Patterson, Bailey - Solid State Physics Introduction to theory

164 3 Electrons in Periodic Potentials

Pauli Spin Paramagnetism (B)

The quasifree electrons in metals show both a paramagnetic and diamagnetic effect. Paramagnetism is a fairly weak induced magnetization in the direction of the applied field. Diamagnetism is a very weak induced magnetization opposite the direction of the applied field. The paramagnetism of quasifree electrons is called Pauli spin paramagnetism. This phenomenon will be discussed now because it is a simple application of Fermi–Dirac statistics to electrons.

For Pauli spin paramagnetism we must consider the effect of an external magnetic field on the spins and hence magnetic moments of the electrons. If the magnetic moment of an electron is parallel to the magnetic field, the energy of the electron is lowered by the magnetic field. If the magnetic moment of the electron is in the opposite direction to the magnetic field, the energy of the electron is raised by the magnetic field. In equilibrium at absolute zero, all of the electrons are in as low an energy state as they can get into without violating the Pauli principle. Consequently, in the presence of the magnetic field there will be more electrons with magnetic moment parallel to the magnetic field than antiparallel. In other words there will be a net magnetization of the electrons in the presence of a magnetic field. The idea is illustrated in Fig. 3.9, where μ is the magnetic moment of the electron and H is the magnetic field.

Using (3.165), Fig. 3.9, and the definition of magnetization, we see that for absolute zero and for a small magnetic field the net magnetization is given approximately by

M = 1	K	E	F	(0) 2μ2μ	0	H .	(3.180)
2

The factor of 1/2 arises because Da and Dp (in Fig. 3.9) refer only to half the total number of electrons. In (3.180), K is given by (1/2π2)(2m*/ћ2)3/2.

Equations (3.180) and (3.174) give the following results for the magnetic susceptibility:

	χ =	∂M	= μ0μ2	EF (0)	3N	[EF (0)]−3 2	=	3Nμ0μ2	,
		∂H			2			2EF (0)

or, if we substitute for EF,

χ =	3Nμ0μ2
		.	(3.181)
	2kT (0)
	F

This result was derived for absolute zero, it is fairly good for all T << TF(0). The only trouble with the result is that it is hard to compare to experiment. Experiment measures the total magnetic susceptibility. Thus the above must be corrected for the diamagnetism of the ion cores and the diamagnetism of the conduction electrons if it is to be compared to experiment. Better agreement with experiment is obtained if we use an appropriate effective mass, in the evaluation of TF(0), and if we try to make some corrections for exchange and Coulomb correlation.

3.2 One-Electron Models 165

E E

Dp μ0μH	μ0μH	Da	Dp	Da
	(a)			(b)

Fig. 3.9. A magnetic field is applied to a free-electron gas. (a) Instantaneous situation, and

(b) equilibrium situation. Both (a) and (b) are at absolute zero. Dp is the density of states of parallel (magnetic moment parallel to field) electrons. Da is the density of states of antiparallel electrons. The shaded areas indicate occupied states

Landau Diamagnetism (B)

It has already been mentioned that quasifree electrons show a diamagnetic effect. This diamagnetic effect is referred to as Landau diamagnetism. This Section will not be a complete discussion of Landau diamagnetism. The main part will be devoted to solving exactly the quantum-mechanical problem of a free electron moving in a region in which there is a constant magnetic field. We will find that this situation yields a particularly simple set of energy levels. Standard statisticalmechanical calculations can then be made, and it is from these calculations that a prediction of the magnetic susceptibility of the electron gas can be made. The statistical-mechanical analysis is rather complicated, and it will only be outlined. The analysis here is also closely related to the analysis of the de Haas–van Alphen effect (oscillations of magnetic susceptibility in a magnetic field). The de Haas– van Alphen effect will be discussed in Chap. 5. This Section is also related to the quantum Hall effect, see Sect. 12.7.2. In SI units, neglecting spin effects, the Hamiltonian of an electron in a constant magnetic field described by a vector potential A is (here e > 0)

	H =	1	( p + eA)2 = −	2	2 +	e	A +	e	A +	e2	A2. (3.182)
		2m		2m		2mi		2mi		2m

Using (Aψ) = A ψ + ψ A, we can formally write the Hamiltonian as

	H = −	2	2 +	e	A +	e	A +	e2	A2 .	(3.183)
		2m		2mi		mi		2m

166 3 Electrons in Periodic Potentials

A constant magnetic field in the z direction is described by the nonunique vector potential

	μ0Hy ˆ			μ0Hx ˆ
A = −		i	+		j .
	2			2

To check this result we use the defining relation

μ0 H = × A .

(3.184)

(3.185)

and after a little manipulation it is clear that (3.184) and (3.185) imply H = Hkˆ . It is also easy to see that A defined by (3.184) implies

A = 0 ,

(3.186)

Combining (3.183), (3.184), and (3.186), we find that the Hamiltonian for an electron in a constant magnetic field is given by

2

2

e

μ

0

H

∂

∂

e2μ2H 2

(x2 + y2 ) . (3.187)

H = −

+

x

− y

+

0

2m

2mi

∂y

8m

∂x

It is perhaps worth

pointing

out

that

(3.187)

plus a

central potential is

a Hamiltonian often used for atoms. In the atomic case, the term (x ∂/∂y − y ∂/∂x) gives rise to paramagnetism (orbital), while the term (x2 + y2) gives rise to diamagnetism. For free electrons, however, we will retain both terms as it is possible to obtain an exact energy eigenvalue spectrum of (3.187).

The exact energy eigenvalue spectrum of (3.187) can readily be found by making three transformations. The first transformation that it is convenient to

make is
ieμ		0	H	xy
ψ (x, y, z) = φ(x, y, z) exp				.	(3.188)
	2

Substituting (3.188) into Hψ = Eψ with H given by (3.187), we see that φ satisfies the differential equation

2 2			e μ0H		∂φ		H 2μ02e2	2
−		φ −		x	∂y	+		x φ = Eφ .	(3.189)
	2m		im				2m

A further transformation is suggested by the fact that the effective Hamiltonian of (3.189) does not involve y or z so py and pz are conserved:

φ(x, y, z) = F(x) exp[−i(k y y + kz z)] .	(3.190)
This transformation reduces the differential equation to
d2F + (A + Bx)2 F = CF ,	(3.191)
dx2

3.2 One-Electron Models 167

or more explicitly

2

d

2

F

1

2

2

−

+

[

k

y

− (Hμ

0

)(ex)]2 F

= E −

kz

F .

2

2m dx

2m

2m

Finally, if we make a transformation of the dependent variable x,

x1

= x −

k y

,

eHμ0

then we find

2

d

2

F

e

2

H

2

2

2

2

−

+

μ0

(x1)2 F = E −

kz

F .

1

2

2m

2m

2m d(x )

(3.192)

(3.193)

(3.194)

Equation (3.194) is the equation of a harmonic oscillator. Thus the allowed energy eigenvalues are

En,k	z	=	2kz2		+ ωc (n +			1	) ,	(3.195)
			2m
					2
where n is an integer and
		ωc ≡			eHμ0					(3.196)
					m

is just the cyclotron frequency.

This quantum-mechanical result can be given quite a simple classical meaning. We think of the electron as describing a helix about the magnetic field. The helical motion comes from the fact that, in general, the electron may have a velocity parallel to the magnetic field (which velocity is unaffected by the magnetic field) in addition to the component of velocity that is perpendicular to the magnetic field. The linear motion has the kinetic energy p2/2m = ћ2kz2/2m, while the circular motion is quantized and is mathematically described by harmonic oscillator wave functions.

It is at this stage that the rather complex statistical-mechanical analysis must be made. Landau diamagnetism for electrons in a periodic lattice requires a still more complicated analysis. The general method is to compute the free energy and concentrate on the terms that are monotonic in H. Then thermodynamics tells us how to relate the free energy to the magnetic susceptibility.

A beginning is made by calculating the partition function for a canonical ensemble,

Z = ∑i exp(−Ei / kT ) ,

(3.197)

168 3 Electrons in Periodic Potentials

where Ei is the energy of the whole system in state i, and i may represent several quantum numbers. (Proper account of the Pauli principle must be taken in calculating Ei from (3.195).) The Helmholtz free energy F is then obtained from

F = −kT ln Z ,

and from this the magnetization is determined:

M = ∂F .

μ0∂H

Finally the magnetic susceptibility is determined from

χ = ∂M

∂H H =0 .

The approximate result obtained for free electrons is

χLandau = − 13 χPauli = −Nμ0μ2 / 2kTF .

(3.198)

(3.199)

(3.200)

(3.201)

Physically, Landau diamagnetism (negative χ) arises because the coalescing of energy levels (described by (3.195)) increases the total energy of the system. Fermi–Dirac statistics play an essential role in making the average energy increase. Seitz [82] is a basic reference for this section.

Soft X-ray Emission Spectra (B)

So far we have discussed the concept of density of states but we have given no direct experimental way of measuring this concept for the quasifree electrons. Soft X-ray emission spectra give a way of measuring the density of states. They are even more directly related to the concept of the bandwidth. If a metal is exposed to a beam of electrons, electrons may be knocked out of the inner or bound levels. The conduction-band electrons tend to drop into the inner or bound levels and they emit an X-ray photon in the process. If El is the energy of a conduction-band electron and E2 is the energy of a bound level, the conduction-band electron emits a photon of angular frequency

ω = (E1 − E2 ) / .

Because these X-ray photons have, in general, low frequency compared to other X-rays, they are called soft X-rays. Compare Fig. 3.10. The conduction-band width is determined by the spread in frequency of all the X-rays. The intensities of the X-rays for the various frequencies are (at least approximately) proportional to the density of states in the conduction band. It should be mentioned that the measured bandwidths so obtained are only the width of the occupied portion of the band. This may be less than the actual bandwidth.

3.2 One-Electron Models 169

Conduction

band Occupied portion of band

Typical transition

Empty bound level

Fig. 3.10. Soft X-ray emission

The results of some soft X-ray measurements have been compared with Hartree calculations.12 Hartree–Fock theory does not yield nearly so accurate agreement unless one somehow fixes the omission of Coulomb correlation. With the advent of synchrotron radiation, soft X-rays have found application in a wide variety of areas. See Smith [3.51].

The Wiedeman–Franz Law (B)

This law applies to metals where the main carriers of both heat and charge are electrons. It states that the thermal conductivity is proportional to the electrical conductivity times the absolute temperature. Good conductors seem to obey this law quite well if the temperature is not too low.

The straightforward way to derive this law is to derive simple expressions for the electrical and thermal conductivity of quasifree electrons, and to divide the two expressions. Simple expressions may be obtained by kinetic theory arguments that treat the electrons as classical particles. The thermal conductivity will be derived first.

Suppose one has a homogeneous rod in which there is a temperature gradient of ∂T/∂z along its length. Suppose Q· units of energy cross any cross-sectional area (perpendicular to the axis of the rod) of the rod per unit area per unit time. Then the thermal conductivity k of the rod is defined as

k =	Q
	∂T / ∂z .	(3.202)

Figure 3.11 sets the notation for our calculation of the thermal conductivity.

12 See Raimes [3.42, Table I, p 190].

170 3 Electrons in Periodic Potentials

E(0)

λ

θ

z

Fig. 3.11. Picture used for a simple kinetic theory calculation of the thermal conductivity. E(0) is the mean energy of an electron in the (x,y)-plane, and λ is the mean free path of an electron. A temperature gradient exists in the z direction

If an electron travels a distance equal to the mean free path λ after leaving the (x,y)-plane at an angle θ, then it has a mean energy

E(0) + λcosθ	∂E	.	(3.203)
	∂z

Note that θ going from 0 to π takes care of both forward and backward motion. If N is the number of electrons per unit volume and u is their average velocity, then the number of electrons that cross unit area of the (x,y)-plane in unit time and that make an angle between θ and θ + dθ with the z-axis is

2π sinθdθ	Nu cosθ =	1	Nu cosθ sinθdθ .	(3.204)
4π
	2

From (3.203) and (3.204) it can be seen that the net energy flux is

Q =		∂T		=	1	Nu∫	π		+ λ cosθ	∂E
	k	∂z						cosθ sinθ E(0)		dθ
				2		0				∂z

=12 Nu∫0π λ cos2 θ sinθ ∂∂Ez dθ

=13 Nuλ ∂∂Ez = 13 Nuλ ∂∂TE ∂∂Tz ,

but since the heat capacity is C = N(∂E/∂T), we can write the thermal conductivity as

k = 1	Cuλ .	(3.205)
3

Equation (3.205) is a basic equation for the thermal conductivity. Fermi–Dirac statistics can somewhat belatedly be put in by letting u → uF (the Fermi velocity) where

	1	mu2	= kT	F	,	(3.206)
	2
		F

3.2 One-Electron Models 171

and by using the correct (by Fermi–Dirac statistics) expression for the heat capacity,

C = π 2 Nk 2T . muF2

It is also convenient to define a relaxation time τ:

τ ≡ λ / uF .

The expression for the thermal conductivity of an electron gas is then

k = π 2 Nk 2τT .

3 m

(3.207)

(3.208)

(3.209)

If we replace m by a suitable m* in (3.209), then (3.209) would probably give more reliable results.

An expression is also needed for the electrical conductivity of a gas of electrons. We follow here essentially the classical Drude–Lorentz theory. If vi is the velocity of electron i, we define the average drift velocity of N electrons to be

	v =	1	∑iN=1vi .	(3.210)
		N

If τ is the relaxation time for the electrons (or the mean time between collisions) and a constant external field E is applied to the gas of the electrons, then the equation of motion of the drift velocity is

m	dv	+	v	= −eE .	(3.211)
	dt
			τ
The steady-state solution of (3.211) is
v = −eτE / m .					(3.212)
Thus the electric current density j is given by
j = −Nev = Ne2 (τ / m)E .					(3.213)
Therefore, the electrical conductivity is given by
σ = Ne2τ / m .					(3.214)

Equation (3.214) is a basic equation for the electrical conductivity. Again, (3.214) agrees with experiment more closely if m is replaced by a suitable m*.

Dividing (3.209) by (3.214), we obtain the law of Wiedeman and Franz:

k		π 2	k			2
	=					T = LT ,	(3.215)
σ		3		e

172 3 Electrons in Periodic Potentials

where L is by definition the Lorentz number and has a value of 2.45 10−8 w Ω K−2. At room temperature, most metals do obey (3.215); however, the experimental value of k/σT may easily differ from L by 20% or so. Of course, we should not be surprised as, for example, our derivation assumed that the relaxation times for both electrical and thermal conductivity were the same. This perhaps is a reasonable first approximation when electrons are the main carriers of both heat and electricity. However, it clearly is not good when the phonons carry an appreciable portion of the thermal energy.

We might also note in the derivation of the Wiedeman–Franz law that the electrons are treated as partly classical and more or less noninteracting, but it is absolutely essential to assume that the electrons collide with something. Without this assumption, τ → ∞ and our equations obviously make no sense. We also see why the Wiedeman–Franz law may be good even though the expressions for k and σ were only qualitative. The phenomenological and unknown τ simply cancelled out on division. For further discussion of the conditions for the validity of Weideman–Franz law see Berman [3.4].

There are several other applications of the quasifree electron model as it is often used in some metals and semiconductors. Some of these will be treated in later chapters. These include thermionic and cold field electron emission (Chap. 11), the plasma edge and transparency of metals in the ultraviolet (Chap. 10), and the Hall effect (Chap. 6).

Angle-resolved Photoemission Spectroscopy (ARPES) (B)

Starting with Spicer [3.52], a very effective technique for learning about band structure has been developed by looking at the angular dependence of the photoelectric effect. When light of suitable wavelength impinges on a metal, electrons are emitted and this is the photoelectric effect. Einstein explained this by saying the light consisted of quanta called photons of energy E = ω where ω is the frequency. For emission of electrons the light has to be above a cutoff frequency, in order that the electrons have sufficient energy to surmount the energy barrier at the surface.

The idea of angle-resolved photoemission is based on the fact that the component of the electron’s wave vector k parallel to the surface is conserved in the emission process. Thus there are three conserved quantities in this process: the two components of k parallel to the surface, and the total energy. Various experimental techniques are then used to unravel the energy band structure for the band in which the electron originally resided (say the valence band Ev(k)). One technique considers photoemission from differently oriented surfaces. Another uses high enough photon energies that the final state of the electron is freeelectron like. If one assumes high energies so there is ballistic transport near the surface then k perpendicular to the surface is also conserved. Energy conservation and experiment will then yield both k perpendicular and Ev(k), and k parallel to the

3.2 One-Electron Models 173

surface can also by obtained from experiment—thus Ev(k) is obtained. In most cases, the photon momentum can be neglected compared to the electron’s k.13

3.2.3The Problem of One Electron in a Three-Dimensional Periodic Potential

There are two easy problems in this Section and one difficult problem. The easy problems are the limiting cases where the periodic potential is very strong or where it is very weak. When the periodic potential is very weak, we can treat it as a perturbation and we say we have the nearly free-electron approximation. When the periodic potential is very strong, each electron is almost bound to a minimum in the potential and so one can think of the rest of the lattice as being a perturbation on what is going on in this minimum. This is known as the tight binding approximation. For the interesting bands in most real solids neither of these methods is adequate. In this intermediate range we must use much more complex methods such as, for example, orthogonalized plane wave (OPW), augmented plane wave (APW), or in recent years more sophisticated methods. Many methods are applicable only at high symmetry points in the Brillouin zone. For other places we must use more sophisticated methods or some sort of interpolation procedure. Thus this Section breaks down to discussing easy limiting cases, harder realistic cases, and interpolation methods.

Metals, Insulators, and Semiconductors (B)

From the band structure and the number of electrons filling the bands, one can predict the type of material one has. If the highest filled band is full of electrons and there is a sizeable gap (3 eV or so) to the next band, then one has an insulator. Semiconductors result in the same way except the bandgap is smaller (1 eV or so). When the highest band is only partially filled, one has a metal. There are other issues, however. Band overlapping can complicate matters and cause elements to form metals, as can the Mott transition (qv) due to electron–electron interactions. The simple picture of solids with noninteracting electrons in a periodic potential was exhaustively considered by Bloch and Wilson [97].

The Easy Limiting Cases in Band Structure Calculations (B)

The Nearly Free-Electron Approximation (B) Except for the one-dimensional calculation, we have not yet considered the effects of the lattice structure. Obviously, the smeared out positive ion core approximation is rather poor, and the free-electron model does not explain all experiments. In this section, the effects of the periodic potential are considered as a perturbation. As in the one-dimensional Kronig–Penny calculation, it will be found that a periodic potential has the effect of splitting the allowed energies into bands. It might be thought that the nearly

13 A longer discussion is given by Marder [3.34 footnote 3, p. 654].