Download Lecture 21

PHYS 342/555
Introduction to solid state physics
Instructor: Dr. Pengcheng Dai
Associate Professor of Physics
The University of Tennessee
(Room 407A, Nielsen, 974-1509)
(Office hours: TR 1:10PM-2:00 PM)
Lecture 21, room 304 Nielsen
Chapter 12: Superconductivity
Lecture in pdf format will be available at:
http://www.phys.utk.edu
1
The London equation
The assumption of London model: in a superconductor at T < Tc ,
only a fraction ns (T ) / n of the total number of conduction electrons
are capable of participating in a supercurrent. ns (T ) is the density
of superconducting electrons (superfluid density). The remaining
fraction of electrons are "normal fluid" density n − ns that cannot
carry an electric current without normal dissipation.
Suppose that an electric field momentarily arises within a superconductor.
The superconducting electrons will be freely accelerated without
G
G
dvs
dissipation: m
= −eE.
dt
Dai/PHYS 342/555 Spring 2006
Chapter 12-2
Since the current density carried by these electrons is
G
d G ns e 2 G
G
j = −evs ns , we have
j=
E.
dt
m
G
G
ns e 2
AC conductivity j (ω ) = σ (ω ) E (ω ), σ (ω ) = i
.
mω
G
G
1 ∂B
Faraday's law of induction, ∇ × E = −
,
c ∂t
G ns e 2 G ⎞
∂⎛
B ⎟ = 0.
⎜∇× j +
mc ⎠
∂t ⎝
G 4π G
Consider Maxwell equation ∇ × B =
j , that determines the
c
magnetic fields and current densities that can exist within a
perfect conductor.
Dai/PHYS 342/555 Spring 2006
Chapter 12-3
Note that any static field B determines a static current density j.
Since any time-independent B and j are trivially solutions to
the first equation, the two equations are consistent with an arbitrary
static magnetic field. This is incompatible with the observed
behavior of superconductors, which permit no fields in their interior.
By restricting the full set of solutions to those that obey
G
ns e 2 G
∇× j = −
B, known as the London equation. The reason for
mc
having the more restrictive London equation is that it leads to
the Meissner effect.
Dai/PHYS 342/555 Spring 2006
Chapter 12-4
G 4π ns e 2 G 2 G 4π ns e 2 G
∇ B=
B, ∇ j =
j . These equatios predict that
2
2
mc
mc
currents and magnetic fields in superconductors can exist only
within a layer of thickness Λ of the surface, where Λ is the
London penetration depth.
2
1/ 2
3/ 2
1/ 2
⎛ mc ⎞
⎛ rs ⎞ ⎛ n ⎞
Λ=⎜
= 41.9 ⎜ ⎟ ⎜ ⎟ .
2 ⎟
⎝ 4π ns e ⎠
⎝ a0 ⎠ ⎝ ns ⎠
Thus the London equation implies the Meissner effect, along
with a specific picture of the surface current that screen out the
2
applied field. These currents occur within a surface layer of
thickness of 102 − 103 Å.
Dai/PHYS 342/555 Spring 2006
Chapter 12-5
Coherence length
The coherence length is a measure of the distance within which the
superconducting electron concentration cannot change drastically in
a spatially-varying magnetic field.
We define an intrinsic coherence length ξ 0 related to the critical
modulation by ξ 0 = 1/ q0 . We have
ξ 0 = = 2 k F / 2mEg .
The coherence length and the actual penetration depth depend on the
mean free path of the electrons measured in the normal state.
Dai/PHYS 342/555 Spring 2006
Chapter 12-6
Microscopic theory: qualitative features
Bardeen, Cooper, Schrieffer theory of superconductivity:
1. A net attractive interaction between electrons in the neighborhood
of the Fermi surface. Although the direct electrostatic interaction is
repulsive, it is possible for the ionic motion to "overscreen" the
Coulomb interaction, leading to a net attraction.
2. The electron-lattice-electron interaction leads to an energy gap of
the observed magnitude. The indirect interaction proceeds when one
electron interacts with the lattice and deforms it; a second electron
sees the deformed lattice and adjusts itself to take advantage of the
deformation to lower its energy. Thus the net attractive force.
3. The penetration depth and the coherence length are natural
consequences of the BCS theory.
Dai/PHYS 342/555 Spring 2006
Chapter 12-7
4. The criterion for the Tc of an element or alloy involves the electron
density of orbital D (ε F ) of one spin at the Fermi level and the electron
lattice interaction U , which can be estimated from the electrical
resistivity. For UD (ε F ) 1 the BCS theory predicts
Tc = 1.13θ e −1/ UD (ε F ) . Where θ is the Debye temperature and U is an
attractive interaction. The result for Tc is satisfied at least qualitatively
by the experimental data. The higher the resistivity at room temperature
the higher is U , and thus the more likely it is that a metal will be a
superconductor when cooled.
5. Mangetic flux through a superconducting ring is quantized and the
effective unit of charge is 2e rather than e. The BCS ground state
involves pairs of electrons.
Dai/PHYS 342/555 Spring 2006
Chapter 12-8
„
BCS superconductors
k BTc = 1.13=ω D ⋅ e
−
1
N (0) V
N (0) is the density of electronic levels for a single spin population
in the normal metal and ω and V0 are the parameters of the model
Hamiltonian. Because of the exponential dependence, the effective
coupling V0 cannot be determined precisely enough to permit very
accurate computations of the critical temperature.
„
Higher Tc is achieved with:
ωD → lighter elements
‹ Larger N(0) → van Hove singularity etc.
‹ Larger |V| → stronger electron-phonon int.
‹ Larger
Dai/PHYS 342/555 Spring 2006
Chapter 12-9
Energy gap
The zero temperature energy gap:
Δ(0) = 2=ω D ⋅ e
−
1
N (0) V
The fundamental formula for the relationship between Tc and gap
independent of the phenomenological parameters:
Δ (0)
= 1.76,
k BTc
The BCS theory also predicts that near the critical temperature the
energy gap vanishes according to the universal law
1/ 2
⎛ T⎞
Δ (T )
= 1.74 ⎜1 − ⎟
Δ(0)
⎝ Tc ⎠
, T ≈ Tc .
Dai/PHYS 342/555 Spring 2006
Chapter 12-10
Dai/PHYS 342/555 Spring 2006
Chapter 12-11
Critical Field
The elementary BCS prediction
for H c (T ) is often expressed in
terms of the deviation from the
empirical law:
2
⎛T ⎞
H c (T )
≈ 1− ⎜ ⎟ .
H c (0)
⎝ Tc ⎠
The quantity [ H c (T ) / H c (0)] −
[1 − (T / Tc ) 2 ] is shown for several
superconductors below.
Dai/PHYS 342/555 Spring 2006
Chapter 12-12
Specific heat
At the critical temperature (in zero magnetic field) the elementary
BCS theory predicts a discontinuity in the specific heat:
cs − cn
cn
= 1.43.
Tc
The low-temperature electronic specific heat can also be cast in a
parameter independent form,
cs
⎛ Δ (0) ⎞
= 1.34 ⎜
⎟
γ Tc
⎝ T ⎠
3/ 2
e −Δ (0) / T ,
where γ is the coefficient of the linear term in the specific heat of
the metal in the normal state.
Dai/PHYS 342/555 Spring 2006
Chapter 12-13
(a ) Take the time derivative of the London equation (10) to show
G
G
G
G
2
2
that ∂j / ∂t = (c / 4πλL ) E. (b) if mdv / dt = qE , as for free carriers
of charge q and mass m, show that λL2 = mc 2 / 4π nq 2 .
Dai/PHYS 342/555 Spring 2006
Chapter 12-14
High-temperature superconductors
„
Parent Cuprate La2CuO4
2D Heisenberg Antiferromagnet
with s = 1/2 on a square lattice
T (K)
300
Néel order is
quickly suppressed
upon hole doping.
200
Hole Doping:
→
(Cu2+ → Cu2+δ )
La3+
3D
Néel
order
100
spin-glass
Sr2+
0
0
0.02
0.05
Hole doping
Dai/PHYS 342/555 Spring 2006
Chapter 12-15
High-Tc Superconductivity
„
Superconductivity shows up when moderately doped.
Layered structure
→ Essentially 2D transport
„
Basic building block: CuO2 plane
Cu
M. Kastner et al., RMP 70, 897 (1998).
Oxygen
Only one band is relevant to
the transport in the CuO2 plane.
(Cu 3d – O 2p mixture)
Dai/PHYS 342/555 Spring 2006
Chapter 12-16
Electrons form Cooper pairs
Conventional low
temperature SC
z
The binding of
electrons into Cooper
pairs is essential.
S=
z
z
=0
Long-range phase
coherence among the
pairs is also required to
have superconductivity.
Electron pairing is
mediated by electronphonon interaction.
Dai/PHYS 342/555 Spring 2006
Chapter 12-17
Single layer families of high-Tc
cuprates
Nd2-xCexCuO4 (Tc~25K)
La2-xSrxCuO4 (Tc~40K)
Pr1-xLaCexCuO4 (Tc~25K)
La2CuO4+y (Tc~40K)
Dai/PHYS 342/555 Spring 2006
Chapter 12-18
Real, reciprocal space of CuO2 plane
La2-xSrxCuO4 single layer, orthorhombic
Nd2-xCexCuO4 single layer, tetragonal
Dai/PHYS 342/555 Spring 2006
Chapter 12-19
How SC is established from a long range
AF ordered insulator?
Diagonal SDW
coexists with 3D
AF order in lightly
doped region and
survive in the
insulating phase
with doping.
At the transition
to SC, SDW
changes its
direction and
coexists w/ SC.
SDW has 2D
characteristic.
Kang et al. (2005) PRB
Wakimoto et al.
Dai/PHYS 342/555 Spring 2006
Chapter 12-20
SC Tc is intimated related to the
incommensurability of the SDW order
Dai/PHYS 342/555 Spring 2006
Chapter 12-21