Superconductivity: from BCS to
modern electronic structure theory
Sandro Massidda
Physics Department, University of Cagliari
sandro.massidda@dsf.unica.it
Outline
•Introduction: Basic concepts of the BCS theory:
1.Concepts of Fermi liquid theory and instability of the normal state.
2.Phonon mediated attractive interaction and the Cooper pairs.
3.The BCS gap equation.
•Limitations of BCS theory.
•The introduction of Coulomb repulsion.
•The electron-phonon interaction in real materials.
•Applications to real materials: MgB2, alkali under pressure, CaC6
•Kohn anomaly,
•Fermi surface nesting,
•two-gap superconductivity.
Superconductivity: 100 years
ρ
Kamerling Onnes
Leiden 1911
Zero resistivity below Tc
Expulsion of magnetic field
2
c
H
FN − FS =
V
8π
Application needs: high critical temperature, currents e fields
Tc as a function of time
140
- Critical temperature
-
130
-
150
~ 155K
(K)
pressure
HgBaCaCuO
TlBaCaCuO
Bi2Sr2Ca2Cu3Ox
110 100 YBa2Cu3O7-d
90 Liquid N2
80 70 conventional
60 PrFeAsO0.9F0.1
superconductors
50 pressure
SmFeAsO0.9F0.1
Bednorz and Muller
40 CeFeAsO0.84F0.16
(La,Ba)CuO
NbAlSi
Y.Kamihara et al.
30 NbGe
V Sn
LaFeAsO0.89F0.11
20 NbN
Nb Sn
Pb Nb
LaFePO LaNiPo
Liquid He
Ba(Pb,Bi)O
10 - Hg
NbO
0 120
3
3
3
3
1910
1930
1950
1970
1975
1980
1985
1990
1995
2000
2005
2006
2007
2008
2009
2010
Conventional superconductivity (year < 1986)
Elemental superconductors and alloys with Tc ≤ 25 K
Conventional
superconductors
BCS: theory for conventional SC (1957)
Nobel prize in
1972
J. Bardeen, L. Cooper, R. Schrieffer

+k
Cooper pairs
Cooper pair

−k
Cooper pairs form zero-spin (singlet) bosons who condense and
transport electric current without dissipation
How can electrons possibly attract each other to form a pair?
Normal state: Fermi liquid (Landau)
εk− µ
εk− µ
•Holes and electrons: basis of elementary excitations
•e and h interaction is screened by the e-gas,
•e and h become quasi-particles with a very long life-time
close to EF
The electrons occupying low lying energy states appear
to form a very stable system
Na
Cu
Cooper Problem: The Fermi liquid is unstable towards
any arbitrarily small attractive e-e interaction
How can such a small attraction mess up an
apparently very stable system?
1. Cooper problem
 


2
2
∇1−
∇ 2 + V (r12 )  ϕ (r1 , r2 ) = E ϕ
−
2m
 2m

2
ϕ ( r1, r2 ) =
2
∑
gk e
ik⋅r1
e
− ik⋅r2
χ ( 1, 2 )
k> kF
gk = g− k
Singlet wavefunction:
χ ( 1, 2 ) = α 1 β 2 − α 2 β 1
k ↑ ,− k ↓
The two-e Schrödinger Equation yields to a bound state: E < 2 EF
Approximate interaction
(BCS):
Vkk ' = − V if ε k , ε k ' < ω
= 0 otherwise
cut
The solution of SE gives:
E − 2EF = − 2ω
cut
e
− 1λ
λ = N(EF )V
• A solution exists independently of the coupling strength λ
• It can not be developed in Taylor series (perturbation th.)
• If ωcut is a phonon frequency, the isotope effect is
explained: (ωph~M-1/2)!
2. Source of electron-electron attraction
BCS theory in 1957!
How can phonons produce an attraction
among electrons? (classical view)
Lattice deformation
Overscreening of e-e
repulsion by the lattice
The lattice deformation by
the first electron attracts the
second
Attractive phonon-mediated interaction
k+q
k’-q
phonon
ω
k
k’
qj
Exchange of virtual phonons produces an
attraction for electrons close to EF ( k = k,n )
Vk,k ' = gqj
2
2ω
qj
2
ε
−
ε
−

ω
( k k' )
2
2
qj
Bardeen-Pines,
attraction when
ε k − ε k ' < ω
qj
BCS theory: language
ck↑† .. 0 k.. = ..1k↑ .. creation of an e
ck↑ ..1k↑ .. = .. 0 k.. destruction of an e
ck†σ ckσ .. nk .. = n̂kσ ..nk .. = nkσ .. nk ..
number of electrons
ck↑† c−† k↓ .. 0 k.. ≡ bk† .. 0 k.. = ..1k↑ 1− k↓ .. Cooper pair
BCS Hamiltonian: scattering of pairs only
Ĥ =
∑
kσ
ε k n̂kσ +
Kinetic energy
∑
k,k '
Vkk ' bk†' bk
Scattering of pairs
from k to k’
BCS wavefunction:
Ψ
BCS
=
∏ (u
k
k + v b
†
k k
)0
u k + vk = 1
2
2
•Linear combination of e and h
•No space left for singles (excitations of the system)
• Number of e fluctuates around N0
•All pairs added with the same phase (coherence)
Coherence factors uk,vk:
determine how much of an e or
of a h is travelling as a quasiparticle in the superconductor
uk = uk
vk = vk eiφ
Solution:

u k = 12  1 +



2
2
ξk + ∆ k 

vk 2 = 12  1 −



2
2
ξk + ∆ k 
2
ξk
ξk = εk − µ
ξk
ukvk ≠0 close to EF in the SC
ukvk =0 in the normal state
Conservation of electron number
•The number of particles is not fixed, but the relative
fluctuation is proportional to N0-1/2
•Phase coherence implies release of number conservation
What is Δk? BCS gap equation
∆ k = − ∑ Vkk '
k'
Ek =
ξ k2 + ∆
∆
k'
2 ξ k '2 + ∆
2
k
2
k'
excitation energies
Solution of gap equation, within the BCS approximation
∆ = 2ω D e
− 1λ
λ = N ( EF ) < Vkk ' > EF
Vkk’ pairing potential
results from competition of
phonon attraction and
Coulomb repulsion
Meaning of Δk
Normal
state
Superconducting state: minimum
energy Δk to add a quasi-particle
δk
Ek≈|εk-μ| far from FS
ξ0 ≈ 2π/δk coherence length
Elementary excitations: Bogoliubov-Valatin transformations
γ
†
k↑
= uc
− vk c− k↓
γ
†
− k↓
= uc
+ vk ck↑
†
k k↑
†
k − k↓
vk
Quasi-particles have an energy Ek and correspond to:
•pure h or pure e far from EF
•a combination of them close to EF
Ek =
ξ + ∆
2
k
2
k
uk
Spectral function A(k,ω)
R14 738
π A(k, ω ) =
(ω
Γ uk2
− Ek ) + Γ
2
2
+
Γ vk2
J. C. CAMPUZANO et al.
(ω
+ Ek ) + Γ
2
53
2
Campuzano et al, PRB 53, R14737 (1996)
R14 738
J. C. CAMPUZANO et al.
53
uk2
FIG. 1. Schematic dispersion in the normal thin line and superconducting thick lines states following BCS theory. The thickness of the superconducting state lines indicate the spectral weight
given by the BCS coherence factors ( v 2 below E F and u 2 above .
from E f , although with a decreasing intensity Fig. 1 . This
is the signature of particle-hole mixing.
In the remainder of this paper we will present high reso2
lution ARPES
data on the high temperature superconductor
Bi2212. k
We shall find that there is clear experimental evidence for the anomalous dispersion described above indicaof p-h mixing.
Weinemphasize
FIG. 1.tive
Schematic
dispersion
the normalthat
thinthis
lineis the
and only
su- way
known
to lines
us ofstates
asserting
that BCS
the gap
seenThe
bythickARPES is
perconducting
thick
following
theory.
due
to
superconductivity
rather
than
of
some
other
ness of the superconducting state lines indicate the spectral weightorigin,
2 Finally, we
e.g.,BCS
chargeor spin-density
wave Eformation.
given by the
coherence
factors ( v 2 below
.
F and u above
v
FIG. 2. Superconducting state a and normal state b EDC’s
for the same Bi2212 sample for the set of k values 1/a units which
are shown at the top. Note the different energy ranges.
T≠0 properties: the gap
∆ k = − ∑ Vkk '
k'
∆
k'
2 ξ k' + ∆
2
2
k'
2∆
= 3.52
k BTc
Tc = 1.14 Θ e
− 1λ
ξ k '2 + ∆
tanh
2k BT
2
k'
1− f (Ek ) − f (Ek ' )
Tc as the max T
allowing a non-trivial
solution
The nuclear-mass dependence of
Debye temperature Θ implies an
isotopic effect α ≈ 0.5
Tc ∝ M
−α
Thermodinamic properties
s
Entropy
Specific heat
jump at Tc
Ces − Cen
= 1.43
Cen
Free energy
(per unit volume)
H c2
Fs − Fn = −
8π
Existence of an energy gap : photoemission, tunneling
MgB2
Problems of BCS theory
•
•
1.
2.
•
It is a weak coupling theory (λ<<1)
It neglects:
electron-electron Coulomb interaction
e-ph retardation effects
Complete lack of predictive power
Tc = 1.14 Θ D e
− 1
λ
Example: Nb (λ=1.18)
Tc exp 9.5 K, BCS 134 K
Despite these problems, B, C & S largely deserved
their Nobel prize: the qualitative behavior of most
physical quantities is correctly described!
Repulsive electron-electron interaction
V (r, r') =
∫
1
3
ε (r, r'', ω )
d r''
r''− r'
−1
where ε-1 is the frequency dependent dielectric function
ε
−1
G,G'
( q, ω ) = δ G,G' +
vq+ G χ G,G' ( q, ω
)
• Bare Coulomb potential: instantaneous
• Screened Coulomb potential: plasma frequency, few eV
• Phonon-mediated attraction: phonon frequencies ~100 meV
Different time scales allow superconductivity to occur
Interaction potential V=V(ph)+V(el) includes:
•Phonon-mediated attraction
•Direct electron-electron repulsion
The total
interaction is
attractive only in
a narrow region
around EF
To get superconductivity, David needs to defeat Goliath
Retardation effects: Nb
Log scale
Linear scale
ξ=ε−EF
Phononic range
∆ nk = − ∑ Vnk ,n 'k'
n ',k'
∆
n ',k'
2 En ',k'
 En ' k ' 
tanh 

 2 k BT 
Constructive interference
from a repulsive
interaction with states far
from EF
In ELIASHBERG theory:
• repulsive Coulomb interaction (Morel Anderson):
µ = Vel − el
∗
FS
µ =
µ
EF
1 + µ ln
ω D
The difference between electron (h/EF) and nuclear (2π/ωph) time
scales reduces the coulomb repulsion (retardation)
Superconductivity results from the competition
of opposite effects: λ-μ*
Condensed Eliashberg:
McMillan Equation
Tc =
ω
1.2
e
− 1.04
1+ λ
λ − µ * (1+ 0.62 λ )
μ* is normally fitted to experimental Tc
e-ph coupling: ELIASHBERG spectral function
F (ω ) =
∑
(
δ ω −ω
q, j
qj
1
α F (ω ) =
N EF
)
∑
( )
2
Phonon density of states
(
δ ω −ω
q, j
qj
)∑
g
nk ,n 'k '
qj
nk ,n 'k '
2
( ) (
δ k + q,k 'δ ε nk δ ε n 'k '
)
Phonon density of states weighted by the coupling with electrons at EF
g
qj
nk ,n 'k '
= ψ
εˆ ⋅
qj
n'k '
δ Veff
δ u(qj)
k’
q= k’-k
k
ψ
nk
Material specific e-ph coupling
α F (ω )
λ = 2∫
dω
ω
2
Kohn anomaly
The electronic screening is discontinuous at 2kF
(log singularity in the derivative of the response )
k’
q
For q>2kF it is not possible
to create excitations at zero
energy
( )
dχ q
k
dq
2
mk
χ ( q ) = − e 2 2 F2
π
 1 1 − x2 1 + x 
 2 + 4 x ln 1 − x 


→ −∞
q→ 2k F
x = q / 2k F
phonons
Baroni et al. Rev. Mod. Phys. 73, 515 (2001)
C
αα '
ss',elettr
∂ V ( r)
(
)
( q) = ∫ ∫ dr dr ' ∂ u ( q) χ (r,r ') ∂ u ( q) +
∂ V ( r)
∫ dr n ( r ) ∂ u ( q) ∂ u ( q)
1
Nc
∂ Vion r '
ion
α '
s'
α
s
2
1
Nc
ion
α
s
α '
s'
The response of the electrons to ion
displacement is a fundamental ingredient
Electron-phonon
spectral function
α2F(ω)
MgB2 is a superconductor, AlB2 is not
Phonon density of states
Spectral function α2F(ω)
α 2 F (ω )
λ = 2∫
dω
ω
Example: MgB2
Superconductor, Tc=39.5 K
Energy bands of MgB2
3D π bands (strongly dispersed along Γ-A (k z))
2D σ bands (weakly dispersed along Γ-A)
σ bonding
(px,py)
π bonding &
antibonding
(pz orbitals)
s
k=(kx;ky;0)
(0,0,kz )
(kx;ky;π/c)
sp2
Electronic
σ
p r o p e r t i e s o f MgB2
π
B
B
B
2-D σ-bonding bands
3-D π bands
Fermi surface of MgB2
Blue and green warped
cylinders derive from px and
py.orbitals (σ bonds)
Blue and red “nets” come
from pz orbitals (π bonding
and antibonding respectively)
B1g
Phonons in MgB2
E2g
Anomalously low frequency E2g branch
(B-B bond stretching) – Kohn anomaly
Kohn anomaly: LiBC, isoelettronic to MgB2 (Pickett)
phonon frequency
Stoichiometric compound
is a semiconductor
Strong renormalization
of phonon frequencies
Metallic upon doping
Kohn anomaly
High Tc predicted
Unfortunately not found
experimentally
Large coupling of the E2g phonon mode
with σ hole pockets (band splitting)
ωE2g=0.075 eV
Δε~1-2 eV !!!
Pb and MgB2 Eliashberg functions
Tc =
ω
1.2
e
− 1.04
1+ λ
λ − µ * (1+ 0.62 λ )
MgB2
Pb
λ=1.62
Tc=7.2 K
λ=0.87
Tc=39.5 K
Two band model for the electron phonon coupling (EPC)
•λ stronger in σ bands due to the
coupling with E2g phonon mode
• Experiments show the existence
of two gaps: Δσ and Δπ.
σ
π
Fermi surface
Two band model:
experimental
evidence
R. S. Gonnelli, PRL
89, 247004 (2002)
Specific heat evidence of 2 gaps
Two-gap structure associated with σ and π bands
Tunnelling
experiments
Two band superconductivity
Tc depends on the largest
eigenvalue of the inter- and intraband coupling constants, λnm in
place of the average λ
Newest developments:
EXTENSION OF DFT TO THE SUPERCONDUCTING
STATE (SCDFT)
Order parameter
χ ( r, r') = ψ
(↑ r) ψ ↓ ( r')
In the SC state
≠ 0
SCDFT Oliveira Gross Kohn, PRL 60, 2430 (1988)
S. Kurth, M. Lüders, M. Marques, PhD thesis
Normal
The anomalous
density χ represents
the order parameter
ρ ( r) =
∑
ψ
†
σ
Anomalous χ ( r, r ') = ψ
↑
( r) ψ ( r)
( r ) ψ ↓ ( r ')
(
) (
)
Nuclear Γ (R) = φˆ + ( R1 ) ⋅ ⋅ ⋅ φˆ + RN n φˆ RN n ⋅ ⋅ ⋅ φˆ ( R1 )
Hohenberg-Kohn Theorem:
ρ , χ , Γ ↔ V , ∆ ext ,V
e
ext
Ω = F [ ρ , χ ,Γ ] +
*
n


+ ∫ dr ρ (V − µ ) − ∫∫ drdr '  χ ∆ ext + h.c. + ∫ dR Γ Vext
e
ext
PRB B 72, 24545 (2005), ibid 72, 24546 (2005)
n
ext
Universal exchange-correlation functional Fxc
δ Fxc
Vxc (r ) =
δ ρ (r )
δ Fxc
∆ xc (r, r') = −
δ χ * (r, r')
δ Fxc
Vcn (R ) =
δ Γ (R )
Variational eqs.: Bogoliubov-De Gennes-type + nuclear eq
 ∇2

+ Vs (r) − µ  uk n (r)
−
 2

∫
d 3r ' ∆ ∗s (r,r')uk n (r')
+
∫
d 3r ' ∆ s (r,r') vk n (r') = Ek nuk n (r')
 ∇2

− −
+ Vs (r) − µ  vk n (r) = Ek n vk n (r')
 2

SCDFT
physical ρ,χ,Γ minimize the grand-canonical potential
•BCS-like gap equation, with an interaction potential coming
from first-principles: no μ*
∆
∆
n,k
 β En' k ' 
= − ∑ wnk ,n'k'
tanh 

2E
2


n',k'
n',k'
n',k'
We solve self-consistently for Δnk
Static-looking potential, but dynamical effects are
build-in in the spirit of DFT
Superconductivity under pressure
29 elements superconducts under normal conditions
23 only under pressure: Lithium is the last discovered
Tc(P) is a strongly material-dependent function*
* C. Buzea and K. Robbie
Supercond. Sci. Technol. 18 (2005) R1–R8
Aluminium under pressure……
270 GPa
Bonds get stiffer, frequencies higer
…Al becomes a normal metal
N (EF ) I 2
λ =
M ω 2ph
Alkali metal under high pressure
Lithium is a superconductor under pressure
CI16
42
hR1
…
39
…
…
fcc
7
0
9R
Electron states of Li and K under pressure
Charge on
d states
K
27 GPa
Li
30 GPa
Charge on
p states
Spectral Function α F(ω)
2
1
α F (ω ) =
N EF
∑
∑
( )
2
g
q, j nk ,n 'k '
qj
nk ,n 'k '
2
( ) (
) (
δ k + q,k 'δ ε nk δ ε n 'k ' δ ω − ω
qj
)
k’
q
k
k
k’
λ comes from a complex integration over
the Fermi surface
q
parallel pieces of the Fermi surface
enhance the coupling (NESTING)
Phonon dispersion in Li: softening
26 GPa
0 GPa
Why?
Increasing the pressure a lattice instability drive by the
Fermi surface nesting increases the electron-phonon coupling
Pieces of Fermi surface connected
by the same wave-vector q
Phonon softening and
lattice instability
q
λq
Imaginary frequency: instablility
Orbital character at EF and superconductivity
d character
Δ
Κ
Li
p character
Δ
Electron-Phonon Coupling
Pressure
λ
Stiffer bonds (higher ω’s) but higher coupling at low ω
Theoretical predictions
Intercalated graphite: CaC6 Tc=11.5 K
N. Emery et al. Phys. Rev Lett.
95, 087003 (2005)
Fermi surface of CaC6
CaC6
Amount of Ca contribution
FS
Bands similar to graphite
Phonons in CaC6
Electron-phonon coupling
Ca
C-z
C-xy
CaC6
Calculated Tc=9.5 K Experiment Tc =11.5 K
A. Sanna et al. PRB 75, 20511(R) (2007)
Spherical
(intercalant) FS
CaC6 gap anisotropy
Tubular FS
π
Large anisotropy but with a
continuous gap distribution
Anisotropy confirmed by
experiments (tunneling)
CaC6 comparison with experiments
STM experiments Begeal et al
PRL 97, 77003 (2006) on
samples with a slightly reduced Tc
(accidentally very similar to ours)
Specific heat
Comparison with STM
Specific heat by Kim et al.
PRL 96, 217002 (2006)
The agreement improves by
considering gap anisotropy
CaC6 comparison with experiments at 0.4 K; to be
published on PRL
non conventional superconductivity : cuprates, pnictides
140
- Temperature
-
130
-
150
~ 155K
(K)
pressure
HgBaCaCuO
TlBaCaCuO
- “High
Bi2Sr2Ca2Cu3Ox
110 temperature”
100 YBa2Cu3O7-d
“cuprate”
90 80 70 superconductors
60 conventional
50 pressure
superconductors
Bednorz and Muller
40 (La,Ba)CuO
NbAlSi
Ba(K,Bi)O
30 NbGe
V Sn
20 NbN
Nb Sn
organic materials
Pb Nb
Ba(Pb,Bi)O
10 - Hg
NbO
0 120
3
Liquid N2
MgB2 39K
3
Doped buckyballs
A3C60
3
3
3
1910
1930
1950
1970
1975
1980
1985
1990
Liquid He
1995
2000
2005
Summary
• I gave a brief description of BCS theory of
superconductivity
• I tried to give an essential overview of the state of
the art in electronic structure calculations
• I presented an essential description of the properties and
SC mechanisms in a few important materials
• Each real material has plenty of interesting physics