PHYSICS 617 – Jan. 20, 2017 Properties of Fermi Gases (1) Introduction: These notes accompany chapter 2 of the text, covering noninteracting electron gases, or “Fermi gases”. Some terminology: “Fermi gas” refers to a noninteracting or weakly-interacting gas of Fermions, which at low temperatures will exhibit a Fermi surface due to Pauli exclusion. The Fermi gas idea works surprisingly well for simple metals such as sodium or copper, since screening and related effects can effectively reduce the strong Coulomb repulsion of the electrons. “Fermi liquid” is used to refer to a more strongly interacting system; see the discussion of Landau Fermi liquid behavior in chapter 17 of the text. Fermi liquids occupy a regime for which interactions between electrons are strong, yet even for such a case many of the characteristics of non-interacting Fermi gases still persist, for example the existence of a Fermi surface. In addition there is considerable interest in understanding systems for which the electron-electron interaction effects increase to the extent that the Fermi liquid characteristics break down: there is a class of properties referred to as “non-Fermi liquid” behavior, and a particular model sometimes applied to such systems is the “Luttinger liquid”. The metals exhibiting high-temperature superconductivity may fall into these categories. Some techniques to analyze Fermi surfaces of metals are described in chapter 14 of the text. One recent addition is based on photoemission; with the development of high-intensity synchrotron light sources, analysis of the ejected electrons now provides provides an extremely powerful analysis tool for the electron states in solids. Angle Resolved PhotoEmission Spectroscopy (ARPES) is the term for the technique as often applied to solids. Briefly, with the incoming photon energy and momentum known, by detecting the momentum of the outgoing electron, by conservation principles one can extract considerable information about the electrons in the valence band. In many cases this allows mapping of the Fermi surface, and the energy vs. momentum character of the valence band. The references at the end include a review of photoemission techniques1 and applications to high-temperature superconductors2. Also see reference 3 for a description of Fermi surface and related behavior in condensed cold atoms and optical lattices. (2) Fermi Gas low-T behavior, Fermi energy and state counting: For non-interacting Fermions, the thermal-equilibrium occupation function is given by:

�

f (ε) = exp(ε − µ) / kBT + 1[ ]−1, [1]

which is the “Fermi function,” expressing the probability of a Fermion particle occupying a state at a given energy,

�

ε. The parameter µ is the chemical potential, which changes according to the density of the Fermi gas, as shown below. When µ is much smaller than

�

kBT , f(e) goes over to the Boltzmann distribution, proportional to

�

exp(−ε / kBT ) , as in a classical gas. However in the

other extreme (low T or high density of particles), Pauli exclusion strongly controls the result, with the distribution [1] essentially equal to 1 or 0 except for a transition region near

�

ε = µ . The transition region has a width of approximately 4

�

kBT , as in the sketch.

For the zero temperature limit (or very high densities) when the Fermi distribution is a step function in energy, the step position is defined as µ =

�

εF , the Fermi energy. For electron densities corresponding to typical metallic materials, the corresponding temperature ( kTF ≡ εF the Fermi temperature) is on order of 10,000 K; for example see the numbers in section 2 below. For such cases

the step-function picture can be an excellent approximation even at ambient temperatures. Also the surface containing the occupied states in momentum space is called the Fermi surface. To understand how states fill the Fermi surface, consider the free electron plane-wave states to have the form

�

ψ = s exp(i k ⋅ r ) / V , with the spin considered not to interact with the spatial

states (neglecting spin-orbit), and therefore represented as a product state. Also consider a box of dimensions

�

L× L× L to represent the crystal, and assume periodic boundary conditions: the wavefunction is assumed periodic with respect to translation by L the x, y, or z direction. That is,

�

ψ( r ) =ψ(

r + Lˆ i ) =ψ(

r + Lˆ j ) =ψ(

r + L ˆ k ). This restricts

�

k to discrete values,

�

k = (2π / L)(n1

ˆ i + n2ˆ j + n3

ˆ k ) , [2] forming a dense array of allowed values in the space of

�

k vectors or k-space. Periodic boundary

conditions are not particularly realistic when applied to the surface of a real crystal, but since we are concerned with the behavior of an essentially infinite crystal, the boundary details should not be important, and we will verify in the end that the measurable quantities do not depend on L. The energy of non-interacting electrons is simply the kinetic energy,

�

ε = 2k 2 2m . Since this is independent of the orientation of

�

k , at T = 0 when the N electrons occupy the N/2 lowest

wavevectors (factor 2 for spin), the states fill a sphere. Thus the Fermi surface in this case is a Fermi sphere, pictured in the sketch above along with the discrete states for the crystal size L. (3) Summations and averaging in k-space and energy – Density of states: According to the expression [2] for allowed k-values, the volume per state in k-space is

�

(2π / L)3. So, given N electrons with spin, the occupied volume is

�

(N / 2)(2π / L)3 , or per electron the k-space volume is

�

4π 3 /V , with L3 = V, the crystal volume. Defining n = N/V as the electron density, the Fermi sphere volume can also be written

�

4π 3n , or for a given region of k-space the density of states (states per real space volume, per volume in k-space) is given by, , [3] sometimes defined as the density of states in k-space (not to be confused with the energy density of states, for which we use the notation g(e)). This result corresponds to the uniform density of

Dk = 1/(4⇡3)

states in k-space, with spin states counted, and as expected the sample dimensions do not appear in the final expression. From [3] it is easy to find the Fermi surface dimensions: setting

�

4π 3n = 43 πkF

3 , with kF the Fermi sphere radius, yields

�

kF = 3π 2n3 . [4]

Correspondingly the Fermi energy is

�

εF = 2kF

2

2m , and the Fermi temperature

�

TF = εFkB

.

A typical metal has a unit cell dimension of 1 nm or smaller, and if (as in the case of Cu) we have 4 atoms/cubic cell each with one valence electron, a cell of this size will correspond to n = 4 ´ 1021 cm-3, a lower bound for the electron density expected for metals. Using [4] this gives kF = 4.9 ´ 107 cm-1, for which we can calculate

�

εF = 0.9 eV. This corresponds to TF equal to approximately 10,000 K, similar to what was quoted above. Since typically a metal will melt or decompose well below its TF, its properties may be reliably obtained from T = 0 results thorugh perturbation in terms of T/TF (the Sommerfeld expansion discussion below). Another measure is the Fermi velocity, vF = kF /m , which is roughly 108 cm/s for the values quoted here. The Fermi velocity is typically several orders of magnitude larger than the classical thermal velocity. Often the quantities of interest depend only on the energy, independent of orientation in k-space. In this case we use the energy density of states [

�

g(ε) ], defined so that

�

g(ε)dε is the number of electron states per volume in the energy range (

�

ε to

�

ε + dε). Within the Fermi sphere, the region between (

�

ε and

�

ε + dε) is a spherical shell of volume

�

4πk 2dk . For free electrons

�

dε = (2k

m)dk

so the volume in k-space for this shell is

�

4πk(m / 2 )dε . Using [3] the number of states in this volume is

�

Vk(m /π 22 )dε , and converting to terms of energy we obtain,

�

g(ε) = 2π 2

m2

⎛

⎝ ⎜

⎞

⎠ ⎟

3 / 2

ε . [5]

The form [5] is the free-electron form for

�

g(ε) . We will continue to use

�

g(ε) later in the course for situations where the energy vs. k has a more complicated structure, but note that the summation procedures outlined below remain generally valid. (4) Series analysis of temperature dependence; Sommerfeld expansion. We start with the average value, summed over all electrons, of some function H which depends on the energy of an individual particle:

�

H = V H (ε)g(ε) f (ε)dε−∞

∞

∫ . [6]

Note that this is N times the average value of H per particle; V is included since g is defined per volume. Integrating by parts:

�

H = −V H ( ′ ε )g( ′ ε )d ′ ε −∞

ε

∫⎡

⎣ ⎢

⎤

⎦ ⎥

dfdε

dε−∞

∞

∫ = +V H ( ′ ε )g( ′ ε )d ′ ε −∞

ε

∫⎡

⎣ ⎢

⎤

⎦ ⎥

dfdµ

dε−∞

∞

∫ . [7]

For the last term in [7], since

�

f (ε − µ) = exp(ε − µ) / kBT + 1[ ]−1,

�

dfdε

= − dfdµ

. However, note that

�

dfdµ

is still a function of the energy e, and

�

dfdµ

is zero except for a delta-function-like peak in e

near the Fermi energy, so only energies very close to µ will contribute in the integration. The Sommerfeld expansion is a Taylor expansion in (e – µ) of the square bracket in [7], starting from the case e = µ. Proceeding we have:

�

H = +V H ( ′ ε )g( ′ ε )d ′ ε −∞

µ

∫⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

dfdµ

dε−∞

∞

∫

+ V H (µ)g(µ)[ ](ε − µ) dfdµ

dε−∞

∞

∫

+ V ddµ

H (µ)g(µ)( )⎡

⎣ ⎢

⎤

⎦ ⎥ (ε − µ)2

2dfdµ

dε−∞

∞

∫+ ...

. [8]

Note that the terms written with square brackets in [8] do not contain the energy, and could be placed outside the integrals. The middle term is odd in (e – µ) and it integrates to zero. We keep the first and third terms, neglecting higher-order terms. A more general form for the Sommerfeld expansion can be seen in a Statistical Mechanics text such as Huang, and also in Ashcroft and Mermin. The energy integrals for the remaining terms in [8] can be evaluated to yield,

�

H ≅V H (ε)g(ε)dε−∞

µ

∫ + π2

6V (kBT )2 d

dµH (µ)g(µ)( ) . [9]

(5) T-dependence of chemical potential, and analysis of specific heat: From the general definition of the density of states and the occupation probability, the total

number of particles must be

�

N = Vg(ε) f (ε)dε−∞

∞

∫ . This is just [6] with H replaced by unity. So

using [9] we can write,

�

N ≅V g(ε)dε−∞

µ

∫ + V π2

6(kBT )2 dg(µ)

dµ . [10]

There is a thermodynamic relation,

�

∂N∂µ

⎞

⎠ ⎟

T

∂µ∂T

⎞

⎠ ⎟

N

∂T∂N

⎞

⎠ ⎟

µ

= −1 by which we can obtain the

following, where the final form is obtained by directly evaluating derivatives of equation [10]:

�

∂µ∂T

⎞

⎠ ⎟

N

= −

∂N∂T

⎞

⎠ ⎟

µ

∂N∂µ

⎞

⎠ ⎟

T

= −V π2

62kB

2T dg(µ)dµ

Vg(µ) + (small term with ′ ′ g ). [11]

Given that the T = 0 value is µ = eF, from [11] we obtain by a 1st order expansion,

�

µ ≅ εF − π2

6(kBT )2 dg(µ)

dµg(µ)

⎧ ⎨ ⎩

⎫ ⎬ ⎭

. [12]

This shows how µ changes with the temperature; for example for free electrons, where

�

dg / dε is positive (see [5]), µ decreases with increasing T. However near the top of a band, as we shall see for example with a hole pocket, the opposite temperature dependence holds. Note that the departure from

�

µ = εF is small for a good metal with T<<TF: e.g. for free electrons specifically,

�

g(ε) = (Const.) ⋅ ε , so that we find that

�

′ g / g = 12ε , and [12] reduces to,

�

µ ≅ εF − π2

12kBT T

TF

⎛

⎝ ⎜

⎞

⎠ ⎟ . [13]

The last term is indeed very small in a metal, and in very many cases one may simply use

�

µ = εF . We can similarly examine the average total energy, replacing H in our expansion by e:

�

E ≅V εg(ε)dε−∞

µ

∫ + V π2

6(kBT )2 d µg(µ)[ ]

dµ , [14]

I wrote

�

E to emphasize that [14] yields the average total energy, which we could write as N times a single-particle average,

�

E ≡ N ε . With this replacement, equation [14] expands to (dividing by V to get the energy density, using n = N/V, and keeping lowest-order terms),

u = n ε ≅ εg(ε )dε−∞

εF

∫ + (µ − εF )εFg(εF )+π2

6(kBT )

2g(εF )+π2

6(kBT )

2 dgdµ

⎞⎠⎟ εF

εF . [15]

The second and fourth term on the right are identical and cancel, which can be seen by using [12] to simplify the fourth term. (Also making the approximation

�

µ ≈ εF , which is true to lowest order in T.) The first term on the right is the T = 0 energy density, which evaluates to 35 nεF for a free-electron gas. The third term contains all the temperature dependence:

u ≅ 35 nεF +

π2

6(kBT )

2g(εF ) , [16]

so that the volume heat capacity, which is the temperature derivative of the energy density, is:

�

c = π2

3(kB )2 g(εF )T ≡ γT . [17]

This is the characteristic linear-T heat capacity seen experimentally in metals at very low temperatures. The T-linear coefficient is traditionally denoted g, and its measurement is one way of probing the electronic density of states in metals. References: 1. Friedrich Reinert and Stefan Hüfner, “Photoemission spectroscopy - from early days to recent applications” New J. Phys. 7 (2005) 97 2. W S Lee, I M Vishik, D H Lu and Z-X Shen, “A brief update of angle-resolved photoemission spectroscopy on a correlated electron system”, J. Phys.: Condens. Matter 21 (2009) 164217. 3. Immanuel Bloch, “Ultracold quantum gases in optical lattices”, Nature Physics 1, 23 - 30 (2005).