Marder M.P. Condensed Matter Physics

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Noninteracting Electrons in

Electric Field 463

£-

/eE

Figure 16.6. An electric field can

thought

as shifting energy bands up and down

function

spatial location. An electron that tunnels through the energy gap over

distance

can succeed in changing bands.

exp

f J I /-> /7c c\7c cTl

Want something with dimensions

-dX y Zmy {t

—

t){t

— t) of

energy

that is negative in the (16.43)

« JO '

gap, and that vanishes at

and

exp

2mE

Very rough estimate.

I2m£

eE\

(16.44)

(16.45)

This simple estimate

borne out surprisingly well

the detailed calculation

that follows.

Detailed Calculation.

solution

Schrödinger's equation with Bloch index

must be some linear combination

Houston states with the same index:

implies that

M0>

»'(')l^'*<.)>-

ih-

= W>

^IV')

= z2

Cn'(.

)£n'k(.t)\

(

)

n'k(t))) Because

<f>

obeys Eq. (16.33).

^2 -^-\4>n'k(,))+C„'(t)

—

\4)

kil)

Using Schrödinger's equation, and the fact that

(j>

depends upon

only through

(16.46)

(16.47)

(16.48)

(16.49)

464 Chapter 16. Dynamics of Block Electrons

{<l>nk(,)\W\lp)

(t)£.

nk(l)

9Cn

Pnk(t) I

ß(j)

'*(<)

eE.

(16.50)

(16.51)

For simplicity, restrict attention to a two-band model so that there are only two

coefficients C\, and C

. The first of these begins at amplitude 1 when t = 0; the

second begins at 0, and the goal is to see how fast it will grow. To leading order

one has

dC\

C\£\k(,)

= ih-

=> C\ = exp

Now look at the equation for C

. Defining

dt' £

\k(,')

(16.52)

(16.53)

«2(0 = C2W e

one has

dt'l

2k(,')

Because a and C are related by a phase factor, ( 16.54)

they can be used equally well to determine the

probability that the electron arrives in band 2.

a.2

2*(0

exp

lk(,,)

■\k(i'))

(16.55)

dk ' h

Equation (16.55) is sufficiently difficult to solve exactly that the greatest benefit

comes from evaluating its leading behavior. Krieger and Iafrate (1986) evaluated

the matrix element between d(p/dk and

and found that it was of order L/N, where

L is the system length and N is the number of lattice points. This matrix element is

largest when k is near the Brillouin zone boundary, and oscillates as a function of

k. However, variations in the matrix element are unimportant in comparison with

much more rapid oscillations of the exponential that make a exponentially small.

It is reasonable to treat the matrix element as a constant, and to estimate the rate of

tunneling from band 1 to band 2 after time 7 as

dt {&2k(i') — &\k(,'))

{7)

dt -r- exp

(16.56)

Take 7 to be the time needed for k to advance by one reciprocal lattice vector

K = 2nN/L. Changing the variable of integration from t to k gives

(7)

2-KN/L

dk exp

dk?(£

2k'

■\k'

Use Eq. (16.42) for the change of

variables.

— I

\_eE JO

(16.57)

Assume, as in Figure 16.6, that the tunneling is to take place between two

parabolic bands, the lower of the form £ = E

—

ft k

/2m* and the upper of the

form £ = £,

+ h

/2m*. Then defining the reduced mass

gives

-ml

1 i

m*-

(16.58)

Semiclassical Equations from

Wave

Packets 465

2t/2

'■■2k'

■Ik' £,+

2m*

(16.59)

Placing Eq. (16.59) into Eq. (16.57) produces an integral that is impossible to per-

form exactly, but that easily can be estimated by the method of steepest descents

(Appendix B.4). Let q be the point in the complex plane where

2^2

£,+

¥q

2m*

(16.60)

Then the method of steepest descents gives immediately that

&2{T)

~exp

exp

' -i H _,,, .

-2I

3eE

-28

3/2

12m*

3eE

2m*

In the original calculation by Zener (1932),

the energy bands did not have the form in

Eq. (16.59), but were instead linear except for

a small region where they avoid one another.

The tunneling rate is then

exp[-2TTE

/(eE rf(£, - E

)/dk)].

exp f—3.41

■

[£,

/éV]

[m*/m]

/[E-cmV-

(16.61)

(16.62)

(16.63)

(16.64)

Equation (16.64) shows that because band gaps are typically 1 eV for metals,

and because it is difficult to obtain voltages larger than 10

V cm

', the probability

for an electron to tunnel from one band to another is on the order of exp[—10

This probability is negligible. In semiconductors, on the other hand, electric fields

can reach 10

V cm

-1

, effective masses may be one-tenth of the electron mass,

band gaps are typically less than

eV, and Zener tunneling is therefore a common

feature in the reverse bias of heavily doped diodes, as discussed by Sze (1981),

pp.

516-536.

16.4 Semiclassical Equations from Wave Packets

16.4.1 Formal Dynamics of Wave Packets

As pointed out in Section 7.2.4, in any situation described by a wave equation

where one wishes to speak of "particles," in fact one is referring to wave packets.

The most powerful and general way to derive a semiclassical equation of motion

for electrons is to consider carefully how wave packets evolve.

While formal analysis essentially recovers the results of Section 7.2.4, it does

not exactly reproduce them. The dynamical equations for electrons acquire terms

not presented in Eqs. (16.11) or (16.12). These extra terms should in principle

always be used to describe electron dynamics. In practice, they are frequently

neglected, for two reasons. First, they vanish by symmetry in any centrosymmetric

crystal. Second, they are unfamiliar and difficult to compute.

Sundaram and Niu (1999) discuss a formal method that can be used systemati-

cally to find electron dynamics. It is the same as the one used in Section 15.5.4 to

466 Chapter 16. Dynamics of Block Electrons

find an equation of motion for the wave function of superfluid helium, the effective

Lagrangian of Appendix B.

It is first necessary to define carefully a wave packet W-

(r) centered in space

at r

and dominated by wave vector k

. In the presence of a magnetic induction

whose vector potential is A, the proper definition is

]

• 2/—i-/s

-Z

~ N is the number of unit cells in the system. £

W- r (r) = / Wrr e~ '

''ll>7

(V). '

summed over a single Brillouin zone, ip];

' ' vAf _ ' are solutions of Schrödinger's equation, all

k sharing band index n.

(16.65)

The phase factors in

(

16.65) are needed in order to ensure that W vary slowly as

and

vary. Bloch functions naturally oscillate with phase factor e'

and, in the

presence of a magnetic field, acquire an additional phase factor e~'

drA

(

incorporating these phase factors into the definition of the wave packet, the hope is

to eliminate wild oscillations in W. The wave packet must be normalized, so

i =

KiKi) =4 E /0^-

^^T&wm

kk' ' kk'

(16.66)

=> 1 = \ \wrr I . ^ is normalized over the volume V of the crystal. (16.67)

Reciprocal space Real space

Figure 16.7. A wave packet involves a range of wave vectors that is considerably smaller

than the width of the Brillouin zone, so the spatial extent of the wave packet is much

larger than an atomic spacing. By taking the spatial extent to be on the order of 100 Â,

it is possible to have the packet tightly localized in comparison with external potentials,

making it possible to speak simultaneously of the wave number k

and position ?

. of an

electron.

The shape of the wave packet is dictated by the weighting function w, which

in addition to being normalized, must be taken to have a rather special form. Its

amplitude

|^|j_^

depends only upon k

—

. This amplitude determines the range

A^ of wave numbers involved in the wave packet, and also determines its spatial

extent Ar ~ l/Ak. One should think of Ak extending about 1% of the spacing

of the reciprocal lattice, which implies that Ar extends over about 100 unit cells

Semiclassical Equations from

Wave

Packets 467

as shown in Figure 16.7. Electrical fields, magnetic fields, and other external po-

tentials must therefore vary slowly on a scale of around 100 Â, a limitation being

violated more and more frequently in modern electronics.

The phase of w carries information about the actual spatial location of

the

wave

packet, and it must be chosen carefully if it truly is to be centered at r

. In particu-

lar,

lt-*

(

*~*

'^'

(16

68)

where

is precisely the Berry connection of

Eq.

(8.59),

%l=i

d7ui(7)—u

(7). (16.69)

< 'dk

The demonstration that such a wave packet truly is centered at

follows from the

computation

î\r-r

)

? \r\W

?)-r

Because the packet is

(16.70)

rc^c

normalized.

Use Eq. (7.45) for the periodic function u to replace ip in (16.65). (16.72)

/ —Y

v& m,z w^rW,(r)^lv(*'-*H^) (16.73)

N tr!

kkc k k

dik'

k'k

- j

^«î^^^-^^^^Wl

(16-V4)

k'k

Change the

sum to an integral with (6.11), integrate by parts, and return to the sum.

E hk'

t^^

(16

75)

k'k

As in Eq. (7.26),

indicates integration over a single unit cell. No reciprocal lattice

vectors K needed because k and k' lie in a single zone.

- [ drY Hr

uUr)

^\w

ur(7)}

(16.76)

KKc

dr m- (7)—U7 (7) In

, , =

ftr

-Dk

(16.77)

< '

' dik

kkc

k=k

<<c

\u\

integrates to 1; \w\

sums to 1; and

d\w\-

/dk

0 because of the maximum

a.tk

=> (W-r \r\W~7

) =

Hence the definition in

(16.78)

> r

i (16.69).

16.4.2 Dynamics from Lagrangian

Given this wave packet, one can obtain equations of motion for r

and k

by evalu-

ating the Lagrangian

468 Chapter 16. Dynamics of Block Electrons

$i=—[P+^Q\

+ U{r) (16.80)

[

+ U{r)]ll>i

fijV^- Suppress the band index

rc.

(16.81)

Magnetic fields are included through the vector potential A, and electric

fields

are

included through the scalar potential V. Because the focus is on wave packets,

which are localized in space, one does not have to worry about the impossibility of

incorporating the scalar potential rigorously into periodic boundary conditions.

Evaluation of the expectation values needed to find the Lagrangian (16.79) is

a matter of evaluating various integrals. Whenever \w\Z

appears, it should be

treated as a delta function centered at k

. Details are left to Problem 5, and the

results are

€,Y

dA(Y

) —* • —■ —*

<rç&

/Ä

}

~f '

+hkc

?c+hkc

►

(16

82a)

Tcl

\%-eV(r)\W

Tcl

) = £

-B.m

-eV(r

) (16.82b)

To obtain this result in the presence of the magnetic field requires the phase factor in-

volving A in Eq. (16.65), and it requires that the wave packet not be spatially extended

on scales where the electric and magnetic fields vary noticeably. The term m^ is an

intrinsic magnetic moment of the wave function and comes formally from expanding

A about r

. c.c below is "complex conjugate."

with the orbital magnetic moment of the wave packet

m-^

given by

eft

1 f

dm *

in?

= /

dr\—Z--%i uy

x [— +k

] u? +c.c. (16.82c)

*< 2mc2 Jn

[

dik

dir

Finally, employing Lagrange's equations, one obtains

ö£

_, du d

„

— = -—r-

and — =

—

^. (16.83)

dtdr

Problem 6 shows that after dropping the subscript c on k and r, one has the semi-

classical equations of motion

nk=-eE--7xB

(16.84a)

•

\dh i -,

r= ^-kxQ, (16.84b)

n Ok

where

£j = £

-B-%, (16.85a)

B(r) = — xA(r),

and

(16.85b)

- -

n(k) = —

x%(k).

(16.85c)

Semiclassical Equations from

Wave

Packets 469

The terms involving the anomalous velocity Q and wave packet magnetic mo-

ment rhp can sometimes be omitted from semiclassical equations

motion

for

electrons. One reason is that they often vanish by symmetry. In particular, they

vanish for crystals that have both time-inversion symmetry, and spatial inversion

symmetry. Under time inversion, k, d/idr, r, and B flip sign while

and

are

unchanged; therefore Ù(k)

—Çl(—k)

and

—«*_-£. Under spatial inversion,

r, k,

%£

and

flip sign while B does not, so Q(k) = Q(—k) and

in-^

= m_^. These

two symmetries are only compatible

Ù and

vanish for all k. Thus the terms

can be omitted for many monatomic solids. However, they should not be omitted in

crystals without inversion symmetry, such as GaAs, or in crystals with spontaneous

magnetic moments such as iron. The existence of such terms has been known for

a long time, since work

Karplus and Luttinger (1954) and Blount (1962), and

the fact that they nevertheless have often been left off is due to

desire for sim-

plicity. The careful development of spintronics (Section 26.5) will however require

these terms to provide an accurate account of how spin currents respond to external

fields.

Limitations of Semiclassical Dynamics. Validity of the semiclassical equations

of motion requires four conditions:

The spatial scales

all external potentials must be much larger than inter-

atomic spacing, making it possible to construct wave packets spanning many

unit cells, but seeing the external potentials as very slowly varying.

The magnitudes of the electric fields cannot be too large, or else they induce

Zener tunneling between bands. To prevent this tunneling in a solid with band

gap £„, one must require, according to Eq. (16.64),

(16.86)

Electric fields this large are impossible to obtain in metals except near points

of degeneracy where band gaps shrink to zero. With energy gaps on the order

eV, and

\/kp

on the order

1 Â, the electric field needs to be on the

order of

V/Â, or 10

V/m. The largest fields that actually can be generated

metal are six orders

magnitude smaller, because electrons are quite

effective in screening external fields out of existence.

The magnitudes

magnetic fields cannot be too large. The characteristic

energy of electrons in magnetic fields

derived from their orbital period

and is 2TTH/7. For free electrons, as shown in Section 21.2, T

2irmc/eB,

which can used to obtain the estimate 2nh/7

1.16- 10

[Z?/T]eV, measur-

ing the magnetic induction in Tesla. By analogy with Eq. (16.86), estimate

that magnetic fields will not induce interband transitions so long as

2irh/7<^E

J-f-.

(16.87)

470

Chapter 16. Dynamics of Block Electrons

= hk-

—*

= #%,

In magnetic fields of 10 T, electrons are therefore able to jump across band

gaps on the order of

leV.

As will be shown in Section 23.4, when incoming electrical fields oscillate at

frequencies

such that

Hu>

= £

, they excite electrons between bands. There-

fore semiclassical electron dynamics only applies in the presence of slowly

varying external fields.

Hamiltonian Dynamics. Whenever classical dynamics are described by a La-

grangian £(ß, Q), it is possible to derive also a Hamiltonian, using the formulae

"K = y^

QlPl

— £; Pi = ——

•

The

s here are the three

components of r

(16.88)

I OQi and the three components of ï

In the present case, dropping the subscript c from r

and

one has the canonical

momenta p and

hk = p + eA/c (16.89a)

(16.89b)

and from Eq. (16.88) obtains the Hamiltonian

■X = Z-

-eV(r) + {e/2mc)B-L

8,(p

+ eA/c) -eV(r) + {e/2mc)B-L

. (16.90)

The Hamiltonian is a constant of the motion, a fact that is especially useful in

visualizing electron dynamics when the electrical potential V and L vanish. One

simply draws contours on the energy surface £

and concludes that k travels along

these contours. Closed orbits are those for which k is periodic in the repeated zone

scheme, and open orbits are those for which k continually increases in the repeated

zone scheme, as illustrated in Figure 16.8.

16.5 Quantizing Semiclassical Dynamics

Having gone to all the trouble of coercing quantum mechanics into a classical

form, it is interesting to move slightly backwards and quantize the semiclassical

dynamics. This task may be accomplished by considering the overall phase of the

wave packet.

A solution of Schrödinger's equation should obey

ih—\W) = H\W). (16.91)

Because wave packets are not exact solutions of Schrödinger's equation, they do

not exactly obey Eq. (16.91). However, they should approximately obey

ih—\W)=M\W), (16.92)

Quantizing Semiclassical Dynamics 471

Figure 16.8. Energy contours on the Fermi surface of copper, showing open and closed

orbits.

where

is the Hamiltonian of

Eq.

(16.90), so that as time evolves, the wave packet

acquires an additional phase

g~' I . Because

is a constant of the motion. (16.93)

Consider now a case in which after some time t = T the dynamics of the wave

packet brings it back to its location at time t = 0; the position ?(T) returns to r(0),

and the wave number k(7) arrives at Ä:(0) + K, where K is any reciprocal lattice

vector. If the phase factor in (16.93) differs from 1, the wave packet interferes

destructively with its past

self,

but if it equals 1, the packet reinforces itself and

can establish a resonant standing wave. The condition for this resonance to occur

should roughly be of the form

"KJ = 2nhj. Where ; is some integer. (16.94)

More careful examination of the resonance conditions for wave packets, provided

by Littlejohn (1986), leads instead to the Bohr-Sommerfeld quantization condition

the

"old" quantum mechanics, a good approximation for large quantum numbers

2Trh(j

+ v) =

I dtJ2

fj2

(

)

The Maslov index v is a constant, frequently 1/2. This additional constant does

not lead to any physical consequences in any of the following arguments, and it

can be viewed as absorbed into T in all the formulae following Eq. (16.98).

Employing the canonical momenta calculated in Eq. (16.89) gives the quanti-

zation condition

- - eÄ

dk-Jl~

+ dr-[k-—} = 2irj (16.96)

472

Chapter 16. Dynamics of Block Electrons

- - eA

•

($£ — 7) —

■

— =

2-nj. Integrating by parts.

(16.97)

16.5.1 Wannier-Stark Ladders

One way in principle to send electrons into closed orbits is to consider

system

with uniform electric field

E =

—VV and no magnetic field. The wave vector

k evolves linearly in time as

k =

—eEt/H.

Because 7 = d8./dhk flips sign when

—>

—

k, so long as

points in the direction of a reciprocal lattice vector, the

electron is supposed to execute periodic motion. Defining

T= i dk-%,

Because k returns to itself by increasing by a

(16.98)

J reciprocal lattice vector, the integral is a line

integral from 0 to K.

Eq. (16.97) becomes

2TTJ=6

dk-(ül

-r)

dk-r =

T-K

(r)

(7) is defined to be the time average of

and gives the center of the

orbit.

r =

r-2nj

(16.99)

(16.100)

Figure 16.9. The Wannier-Stark

ladder

is a

collection

electrons

trapped in Bloch oscillations by an

intense electric field, and spaced at

intervals of

2TT/K,

where K is a re-

ciprocal lattice vector.

2-K/K

pure metal at very low temperatures, Eq. (16.100) predicts no electrical

conductivity. Each electron should circle around its unit cell in a closed orbit, as

shown in Figure 16.9. The reason this result is never observed in practice is that

electrons cannot execute even one oscillation before undergoing a scattering event

that ruins the necessary quantum coherence, and the huge electric fields would

produce Zener tunneling. Problem 1 is devoted to estimating orders of magnitude

that show how difficult

would be to observe the phenomenon in

metal like

copper.

Wannier-Stark ladders have been observed by Mendez et al. (1988) in GaAs

superlattices, although the experiments are difficult, and the results not entirely in

accord with the simplest theoretical account. Cleaner results are reported in optical

lattices by Wilkinson et al. (1996).