Solid State Physics 7506266318

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1A

LEGEND

HYDROGEN

0.089

1.0079

1

H

Name

is1

1 3.75

HEX

14.0 LITHIUM

110

2A

6.941

BERYLLIUM

3

Li

0.53

Density (gm cm-3) (Common Crystal phase)

1731

1s22s'

4

400

0.97

22 9898

Na

11

1550

1.74

3.21

150

371.0

POTASSIUM

39 09

K

19

0.86

4

Rb

1.53

5

85 47 37

312

6.05

56LT

55

Fr

Sr

38

21

HEX

1812

YTTRIUM 4.46

.

.

•

.

-■

6

' sign Hies a low temperature Determination/

3 65

HEX

1796

137 34

LANTHANUM

Ba

56

3-75 110LT

87

(5.0)

226

Ra

1933

39

88

I388

9122 40

Zr

HEX

2125

57

13.1

Hf

1-593

178 49 72

[Xe] 4/’45d26s2

1-619

3 20

ACTINIUM

227

50.942

V

6.1

23

CHROMIUM

3.02

92 91

Nb

8.4

41

les

180.95

73

Ta

11.5

BCC

331

42

2.74 380

TUNGSTEN (WOLFRAM) 19.3 VV

3.16

2495

225

183 85 74

BCC 310

HEX

43

213

Re

HEX

8.9

101.07

44

RU

HEX

1-822

1768

385

102 90

RHODIUM

12.4

Rh

45

[Kr] 4d85s’

FCC

1684

3-80

2583

382LT

2239

350LT

OSMIUM

190.20

IRIDIUM

192.22

2.70

186 2 75

27

Co

2-51 420

123

58 93

[Ar] 3d74s2

BCC

RUTHENIUM

1804

[Xe] 4/M5tf56s2

3453

COBALT

[Kr] 4d75s’

2445

RHENIUM

2 76

3683

98.91

Tc

26

1808

[Kr] 4 c/6 5s2

BCC

Fe

237

400

10.2

5585

[Ar] 3d®4s2

CUB

TECHNETIUM

Mo

25

1518

MOLYBDENUM 95.94

[Xe] 4/’45c/46s2

[Xe]

HEX 1-562

460

275 2890

TANTALUM

7.86

8.89

2130

3.15

2741

Mn

7.43

[Kr] 4r/55s’

BCC

3.30

IRON

[Ar] 3d54s2

BCC

2 88

390

NIOBIUM

24

MANGANESE 54.938

(Ar) 3c/64s’

BCC

2163

52.00

Cr

719

[Kr] 4tT»5s’

250

HAFNIUM

138.91

ID

[Ar] 3d34s2

380

6.49

3.23

1-671

132

10.1

22

VANADIUM

[Kr] 4d25s2

256lt

HEX

HEX

ZIRCONIUM

1193

22.6

HEX

Os

76

[Xe] 4/’45d66s2

1616

2 74

416lt

3318

HEX

1-579 400LT

22.5

lr

77

[Xe] 4/’45d76s2 3 64

FCC

2683

430

89

Ac

[Rn] 6c/’ 7s2

[Rn] 7s2

531 973

295

359LT

La

617

47 90

[Ar] 3 o'2 4 s2

[Xe]5d’6r2

BCC

998

RADIUM

ORC — Orthorhombic HEX - Hexagonal DIA — Diamond RHL - Rhombohedral MCL - Monoclinic

U

Ti

1-594

Y

147lt

223

4.51

[Kr] 4d’5s2

FCC

3.5

TITANIUM

88 91

BARIUM

(BCC) (300)

SC

2 99

1043

[Rn] 7s’

7

230

87.62

5.02 40lt

FRANCIUM

44 956

[Xe] 6s2

BCC

302

SCANDIUM

331

FCC

2 60

Face-Cent sred Cubic Body-Centered Cubic Simple Cubic Cubic Tetragonal

AR

[Ar] 3d’4s2

STRONTIUM

[Xe] 6s’

6

20

1111

608

132.91

Cs

40 08

Ca

i.54

— — —

IO

318

[Kr] 5s2

BCC

CESIUM 1.90

/I T

1624

CALCIUM

[Kr] 5s’

5.59

HEX

558

100

RUBIDIUM

12

[Ar] 4s2

BCC

5.23

Mg

922

[Ar] 4s’

337

FCC BCC SC CUB TET

24.305

[Ne] 3s2

BCC

4.23

318 *—Mean Debye I emperature

Crystal' 'ype (Common Crys tai Phase)

1000

MAGNESIUM

[Ne] 3s’

3

igle a in RHL (see Ch. 7) and b/a Ratio in ORC

1.624

"* 922

1-967

HEX

229

453

SODIUM

HEX

Melting Temperature____

is22s2

BCC

3.49

------ Atomic Config j rat ion

X

(A)

(K)

2

12 •-----Atomic Numbc r

Mg

321

Be

1.85

i.74

Lattice Constant, a 90122

Mass Number

Symbol MAGNESIUM /24.305

FCC

.

RARE EARTHS

1323

140 12

CERIUM

Ce

6.77

LANTHANIDES

6 ◄

[Xe] 4/25d°6s2

FCC

6.18

◄ ACTINIDES

58

PRASEODYMIUM 140 91 6.77 Pr 59

NEODYMIUM 14424

[Xe] 4/35d°6s2

[Xe] 4/45d°6s2

367

HEX

1283

157lt

(1350)

URANIUM

238.03

NEPTUNIUM

1204

PROTACTINIUM

[Rn] 6d27s2

7 6.08

2020

15.4

Pa

231 91

[Rn] 5/26c = 2nvc.

(1.22)

Chapter 1 The Drude Theory of Metals

16

AC ELECTRICAL CONDUCTIVITY OF A METAL To calculate the current induced in a metal by a time-dependent electric field, we write the field in the form E(r) = Re(E(wk~icot). (1.23)

The equation of motion (1.12) for the momentum per electron becomes P

dt

eE.

(1.24)

t

We seek a steady-state solution of the form

p(r) = Re(p(co)e-n

(1.25)

Substituting the complex p and E into (1.24), which must be satisfied by both the real and imaginary parts of any complex solution, we find that p(co) must satisfy -top(w) = - — - eE(co). T

(1.26)

Since j = -nep/m, the current density is just j(r) = ReGHe-n

j(w) = _

(l27) m

(1/t) — ia>

One customarily writes this result as (1.28)

j(co) = (t(co)E(co),

where cr(co), known as the frequency-dependent (or AC) conductivity, is given by

i \ ne2z ff(w) = f---- :—> o-q =------ ■ 1 — icot m

n w

(1-29)

Note that this correctly reduces to the DC Drude result (1.6) at zero frequency. The most important application of this result is to the propagation of electro­ magnetic radiation in a metal. It might appear that the assumptions we made to derive (1.29) would render it inapplicable to this case, since (a) the E field in an electro­ magnetic wave is accompanied by a perpendicular magnetic field H of the same magnitude,20 which we have not included in (1.24), and (b) the fields in an electro­ magnetic wave vary in space as well as time, whereas Eq. (1.12) was derived by assuming a spatially uniform force. The first complication can always be ignored. It leads to an additional term — ep/mc x H in (1.24), which is smaller than the term in E by a factor v/c, where v is the magnitude of the mean electronic velocity. But even in a current as large as 1 amp/mm2, v = j/ne is only of order 0.1 cm/sec. Hence the term in the magnetic field is typically 10“10 of the term in the electric field and can quite correctly be ignored. 20

One of the more appealing features of CGS units.

AC Electrical Conductivity of a Metal

17

The second point raises more serious questions. Equation (1.12) was derived by assuming that at any time the same force acts on each electron, which is not the case if the electric field varies in space. Note, however, that the current density at point r is entirely determined by what the electric field has done to each electron at r since its last collision. This last collision, in the overwhelming majority of cases, takes place no more than a few mean free paths away from r. Therefore if the field does not vary appreciably over distances comparable to the electronic mean free path, we may correctly calculate j(r, r), the current density at point r, by taking the field everywhere in space to be given by its value E(r, t) at the point r. The result, j(r, co) = (j(co)E(r, co),

(1.30)

is therefore valid whenever the wavelength z of the field is large compared to the electronic mean free path t. This is ordinarily satisfied in a metal by visible light (whose wavelength is of the order of 103 to 104 A). When it is not satisfied, one must resort to so-called nonlocal theories, of greater complexity. Assuming, then, that the wavelength is large compared to the mean free path, we may proceed as follows: in the presence of a specified current density j we may write Maxwell’s equations as21

V • E = 0;

V • H = 0;

V x E =

T_ 4ti . V x H = —j c

1 c ct

i aE c dt '

(131)

We look for a solution with time dependence e l(at, noting that in a metal we can write j in terms of E via (1.28). We then find

_ ico „ „ ico/47tcr^ ico \ V x (V x E) = - VE = — V x H = — ---- E------- El, c c \ c c J

„ (1.32)

or -V-E-^fl+^E.

c \

(1.33)

60 /

This has the form of the usual wave equation, — V2E = ^-€(&>)E, c

(1.34)

with a complex dielectric constant given by

If we are at frequencies high enough to satisfy cot

» 1,

(1.36)

21 We are considering here an electromagnetic wave, in which the induced charge density p vanishes. Below we examine the possibility of oscillations in the charge density.

18

Chapter 1 The Drude Theory of Metals

then, to a first approximation, Eqs. (1.35) and (1.29) give

€(co) =1-^4,

(1.37)

co2

where cop, known as the plasma frequency, is given by

When e is real and negative (co < cop) the solutions to (1.34) decay exponentially in space; i.e., no radiation can propagate. However, when e is positive (co > cop) the solutions to (1.34) become oscillatory, radiation can propagate, and the metal should become transparent. This conclusion is only valid, of course, if our high-frequency assumption (1.36) is satisfied in the neighborhood of co - cop. If we express r in terms of the resistivity through Eq. (1.8), then we can use the definition (1.38) of the plasma frequency to compute that copt

fr \3/2/ 1\ = 1.6 x 102 — — . \pj

(1.39)

Since the resistivity in microhm centimeters, pp, is of the order of unity or less, and since rJaQ is in the range from 2 to 6, the high frequency condition (1.36) will be well satisfied at the plasma frequency. The alkali metals have, in fact, been observed to become transparent in the ultra­ violet. A numerical evaluation of (1.38) gives the frequency at which transparency should set in as co / r, \ ~3/2 vp = = 11.4 x — x 1015 Hz (1.40) 2ti yo0 / or z„ = — = 0.26 | — ) ' vp \aoJ

x 103 A.

(1.41)

In Table 1.5 we list the threshold wavelengths calculated from (1.41), along with the Table 1.5 OBSERVED AND THEORETICAL WAVELENGTHS BELOW WHICH THE ALKALI METALS BECOME TRANSPARENT THEORETICAL0 Z

element

(103 A)

Li Na K Rb Cs

1.5 2.0 2.8 3.1 3.5

OBSERVED A

(103 A) 2.0 2.1 3.1 3.6 4.4

a From Eq. (1.41). Source: M. Born and E. Wolf, Principles of Optics, Pergamon, New York, 1964.

AC Electrical Conductivity of a Metal

19

observed thresholds. The agreement between theory and experiment is rather good. As we shall see, the actual dielectric constant of a metal is far more complicated than (1.37) and it is to some extent a piece of good fortune that the alkali metals so strikingly display this Drude behavior. In other metals different contributions to the dielectric constant compete quite substantially with the “Drude term” (1.37). A second important consequence of (1.37) is that the electron gas can sustain charge density oscillations. By this we mean a disturbance in which the electric charge density22 has an oscillatory time dependence From the equation of continuity,

v•j = -

dp

Ct

V • j(/dT) (E • VT), where 8 is the mean thermal energy per electron. (Calculate the energy lost by a typical electron colliding at r, which made its last collision at r - d. Assuming a fixed (that is, energy-independent) relaxation time r, d can be found to linear order in the field and temperature gradient by simple kinematic arguments, which is enough to give the energy loss to second order.)

4.

Helicon Waves

Suppose that a metal is placed in a uniform magnetic field H along the z-axis. Let an AC electric field Ee-*"' be applied perpendicular to H. (a) If the electric field is circularly polarized (Ey = ±iEx) show that Eq. (1.28) must be generalized to(b) *

(b) Show that, in conjunction with (1.61), Maxwell’s equations (1.31) have a solution Ex = EoeWz~m,\

Ey=±iEx,

Ex = 0,

(1.62)

\

(1.63)

provided that k2c2 = ecu2, where e{co} = i

1

(O \O) + (Oc + l/T)

Problems (c)

27

Sketch e(co) for co > 0 (choosing the polarization Ey = iEx) and demonstrate that solu­

tions to k2c2 = eco2 exist for arbitrary k at frequencies a> > cop and co < coc. (Assume the high field condition coct » 1, and note that even for hundreds of kilogauss, (opjwc » 1.)

(d)

Show that when co « coc the relation between k and co for the low-frequency solution is

fk2c2\ co = coj —y . J

(1.64)

This low-frequency wave, known as a helicon, has been observed in many metals.35 Estimate

the helicon frequency if the wavelength is 1 cm and the field is 10 kilogauss, at typical metallic densities.

Surface Plasmons

5.

An electromagnetic wave that can propagate along the surface of a metal complicates the obser­

vation of ordinary (bulk) plasmons. Let the metal be contained in the half space z > 0, z < 0

being vacuum. Assume that the electric charge density p appearing in Maxwell’s equations vanishes both inside and outside the metal. (This does not preclude a surface charge density

concentrated in the plane z = 0.) The surface plasmon is a solution to Maxwell’s equations of

the form:

Ex = Aeiqxe~Kz, Ey = 0, Ez = Be^e’*2, Ex = CeiqxeKz, Ey = 0, Ez = DeiqxeKz, q, K, K'real, K, K' positive. (a)

z > 0; z < 0;

(1.65)

Assuming the usual boundary conditions (E|| continuous, (eE)± continuous) and using

the Drude results (1.35) and (1.29) find three equations relating q, K, and K' as functions of co. (b) (c)

Assuming that cot » 1, plot q2c2 as a function of co2. In the limit as qc » co, show that there is a solution at frequency co = copl^/l. Show

from an examination of K and K' that the wave is confined to the surface. Describe its polarization. This wave is known as a surface plasmon.

35

R. Bowers et al., Phys. Rev. Letters 7,339 (1961).

2 The Sommerfeld Theory of Metals Fermi-Dirac Distribution Free Electrons

Density of Allowed Wave Vectors

Fermi Momentum, Energy, and Temperature Ground-State Energy and Bulk Modulus

Thermal Properties of a Free Electron Gas Sommerfeld Theory of Conduction

Wiedemann-Franz Law

30

Chapter 2 The Sommerfeld Theory of iMetals

In Drude’s time, and for many years thereafter, it seemed reasonable to assume that the electronic velocity distribution, like that of an ordinary classical gas of density n = AT/jz was given in equilibrium at temperature T by the Maxwell-Boltzmann distribution. This gives the number of electrons per unit volume with velocities in the range1 dy about v as/B(v)dv, where

ftM =

\2nkBTJ

e~mii2kBT-

(2J)

We saw in Chapter 1 that in conjunction with the Drude model this leads to good order of magnitude agreement with the Wiedemann-Franz law, but also predicts a contribution to the specific heat of a metal of ?kB per electron that was not observed.2 This paradox cast a shadow over the Drude model for a quarter of a century, which was only removed by the advent of the quantum theory and the recognition that for electrons3 the Pauli exclusion principle requires the replacement of the MaxwellBoltzmann distribution (2.1) with the Fermi-Dirac distribution:

4tt3 exp[(|mt>2 - kBT0)/kBT'] +1’

Here h is Planck’s constant divided by 2ti, and To is a temperature that is determined by the normalization condition4 dyf(y\

(2.3)

and is typically tens of thousands of degrees. At temperatures of interest (that is, less than 103 K) the Maxwell-Boltzmann and Fermi-Dirac distributions are spectacu­ larly different at metallic electronic densities (Figure 2.1). In this chapter we shall describe the theory underlying the Fermi-Dirac distribution (2.2) and survey the consequences of Fermi-Dirac statistics for the metallic electron gas. Shortly after the discovery that the Pauli exclusion principle was needed to account for the bound electronic states of atoms, Sommerfeld applied the same principle to the free electron gas of metals, and thereby resolved the most flagrant thermal anoma­ lies of the early Drude model. In most applications Sommerfeld’s model is nothing more than Drude’s classical electron gas with the single modification that the elec­ tronic velocity distribution is taken to be the quantum Fermi-Dirac distribution 1 We use standard vector notation. Thus by v we mean the magnitude of the vector v; a velocity is in the range Jv about v if its ith component lies between and v, + dvh for i = x, y, z; we also use d\ to denote the volume of the region of velocity space in the range Jv about v: Jv = dvxdvydv2 (thereby following the practice common among physicists of failing to distinguish notationally between a region and its volume, the significance of the symbol being clear from context). 2 Because, as we shall see, the actual electronic contribution is about 100 times smaller at room temperature, becoming smaller still as the temperature drops. 3 And any other particles obeying Fermi-Dirac statistics. 4 Note that the constants in the Maxwell-Boltzmann distribution (2.1) have already been chosen so that (2.3) is satisfied. Equation (2.2) is derived below; see Eq. (2.89). In Problem 3d the prefactor appearing in Eq. (2.2) is cast in a form that facilitates direct comparison with Eq. (2.1).

Fermi-Dirac Distribution

Figure 2.1 (a) The Maxwell-Boltzmann and Fermi-Dirac distributions for typical metallic densities

at room temperature. (Both curves are for the density given by T = 0.01TO.) The scale

is the same for both distributions, and has been normalized so that the Fermi-Dirac distribution approaches 1 at low energies. Below room temperature the differences between

the two distributions are even more marked, (b) A view of that part of (a) between x = 0 and x = 10. The x-axis has been stretched by about a factor of 10, and the /-axis has been

compressed by about 500 to get all of the Maxwell-Boltzmann distribution in the figure. On this scale the graph of the Fermi-Dirac distribution is indistinguishable from the x-axis.

31

32 Chapter 2 The Sommerfeld Theory of Metals rather than the classical Maxwell-Boltzmann distribution. To justify both the use of the Fermi-Dirac distribution and its bold grafting onto an otherwise classical theory, we must examine the quantum theory of the electron gas.5 For simplicity we shall examine the ground state (i.e., T = 0) of the electron gas before studying it at nonzero temperatures. As it turns out, the properties of the ground state are of considerable interest in themselves: we shall find that room tem­ perature, for the electron gas at metallic densities, is a very low temperature indeed, for many purposes indistinguishable from T = 0. Thus many (though not all) of the electronic properties of a metal hardly differ from their values at T = 0, even at room temperature.

GROUND-STATE PROPERTIES OF THE ELECTRON GAS We must calculate the ground-state properties of N electrons confined to a volume V. Because the electrons do not interact with one another (independent electron approximation) we can find the ground state of the N electron system by first finding the energy levels of a single electron in the volume K and then filling these levels up in a manner consistent with the Pauli exclusion principle, which permits at most one election to occupy any single electron level.6 A single electron can be described by a wave function t/r(r) and the specification of which of two possible orientations its spin possesses. If the electron has no interactions, the one electron wave function associated with a level of energy 8 satisfies the time­ independent Schrodinger equation7:

- 5- (T3 + + FJ U 8F.

(2.51)

20 Proof: A level can contain either 0 or 1 electron (more than one being prohibited by the exclusion principle). The mean number of electrons is therefore 1 times the probability of 1 electron plus 0 times the probability of 0 electrons. Thus the mean number of electrons in the level is numerically equal to the probability of its being occupied. Note that this would not be so if multiple occupation of levels were permitted. 21 The chemical potential plays a more fundamental role when the distribution (2.48) is derived in the grand canonical ensemble. See, for example, F. Reif, Statistical and Thermal Physics, McGraw-Hill, New York, 1965, p. 350. Our somewhat unorthodox derivation, which can also be found in Reif, uses only the canonical ensemble.

Thermal Properties of the Free Electron Gas

43

On the other hand, as T -> 0, the limiting form of the Fermi-Dirac distribution (2.48) is lim /ks = 1, S(k) < /.i; (2.52) T-° = 0,

For these to be consistent it is necessary that

lim /i = 8f.

(2.53)

r-o

We shall see shortly that for metals the chemical potential remains equal to the Fermi energy to a high degree of precision, all the way up to room temperature. As a result, people frequently fail to make any distinction between the two when dealing with metals. This, however, can be dangerously misleading. In precise calculations it is essential to keep track of the extent to which /z, the chemical potential, differs from its zero temperature value, &F. The most important single application of Fermi-Dirac statistics is the calculation of the electronic contribution to the constant-volume specific heat of a metal,

u

dll\

~V\fr)v

dTjy

U=F

(2.54)

In the independent electron approximation the internal energy U is just the sum over one-electron levels of 8(k) times the mean number of electrons in the level22: U = 2£s(k)/(s(k)).

(2.55)

k

We have introduced the Fermi function /(8) to emphasize that /k depends on k only through the electronic energy 8(k): (2.56)

If we divide both sides of (2.55) by the volume V, then (2.29) permits us to write the energy density u = U/V as

f Z?V

'

u =

T-jWm). J 471

(2.57)

If we also divide both sides of (2.49) by V, then we can supplement (2.57) by an equation for the electronic density n = N/V, and use it to eliminate the chemical potential: f dk n = |4P/(S(k))-

(2>58)

In evaluating integrals like (2.57) and (2.58) of the form

f dk T^F(8(k)), J

(2.59)

22 As usual, the factor of 2 reflects the fact that each £-leveI can contain two electrons of opposite spin orientations.

Chapter 2 The Sommerfeld Theory of Metals

44

one often exploits the fact that the integrand depends on k only through the electronic energy 8 = h 2k2/2m, by evaluating the integral in spherical coordinates and changing variables from k to 8: ^F(8(k)) =

Af

u ~ Jo

J

6

I 2

+ ^-(kBT)2g(ZF) + 0(T4),

(2.74)

6

0(8) J8 + Mz -f 8F)g(8reF)2 + - (kBT)2ig\&F)[.

n = J°

I

(2.75)

J

6

The temperature-independent first terms on the right sides of (2.74) and (2.75) are just the values of u and n in the ground state. Since we are calculating the specific heat at constant density, n is independent of temperature, and (2.75) reduces to

0 = (Ai-SfMSf) + ^(fcBT)V(Sf),

(2.76)

6

which determines the deviation of the chemical potential from Sf: M

„

~

11 6

t-\2

(^3). -^

I

j=l \K#Kj..Km

(9.18)

& “ ^k-K y

Compare this with the result (9.12) in the case of no near degeneracy. There we found an explicit expression for the shift in energy to order U2 (to which the set of equations (9.18) reduces when m = 1). Now, however, we find that to an accuracy of order U2 the determination of the shifts in the m nearly degenerate levels reduces to the solution of m coupled equations5 for the ck_K.. Furthermore, the coefficients in the second term on the right-hand side of these equations are of higher order in U than those in the first.6 Consequently, to find the leading corrections in U we can replace (9.18) by the far simpler equations: m (S — Sk-K()Ck-K, =

X

^Ky-K^k-Ky,

1 = L • • • , M,

(9.19)

which are just the general equations for a system of m quantum levels.7

ENERGY LEVELS NEAR A SINGLE BRAGG PLANE The simplest and most important example of the preceding discussion is when two free electron levels are within order U of each other, but far compared with U from all other levels. When this happens, Eq. (9.19) reduces to the two equations: (8 — 8k_Kj)ck_Ki = ^K2-K!Ck-K2’ (8 — 8k_K2)ck_K2 = LKi_K2ck_K1>

(9.20)

When only two levels are involved, there is little point in continuing with the notational convention that labels them symmetrically. We therefore introduce vari­ ables particularly convenient for the two-level problem: = K2 - K15

(9.21)

(8 — 8qkq — LKcq_K, (8 ~ S°_Kkq_K = = C*cq.

(9.22)

q = k-K1

and

K

and write (9.20) as

5 These are rather closely related to the equations of second-order degenerate perturbation theory, to which they reduce when all the are rigorously equal, i = 1, . . . , m. (See L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Addison-Wesley, Reading Mass., 1965, p. 134.) 6 The numerator is explicitly of order L2, and since only K-values different from Kt,..., Km appear in the sum, the denominator is not of order U when S is close to the s£_k • z = 1,..., m. 7 Note that the rule of thumb for going from (9.19) back to the more accurate form in (9.18) is simply that U should be replaced by U', where L'k.-K,- — ^K-K, +

£

K*Kj........K„

C'Ky-K C/r-K, 8 ~ 8k_K

Energy Levels Near a Single Bragg Plane

157

We have: « Sq°-K,

|8° - Sq-K | » U,

for K / K, 0.

(9.23)

Now 8® is equal to 8q_K for some reciprocal lattice vector only when |q| = |q - K|. This means (Figure 9.2a) that q must lie on the Bragg plane (see Chapter 6) bisecting the line joining the origin of k space to the reciprocal lattice point K. The assertion that 8q = 8q_K< only for K' = K requires that q lie only on this Bragg plane, and on no other. Figure 9.2

(a) If |q| = |q - K|, then the point q must lie in the Bragg plane determined by K. (b) If the point q lies in the Bragg plane, then the vector q — |K is parallel to the plane.

Thus conditions (9.23) have the geometric significance of requiring q to be close to a Bragg plane (but not close to a place where two or more Bragg planes intersect). Therefore the case of two nearly degenerate levels applies to an electron whose wave vector very nearly satisfies the condition for a single Bragg scattering.8 Corre­ spondingly, the general case of many nearly degenerate levels applies to the treatment of a free electron level whose wave vector is close to one at which many simultaneous Bragg reflections can occur. Since the nearly degenerate levels are the most strongly affected by a weak periodic potential, we conclude that a weak periodic potential has its major effects on only those free electron levels whose wave vectors are close to ones at which Bragg reflections can occur. We discuss systematically on pages 162 to 166 when free electron wave vectors do, or do not, lie on Bragg planes, as well as the general structure this imposes on the energy levels in a weak potential. First, however, we examine the level structure 8 An incident X-ray beam undergoes Bragg reflection only if its wave vector lies on a Bragg plane (see Chapter 6).

158

Chapter 9 Electrons in a Weak Periodic Potential

when only a single Bragg plane is nearby, as determined by (9.22). These equations have a solution when: 8 — Sq

—

(9.24)

This leads to a quadratic equation (8 - 8,)(8 - 8°_k) = |t/K|2.

(9.25)

The two roots

+ KM2

8 = |(8° + 8®_k) ±

(9.26)

give the dominant effect of the periodic potential on the energies of the two free electron levels 8° and 8°_K when q is close to the Bragg plane determined by K. These are plotted in Figure 9.3. Figure 9.3

Plot of the energy bands given by Eq. (9.26) for q parallel to K. The lower band corresponds to the choice of a minus sign in (9.26) and the upper band to a plus sign. When q = |K, the two bands are separated by a band gap of mag­ nitude 2|l/K|. When q is far removed from the Bragg plane, the levels (to leading order) are indistinguishable from their free electron values (denoted by dotted lines).

The result (9.26) is particularly simple for points lying on the Bragg plane since, when q is on the Bragg plane, 8° = 8°_K. Hence

8 = 8q ± |l/K|,

q

on a single Bragg plane.

(9.27)

Thus, at all points on the Bragg plane, one level is uniformly raised by |UK| and the other is uniformly lowered by the same amount. It is also easily verified from (9.26) that when 8° = 8q_K,

Ap

ft2

(9-28) i.e., when the point q is on the Bragg plane the gradient of 8 is parallel to the plane (see Figure 9.2b). Since the gradient is perpendicular to the surfaces on which a

Energy Bands in One Dimension

159

function is constant, the constant-energy surfaces at the Bragg plane are perpendicular to the plane.9 When q lies on a single Bragg plane we may also easily determine the form of the wave functions corresponding to the two solutions 8=8° + |l/K|. From (9.22), when 8 is given by (9.27), the two coefficients cq and cq_K satisfy10 cq = ± sgn (l/K)cq_K.

(9.29)

Since these two coefficients are the dominant ones in the plane-wave expansion (9.1), it follows that if UK > 0, then |