

2γT 2

√

1− 2|ψ|2n

• Near Tc χ ≈ α + β|ψ|2 follows surface potential (Budd-Vannimenus)

eφ0 = 12nβ|ψ|

4

• Without surface dipole, surface potential equals to internal potential

eφ = − 12m∗nψ (−i~∇− e∗A)2 ψ + ∂εcon

∂n2|ψ|2n −

T 2

2∂γ∂n

(

|ψ|2n + |ψ|4

2n2

)

• Inertial and Lorentz forces: neglecting pairing forces

• Khomskii and Kusmartsev approximation adopted by Blatter:

eφBl = γTcn∂Tc∂n |ψ|

2

Effect of surface dipole

0 0.2 0.4 0.6 0.8 1x, y

−2

−1.5

−1

−0.5

0

0.5−φ

[µV

]

inertial + Lorentz

Blatter et al.

present theory

no surface dipole

0 0.2 0.4 0.6 0.8 1x, y

−50

−40

−30

−20

−10

0

10

20

−φ [n

V]

inertial + Lorentz

present theory

Niobium

TTc

= 0.95, κ = 1.5,

BBc2

= 0.7818

0 0.2 0.4 0.6 0.8 1x, y

−2

−1.5

−1

−0.5

0

0.5

−φ [µ

V]

inertial + Lorentz

Blatter et al.

present theory

no surface dipole

0 0.2 0.4 0.6 0.8 1x, y

−5

0

5

10

15

20

−φ [n

V]

inertial + Lorentz

present theory

TTc

= 0.95, κ = 0.78,

BBc2

= 0.7818

• internal potential and Blatter’s result are similar (Clem model, neglect of ∂γ∂n)

• full theory and inertial/Lorentz forces are much smaller

• surface dipole cancels major part of pairing forces

• full theory and inertial/Lorentz forces result in different profiles and sign

1. Summary: Electrostatic potential

• Electrostatic potential above surface of thin superconducting layer with Abrikosov vortex lattice calculated

• Surface dipole strongly modifies magnitude of potential, in particular when GL wave function has a small

magnitude due to φ0 ∝ |ψ|4, while without dipole φBl ∝ |ψ|2

• Possible cases for which presented theory can be tested:

– at vortex core |ψ|2 ∝ r2 so that φ0 ∝ r4 while φBl ∝ r2

–

B close to Bc2 for thin layer

〈ω〉= (1−b)βA

, 〈ω2〉= (1−b)2

βA

with ω = |ψ|2|ψ∞|2

, b = BBc2

mean value becomes

〈eφ0〉 =εcon

nβA

(

1− t2)2

(1− b)2

〈eφBl〉 =εcon

nβA

∂ lnTc∂ lnn

2 (1− t4) (1− b)

Charged vortices in HTSC probed by NMR

K. I. Kumagai, K. Nozaki and Y. Matsuda, PRB 63 (2001) 144502

• NMR frequency depends on B, γCu and number N of holes per Cu/plane

• Polarization of Cu, coupling of spin with electrical field gradient leads to splitting of quadrupole resonance

νNQRQ = E±3/2 − E±1/2 = AN + C.

NQR NMR NMR

lower satellite upper satellite

ν1(B) = E12− E−1

2= γCuB

ν2(B) = E−12− E−3

2= γCuB − νNQRQ

ν3(B) = E32− E1

2= γCuB + νNQRQ

Problems with NMR-lines in YBCO

• Charge accumulated in vortex core per layer (BCS) Q ∼ d lnTcd lnµ ∼ 10−5e− 10−6e exp : 10−2e

• Underdoped regime: Q > 0, overdoped: Q < 0, contrast to experiment

Structure YBa2Cu3O6+x

Ba

Y

CuO plain2

CuO

chain

Cu

apical oxygen

??

Suggestion

• all lines similar width Γ ≈ 200kHz δν ≈ 20kHz

Fi(ν) =1

πΩ

∫

drΓ

(ν − νi(B(r), N(r)))2 + Γ2

• Integral over volume Ω includes vortex cores

charge transfer between planes and chains

• Local shifts of NMR frequencies reflect triangular

structure of Abrikosov lattice

• Space variation of shifts comparable to line width

Space variation of NMR lines

averaging of the NMR line over Abrikosov lattice

F2/3(ν) =1

πΩ

∫

drΓ

(ν − ν2/3(r))2 + Γ2ν2/3(r) = γB(r)∓ C ∓ AN(r)

density N(r) = ΩCue (ρ(r)− ρ∞), given by electrostatic potential via layered structure screening:

Lawrence/Doniach model : ρ(k) =2kε(1 + e−kD)

(1− e−2kDc−p)(1 + e−kDp−p)φ(k) compare 3D : ρ(k) = −εk2φ(k)

B(r) and Ψ(r) from extended GL-theory φ(r) = |Ψ(r)|2n2 γelT

2c∂ lnTc∂ lnn

-1

-0.5

0

0.5

1

x-0.5

0

0.5

y

-250

25

50

75

Ν2@kHzD-1

-0.5

0

0.5

1

x

• Space variation of shifts comparable to line width

→ no approximation by mean value

• Dominant role of magnetic field

Comparison with experiment

0 0.5 1 1.5T/Tc

−55

−30

−5

20

<∆ν>

[kH

z]

Shift 〈∆ν〉 of Γ = 140kHz of a single crystal,

and after averaging over grain orientation

2. Summary

Hall voltage measurements cannot be used - internal probes like NMR or capacitive pick-up are necessary

1. Bulk charge: Transfer of electrons from inner to outer regions of vortices creating Coulomb force to

balance: (P. Lipavsky, K. M., J. Kolacek, J. J. Mares, E. H. Brandt and M. Schreiber, PRB 69 (2004) 024524 )

• Electrons rotate around vortex center, inertial (centrifugal)

• Magnetic field pushes electrons via the Lorentz force outward

• Paired electrons lower free energy, unpaired electrons attracted towards condensate around core

2. Surface charge: Bernoulli potential by charge build up in surface region max[ξ/√

2, λ/2]

(P. Lipavsky, K. M., J. Kolacek, J. J. Mares, E. H. Brandt and M. Schreiber, PRB 71 (2005) 024526-1-7 )

• Surface charge extends over a range L = min[ξ/√

2, λ/2]

• Contrast to former theories: surface charge is not localized on Thomas-Fermi screening length λTF

3. Bulk and surface charge can be measured: first observation of charge transfer with dominant contribution

of pairing forces K. Kumagai, K. Nozaki, and Y. Matsuda PRB 63 (2001) 144502

• Reproduced by assuming charge transfer between planes and chains

P. Lipavsky, J. Kolacek, K. Morawetz, E. H. Brandt, PRB 66 (2002) 134525

4. surface dipole: all contributions of pairing forces are canceled by surface dipole (Morris/Brown),

P. Lipavsky, K. M., J. Kolacek, J. J. Mares, E. H. Brandt and M. Schreiber, PRB 70 (2004) 104518

• resulting observable surface potential eφ0 = −fel/n (Budd-Vannimenus theorem generalized)

Conclusion1. Bardeen’s extension of GL theory provides simple description of electrostatic potentials

(confirmed by BCS and DeGennes)

2. Bulk, surface charge measured by NMR, reproduced by extended GL theory

3. Surface dipole cancels pairing contribution to large extend

4. Suggestion to measure electrostatic potential above vortices to access thermodynamic corrections

5. External electric field creates surface charges and critical temperature can be changed

L2

ξ2(T ∗)= g

(

EL

U

)

with√g(x) tan

√g(x) = x

P. Lipavsky, K. Morawetz, J. Kolacek, T. J. Yang, PRB 73 (2006) 052505

6.

Change of critical temperature due to presence of cavity

• oscillations of critical temperature Tc due to the cavity

• second branch of Cooper-pairing for high densities

• Tc enhanced for increasing opacity Ω

K. Morawetz, M. Schreiber, B. Schmidt, P. Lipavsky, PRB 72

(2005) 174504

7. Forces deforming lattice expressed as electrostatic forces, gradient corrections, effective mass

P. Lipavsky, K. Morawetz, J. Kolacek, T. J. Yang, PRB in press (2007), cond-mat/0609669

Annales de physique 26,1 (2001)

ISBN 2-86883-541-4

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