Download Specific heat: Evidence for an energy gap and its symmetry 1

Phys 735 Superconductivity
Lecture 2 - Sept. 11, 2013
Specific heat: Evidence for an energy gap and its symmetry
Lecturer: Ed Taylor
1
Specific heat and the energy gap
Of all the experimental probes of superconductivity, why start here? Well, it’s simple and can be covered
in a relatively short period of time. Moreover, it was the experimental measurements of the specific heat
in superconducting Niobium that first revealed one of the most fundamental properties of superconductors:
the elementary excitations close to the Fermi surface are gapped, even though the system is
a metal (!) [1]. (In condensed-matter speak, there is a suppression of the low-energy density of states.) It
also gives us (not very high-quality) information about the symmetry of the gap (i.e., s-wave, p-wave, etc.).
Let’s start by reviewing the specific heat of a normal metal.
The specific heat at constant volume is defined as
∂U
(1)
CV ≡
∂T V
where U is the internal energy. For a noninteracting electron gas, this is
Z ∞
X Z d3 k
U=
n (k)f (n (k)) =
dN ()f ().
(2π)3
0
n
(2)
This expression has a simple physical interpretation. It is counting up the number N () = N ()f () of
electrons with energy multiplied by that energy; i.e., the total energy. f () is the Fermi-Dirac distribution,
f () =
1
eβ(−µ)
while
N () =
X
Z
V
n
; β ≡ (kB T )−1 ,
(3)
d3 k
δ[ − n (k)]
(2π)3
(4)
+1
is the density of states, characterizing the number of states with energy between and + d. Here µ ' EF
is the chemical potential which, at low temperatures and weak interactions, is essentially the Fermi energy
EF (by definition, the maximum occupied energy level for noninteracting electrons due to Pauli repulsion).
n (k) is the energy of a Bloch state with quasi momentum k and band n. The density of states (DOS) plays
an important role in the discussion of superconductors, so let’s start with a very simple calculation, the DOS
of an ideal Fermi gas in three dimensions with “parabolic dispersion” (no ionic potential) n (k) = ~2 k2 /2m.
Using the fantastically useful Dirac-delta function identity
δ[f (x)] =
X δ(x − xi )
i
|f 0 (xi )|
,
(5)
where f (x) is some function with zeroes at x = xi , the density of states becomes
V
N () =
2π 2
Z
dkk
2 δ(k
√
3/2
√
− 2m/~)
V
2m
=
.
2
2
2
(~ k/m)
2π
~
1
(6)
Combining (1) and (2), the specific heat is
Z ∞
∂f ()
CV =
dN ()
∂T
0
Z
2
kB
T ∞
dxN (x)sech2 (x/2)x(x + µ/kB T ),
=
4
−µ/kB T
(7)
where we have defined the dimensionless variable x ≡ µ/kB T . At low temperatures, kB T µ (equivalently,
T TF ), we can make use of a couple of approximations to make this integral analytically tractable. First,
sech2 (x) is strongly peaked about x = 0; i.e., the Fermi surface, and one can remove the value N (0) of the
density of states at the Fermi surface from the integral1 Second, the lower limit on the integral can be taken
to be −∞. This gives
Z
2
2
kB
T N (0) ∞
π 2 N (0)kB
CV (T TF ) '
dxsech2 (x/2)x(x + µ/kB T ) =
T ≡ γT,
(8)
4
3
−∞
2
/3 of proportionality is called the Sommerfeld coefficient.
i.e., it’s linear in T . The coefficient γ = π 2 N (0)kB
It turns out that in real metals (in Landau Fermi liquids at least), such a linear contribution is found,
although to see it clearly one usually has to remove a T 3 phonon contribution:
CV (T TF ) ' γT + βT 3
(Landau Fermi liquid).
(9)
In some metals (e.g., Al, K), the phonon contribution is very small at low temperatures and one can clearly
make out the linear-T part. In either case, concentrating on the contribution from electronic excitations, at
low temperatures, the specific heat is sensitive to the density of states at the Fermi surface. In
other words, the availability of excitations near the Fermi surface. If the Fermi surface is completely gapped
(no zero energy excitations), N (0) will be strongly suppressed and so will the low temperature specific heat.
Figure 1: Heat capacity in superconducting and normal Niobium. From Ref. [1].
This is what was first observed in superconducting Niobium in 1953 [1]. Experiments compared the
low-temperature specific heat of Niobium without a magnetic field to the case where a large magnetic field is
1 As a matter of convention, in about half the literature the density of states at the Fermi surface is denoted by N (E ) while
F
in the other half, it’s denoted by N (0) since − EF = 0 there.
2
Figure 2: Specific heat for YBCO, plotted as C/T vs. T 2 . A magnetic field is applied either parallel or perpendicular
to the ab plane. The T 3 phonon contribution to the specific heat is independent of the magnetic field and so the
change in the curves with increasing magnetic field can be interpreted as the H-dependent electronic contribution
CV,DOS ∝ T to the specific heat. From Ref. [4].
applied (killing off superconductivity); see Fig. 1. Without superconductivity, the specific heat exhibits the
linear-T dependence predicted for an ideal gas. In contrast, when superconductivity is present, the specific
heat exhibits two noteworthy features. The first is the very strong, exponential suppression of specific heat
at low temperatures:
CV ∝ exp(−∆0 /T ).
(10)
Later, after we have become acquainted with BCS theory, we will derive this result. For now, we have to
appeal to intuition. If the excitation spectrum has a gap ∆0 in the vicinity of the Fermi surface, N () will
be zero (at zero temperatures) for energies EF − ∆0 /2 . . EF + ∆0 /2 in this region. At nonzero temperatures, there will be a small number of exponentially activated excitations (remember that thermodynamic
distributions are always exponential), N ( ' EF ) ∼ exp(∆0 /T ). This leads to the result (10).2
The other noteworthy feature is the “jump” in the specific heat at Tc . As we will discuss later on when
we come the Ginzburg–Landau theory of superconductivity, this is telling us that the superconducting phase
transition is second order.
In contrast to the specific heat in s-wave superconductors such as Niobium and Vanadium, the situation
in d-wave superconductors is much more complex, in large part because the electronic (“density-of-states”)
contribution to the specific heat is masked by other, usually much larger contributions. In order to extract
this contribution, experiments are usually carried out at nonzero magnetic fields since the electronic densityof-states contribution depends on the magnetic field (not surprisingly: electrons are obviously sensitive to
the magnetic field whereas non-electronic contributions will be less so). Although this complicates the
interpretation of the data, for a clean d-wave superconductor with “line nodes” in the excitation gap,
∆(k) ∼ ∆0 (cos kx a − cos ky a),
(11)
2 As far as I know, the first experiment to actually report this exponential dependence was carried out on Vanadium the
following year [2].
3
Figure 3: Specific heat in Sr2 RuO4 exhibiting the characteristic CV ∝ T 2 dependence at low temperatures associated
with line nodes. From Ref. [5] using the experimental data of Ref. [6].
there will be excitations at certain points of the Fermi surface and there will no be exponential dependence
of the form (10). Instead, the density of states can be shown to vary linearly with energy, resulting in
an electronic contribution to the specific heat that varies as T 2 [3, 4] (intermediate between the node-less
exponential suppression and the linear-T behaviour without a superconducting gap). Because experiments
are done in a field, however, and the density-of-states is field-dependent, this T 2 dependence evolves into
a field-dependent T dependence[4]. This is shown in Fig. 2 for YBCO (the results of this work were taken
to be evidence in favour of line nodes consistent with (11)). We will return to the specific heat of d-wave
superconductors later in this course.
Clearer evidence for T 2 behaviour in the low-temperature specific heat arising from line nodes can be seen
in Sr2 RuO4 ; Fig. 3. (Note that the order parameter symmetry in Sr2 RuO4 is not yet known with complete
confidence, although it is generally believed to be triple p-wave. Reconciling p-wave order with line nodes is
a major theoretical challenge in this material.)
References
[1] A. Brown, M. W. Zemansky, and H. A. Boorse, Phys. Rev. 92, 52 (1953);
http://prola.aps.org/abstract/PR/v92/i1/p52_1.
[2] W. S. Corak, B. B. Goodman, C. B. Satterthwaite, and A. Wexler, Phys. Rev. 96, 1442 (1954);
http://prola.aps.org/abstract/PR/v96/i5/p1442_2.
[3] G. E. Volovik, JETP Lett. 58, 469 (1993).
[4] K. A. Moler, D. J. Baar, J. S. Urbach, R. Liang, W. N. Hardy, and A. Kapitulnik, Phys. Rev. Lett. 73,
2744 (1994);
http://prl.aps.org/abstract/PRL/v73/i20/p2744_1.
[5] T. Nomura and K. Yamada, J. Phys. Soc. Jpn. 71, 404 (2002);
http://jpsj.ipap.jp/link?JPSJ/71/404/.
[6] S. Nishiaki,Y. Maeno, and Z. Q. Mao, J. Phys. Soc. Jpn. 69, 572 (2000).
4