Optical conductivity, Drude model.

Lecture 13 - Oct. 28, 2013
Phys 735 Superconductivity
Optical conductivity and BCS theory
Lecturer: Ed Taylor
ΣHΩL
Another major piece of experimental evidence confirming BCS theory were the spectroscopic conductivity
measurements carrier out of Glover and Tinkham [1] using far-infrared radiation and terahertz radiation
(Tc ∼ ∆/kB ∼ 10K and hence, ∆ ∼ 1.4 × 10−22 J ∼ 1meV. This corresponds to radiation of wavelength
λ = hc/∆ ∼ 1mm.) Although arguably not as direct a probe of the superconducting gap as tunnelling
experiments, optical conductivity measurements have played a crucial role in unravelling many properties of
superconductors, especially in the high-Tc cuprate compounds [2].
In this Lecture, I will calculate discuss the Drude theory of conductivity (a review of what we covered
in Lecture 3) and then carry out an explicit calculation of the optical conductivity using BCS theory (along
the same lines as the tunnelling calculation we did in Lecture 12). Even though Drude theory is for normal
metals, by combining insights from Drude theory and the BCS calculation, you should acquire at least a
qualitative understanding of the physics illustrated in Fig. 1. Namely, the Drude-like form for the optical
conductivity above Tc , the disappearance of spectral weight below 2∆ in the superconducting phase, and the
appearance of a Dirac delta function at zero frequency with weight given by the superfluid density, signifying
a fraction of electrons that exhibits perfect conductivity.
0
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hHِ2ΠL2D
Figure 1: Cartoon of the optical conductivity at temperatures above Tc (red dashed line) and T = 0 (blue solid
line). Above Tc , the optical conductivity exhibits a characteristic “Drude-like” form. As T is lowered through Tc ,
spectral weight begins to disappear in the region ω ≤ 2∆. Finally, at T = 0, there
R ∞is no spectral weight (for a clean
superconductor) below 2∆: σ(ω < 2∆) = 0. Because the total spectral weight 0 dωσ(ω) = ne2 /2m is conserved,
the missing spectral weight in the region 0 < ω < 2∆0 must reappear somewhere, namely a Dirac delta function at
zero frequency with weight given by the superfluid density ns : σ(ω < 2∆) = (πns e2 /m)δ(ω).
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Drude theory and the Ferrell–Tinkham–Glover sum rule
We introduced Drude’s theory of conductivity in Lecture 3 and will recall some of its features here. Strictly
speaking, it only applies to a normal metal (i.e., not superconducting). However, we will use it here to
calculate the conductivity of a perfect conductor and also derive a sum rule which will be of considerable
use in understanding the BCS prediction for the optical conductivity.
1
Regular metals exhibit conductivity (i.e., fealty to Ohm’s law) as a consequence of collisions of electrons
with ion cores or impurities. In the absence of such scattering, electrons subject to an electric field E will
accelerate in accordance with the Lorentz Force law F = eE:
m
dv
= eE(t).
dt
(1)
Now suppose that there are ions or impurities and the electrons scatter off these. While one can construct
a serious kinetic theory of this process, the simplest approximation is to assume that
m
dv
mv
= eE(t) −
,
dt
τ
(2)
where τ is the mean time between collisions. Without the driving electric field, this would result in an
exponential decay v(t) = v(0) exp(−t/τ ) of the velocity of an electron as a result of collisions. For a
time-dependent electric field E(t) = E exp(iωt), the steady-state solution of these equations is
v(t) =
(τ e/m)
E(t).
1 + iωτ
(3)
Hence, the electric current j = nev can be written as
j = σ(ω)E
(4)
with
(ne2 τ /m)
.
(5)
1 + iωτ
Equation (4) is Ohm’s law (which provides the effective definition of the conductivity, even when Drude
theory does not apply) and (5) is Drude formula for the conductivity. In Fig. 2, we plot the real part of
the Drude conductivity, showing the characteristic decay of the optical conductivity in a normal metal at
frequencies greater than the time τ between collisions.
ΣHΩL
σ(ω) =
0
1
2
ΩΤ
3
4
Figure 2: The real part of the conductivity σ(ω) from the Drude model (5). The frequency-dependent optical
conductivity decays over a characteristic frequency scale ω ∼ τ −1 where τ is the mean time between collisions of
electrons with something (other electrons, ion cores, impurities,...). Independent of the temperature, interactions,
etc., the area under this curve is fixed by the exact result ne2 /2m [see (10)].
A perfect conductor can be thought of as one in which τ is infinite. Taking this limit in (5) gives an
imaginary conductivity
(ne2 /m)
σ(ω) =
.
(6)
iω
2
Since the conductivity is an analytic complex function, it also has a real part which can be determined by
the Kramers-Kronig relations. For an arbitrary analytic complex function F (ω), these are
Z ∞
ImF (ω ′ )
1
dω ′ ′
(7)
ReF (ω) = P
π
ω −ω
−∞
and
1
ImF (ω) = − P
π
Z
∞
dω ′
−∞
ReF (ω ′ )
.
ω′ − ω
(8)
Using F (ω) = σ(ω) and (6) in (8), one sees that the real part of the conductivity of a perfect conductor is a
Dirac-delta function at zero frequency:
Reσ(ω) = (πne2 /m)δ(ω).
(9)
Equations (6) and (9) tell us that the optical conductivity of a perfect conductor with no dissipation has
a real part which is a Dirac delta function and an imaginary part which varies as ω −1 . A superconductor
will exhibit the same features although, measuring the real part of the conductivity, it’s not possible to pick
out the Dirac delta function. The existence of this feature is determined instead using a sum rule for the
real part of the conductivity:
Z
1 ∞
ne2
.
(10)
dωReσ(ω) =
π −∞
m
This result can be derived by integrating the real part of the Drude formula for σ(ω) in (5) over all frequencies.
It is an exact result however1, valid for all systems, independent of temperature, presence of impurities,
whether the system being studied is superconductor or normal, etc. Since the real part of the conductivity
is an even function Reσ(−ω) = Reσ(ω) of frequency, we can also write this sum rule as
Z
ne2
1 ∞
.
(11)
dωReσ(ω) =
π 0
2m
Equation (6) corresponds to the conductivity that freely accelerating electrons would have under the
influence of an electric field. It is thus reasonable to assume that for a superconductor, there is a component
of the optical conductivity that follows (6), but with the density n replaced by the superfluid density ns (as
much is also implied by the London equations). The optical conductivity in a superconductor can thus be
written as
πns e2
σ(ω) =
δ(ω) + σI (ω).
(12)
m
Here σI (ω) denotes the dissipative contributions to the conductivity. We will calculate this quantity within
BCS theory next. Before doing this, we note that this result implies a very famous semi-empirical sum rule,
Z
Z
1 ∞
ns e 2
1 ∞
,
(13)
dωσN (ω) =
dωσS (ω) +
π 0+
π 0+
2m
the so-called Ferrell–Tinkham–Glover sum rule, relating the spectral weight in the superconducting conductivity σS (ω) to that of the normal conductivity σN (ω)2 . It tells us that below Tc , spectral weight will begin
to disappear from the Drude form shown in Fig. 2. Since the total spectral weight is fixed by the exact sum
rule (11), the “missing” spectral weight has to go somewhere. That somewhere is a Dirac delta function at
zero frequency with weight given by the superfluid density ns . It corresponds to the fraction of electrons
that accelerates under the influence of an electric field, i.e., the fraction obeying London’s equation
djs
ns e 2
=
E.
dt
m
(14)
1 It can be derived exactly from the Kramers–Kronig relations as well as the fact that at asymptotically high frequencies the
conductivity is purely reactive; i.e., σ(ω → ∞) = (ne2 /m)/iω.
2 The 0+ in the lower limit of integration is a positive infinitesimal. Experimentally, this is the relevant sum rule since you
can’t probe things at zero frequency!
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2
BCS calculation of σ(ω)
We now turn to a microscopic calculation of the non-Dirac-delta part of the optical conductivity in a BCS
superconductor. The calculation will proceed similarly as the calculation of the tunnelling rate we did in the
last lecture. Some details will be omitted since this will be your homework problem for the week!
As before, we are going to use Fermi’s golden rule to calculate some rate in response to some perturbation
applied to the system. In the case of tunnelling, we used Fermi’s golden rule to calculate the rate at which
electrons tunnelled through a barrier in response to a tunnelling barrier (which we wrote down as an extra
term in the hamiltonian). Here we will do essentially the same thing except that now the perturbation we
are applying in an external electromagnetic field (photons!), which couple to the electrons via the term
i
Xh
δ Ĥ =
g(k, k′ )â†k′ σ âkσ + H.c.
(15)
k,k′
and calculate the rate
W = 2π
X
f
|hf |δ Ĥ|ii|2 δ(Ef − Ei − ~ω).
(16)
at which photons of energy ~ω are absorbed. Note that (15) corresponds to an operator which describes the
absorption of a photon (we could have explicitly included a photon destruction operator but its presence
does not change anything), leading to the annihilation of an electron in the state kσ and the creation of one
in the state k′ σ. Microscopically, the “matrix element” g is
g(k, k′ ) = −
e~
k · A(k′ − k),
mc
(17)
where A(q) is the Fourier transform of the vector potential. Apart from one feature (to be discussed below),
the details of this matrix element are unimportant for the calculation to follow since we will take the ratio
of the conductivity in the superconducting and normal phases (as we did for the tunnelling rate).
Before proceeding with our calculation, it’s useful to note how ubiquitous perturbing terms of the form
(15) are. The tunnelling term [Eq. (5) in Lecture 12] has precisely this form. Virtually any experimental
probe of a system can be described by a hamiltonian of this form. In essence, an experiment involves
the destruction of electrons from some state via the absorption of some “probe particle” (neutron, photon,
phonon,...) and the creation of an electron in some new state. The rate at which electrons are created in this
new state corresponds to some experimentally measurable current (in the present case, a current of photons
reflected from the superconductor).
With this is mind, we derive a general expression for the transition rate W given a perturbation of the
form (15). We first note that the contributions of the two spin states being summed over in (15) have some
overlap in the superconducting state. This can be seen by expressing the relevant operators in terms of BCS
quasiparticle operators using the Bogoliubov-Valatin transformations:
and
†
†
+ vk′ uk γ̂k′ γ̂k0
γ̂k′ 1 + uk′ vk γ̂k† ′ 0 γ̂k1
â†k′ ↑ âk↑ = uk′ uk γ̂k† ′ 0 γ̂k0 − vk′ vk γ̂k1
(18)
†
†
+ uk′ vk γ̂k′ γ̂k0 .
γ̂k′ 1 + vk′ uk γ̂k† ′ 0 γ̂k1
â†−k↓ â−k′ ↓ = −vk′ vk γ̂k† ′ 0 γ̂k0 + uk′ uk γ̂k1
(19)
Without superconductivity, the contributions coming from the two spins to some observable in the normal
state would add incoherently. In contrast, when superconductivity is present the two spins both contribute
to certain to the same processes and must be added coherently in evaluating (16). (This is why the us and
vs are called Bogoliubov coherence factors.)
Using (18) and (19) in (16) and specializing again to zero temperature (where only pairs of quasiparticle
creation operators contribute), the rate at which photons are absorbed becomes
X
2
W = 2π
(20)
[uk′ vk g(k, k′ ) + vk′ uk g(−k′ , −k)] δ(Ek + Ek′ − ~ω).
k,k′
4
Using (17), we see that g(−k′ , −k) = −g(k, k′ ) and this becomes
X
W = 2π
g 2 (k, k′ ) [uk′ vk − uk vk′ ]2 δ(Ek + Ek′ − ~ω).
(21)
k,k′
Following the same approach as we used to calculate the tunnelling rate, this can be written (when expressed
as the ratio of the tunnelling rates in the superconducting and normal phases) as
σs
1
Ws
=
=
Wn
σn
~ω
Z
~ω−∆
∆
dE √
E(~ω − E) − ∆2
p
.
E 2 − ∆2 (~ω − E)2 − ∆2
(22)
Plotting the results of this integration in Fig. 3 reveals that there is no optical response at T = 0 below 2∆,
a feature seen in early experiments [1]. (See also Ref. [3] for more refined measurements).
ΣsΣn
1
0
0 1 2 3 4 5 6 7 8 910
hHِ2ΠL2D
Figure 3: BCS prediction for the ratio of the optical conductivities in the superconducting and normal states at
T = 0.
The absence of any spectral weight (in an s-wave superconductor) below twice the gap can understood
as a simple manifestation of the fact that a photon cannot excite any excitations if its energy is below twice
the gap (the factor of two derives from the fact that quasiparticles are always excited in pairs—see (18) for
instance.) Finally, combining the Drude result in Fig. 5 with this leads to the picture in Fig. 1. Namely, the
disappearance of spectral weight below 2∆ when the temperature is lowered below Tc and the appearance
of a Dirac delta function to make up the difference.
References
[1] R. E. Glover and M. Tinkham, Phys. Rev. 104, 844 (1956);
http://prola.aps.org/abstract/PR/v104/i3/p844_1.
[2] D. Basov and T. Timusk, Rev. Mod. Phys. 77, 721 (2005);
http://rmp.aps.org/abstract/RMP/v77/i2/p721_1.
[3] D. M. Ginsberg and M. Tinkham, Phys. Rev. 118, 990 (1960);
http://prola.aps.org/abstract/PR/v118/i4/p990_1.
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