Mitin V.V., Sementsov D.I., Vagidov N.Z. Quantum mechanics for nanostructures

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C.5 The energy spectrum of an electron in a crystal 403

and the atomic state with the set of quantum numbers {q}:

(r) =



(r − a

). (C.76)

Here, a

is the translation vector of the crystalline lattice, and the coefﬁcients C

deﬁne

the probability amplitudes of ﬁnding an electron at a corresponding atom. Let us note that

the electron wavefunction in a periodic inner-crystal potential has, according to Bloch’s

theorem, in the general case the form

(r) = G

(r)e

ik·r

, (C.77)

where k is the electron wavevector and the function G

(r) is a periodic function. The

translation by a lattice vector a

leads to multiplication of the electron wavefunction in

the crystal by the exponential factor corresponding to this shift:

(r +a

) = e

ik·a

(r). (C.78)

From this expression it follows that the expansion coefﬁcients, C

,inEq.(C.76) must

have the following form:

= e

ik·a

. (C.79)

The form of the Bloch electron wavefunction in a crystal (Eq. (C.77)) indicates the

commonality with free-electron character of its behavior. The factor G

(r) modulates the

wavefunction of a free electron e

ik·r

with the periodicity of the crystalline lattice potential.

Taking into account the temporal dependence, the electron wavefunction in a crystal can

be presented in the following form:

(r, t) = G

(r)e

i(k·r−E

t/h

)

. (C.80)

C.5.2 Energy bands in crystals

The one-electron wavefunction (C.76) is a solution of the Schr

odinger equation with a

self-consistent inner-crystal potential, U

eff

(r), i.e.,

Hψ

(r) = E

(r), (C.81)

H =−

∇

eff

(r). (C.82)

The eigenvalue of the electron energy for the non-normalized wavefunction ψ

(r)is

deﬁned by the general expression



∗

Hψ



∗

, (C.83)

404 Appendix C. Crystals as atomic lattices

where dV = dx × dy × dz. By calculating the integrals in Eq. (C.83) and taking into

account orthonormalization of atomic wavefunctions, we obtain

= E

−





−



ik · a

. (C.84)

Here E

is the corresponding atomic level, and the average value of the perturbation of

an atomic potential calculated using the unperturbed state is deﬁned as







∗

(r)[U

eff

(r) − U

(r)]ψ

(r)dV, (C.85)

where U

(r) is the atomic potential. The summation in Eq. (C.84) is done as a rule over

the nearest-neighboring atoms which surround the atom which is chosen as the origin.

This is because the exchange integral, A

, responsible for the splitting of the level E

for a neighboring atom signiﬁcantly exceeds the exchange integrals for non-neighboring

atoms.

Calculation of Eq. (C.84) for a simple cubic lattice gives the following expression for

the energy dispersion in the band formed from atomic level E

= E

−





− 2 A

[cos(ak

) +cos(ak

)]. (C.86)

When the crystal is formed from individual atoms, the electron energy it had in an isolated

atom, E

, is decreased by





, and is spread out into an energy band as a result of

interaction with neighboring atoms. At the limits of the band the electron energy changes

periodically with the components of the wavevector k. The same decrease and spreading

into a band happen to each stationary energy level of an isolated atom. The width of each

allowed energy band is proportional to the exchange integral, E

= 4 A

, and is mostly

deﬁned by the overlap of wavefunctions of neighboring atoms. For valence electrons this

overlap is signiﬁcant and thus the width of the allowed energy band can reach several

electron-volts. For electrons of inner shells the splitting is small. Thus, for K-electrons

in a crystal of metallic sodium, Na, the splitting is just 2 ×10

−19

eV! Therefore, the

corresponding band has the form of a narrow energy level.

Let us discuss the dispersion relation for electrons in crystals using expression (C.86)

for a cubic lattice. It has very much in common with the equation for elastic waves in

a discrete chain of atoms. First of all, let us note that the substitution of wavevector

projection

→ k

2π

,α= x, y, z, (C.87)

where n

is an integer number, changes neither the dispersion relation nor the wavefunc-

tion. Therefore, instead of considering all possible values of the wavevector, it sufﬁces to

limit our consideration to the interval

−

≤ k

≤

, (C.88)

C.5 The energy spectrum of an electron in a crystal 405

+ 2A

E 2A

Figure C.10 The ﬁrst

Brillouin zone. Here



= E

−





(a)

Empty

Filled

(b) (c)

Empty

Filled

Empty

Overlap

Figure C.11 Energy bands

in the cases of (a) a

dielectric (E

> 5eV),

(b) a semiconductor

< 5eV),and(c)a

metal (E

= 0eV).

which for electron waves in crystals is called the ﬁrst Brillouin zone (see Fig. C.10). Thus,

the electron wavevector, k, in a crystal is not deﬁned unambiguously and therefore it is

called the quasiwavevector. This vector is related to the quasimomentum by

p = h

k. (C.89)

The allowed energy values for stationary states, E

(k), are limited to a band delimited

−





− 2 A

≤ E

−





+ 2A

. (C.90)

The energy intervals, E

, that lie between the allowed energy bands are forbidden for

electrons in stationary states (see Fig. C.11). These forbidden energy bands are called

bandgaps.

Using periodic boundary conditions, we can show that inside of allowed bands the

projections of the wavevector k

have discrete values

2π

,α= x, y, z, (C.91)

where integer numbers n

independently take on the values ±1, ±2,...,±N

/2, and

thesizeofthesampleisL

= N

a. Since the number of atoms in the sample is

406 Appendix C. Crystals as atomic lattices

enormous, i.e.,

N =

= nV, (C.92)

where n ≈ 10

−3

, for a macroscopic sample the distance between the adjacent levels

even in the widest band is not greater than 10

−15

eV. For this reason the discreteness of

the parameter k

and the energy E

(k) in the allowed bands can be neglected.

C.5.3 Filling energy bands

As shown previously, each allowed band with the set of quantum numbers {q}corresponds

to N stationary states, which can, according to the Pauli principle, contain no more than

2N electrons with opposite spins. Therefore, the way of ﬁlling bands signiﬁcantly depends

on the valency of the atoms forming the crystal. Let us ﬁrst consider a crystal of univalent

lithium, Li, whose atoms have the electron conﬁguration 1s

. N sublevels of the lowest

allowed band, which is formed from atomic levels with quantum numbers n = 1and

l = 0, are completely ﬁlled by 1s-electrons. N atoms have 2N such electrons. The next

allowed band is formed from atomic levels with quantum numbers n = 2andl = 0. Since

there is only one 2s-electron, N electrons can ﬁll only N/2 sublevels, i.e., this band is

only half-ﬁlled. The picture for the ﬁlling of the two highest energy bands (completely or

partly ﬁlled) of univalent metals is similar.

If the valency of an element is equal to two (or is even), then the last band must be

completely ﬁlled. However, if the adjacent bands overlap, which is widely encountered

in real crystals, then incompletely ﬁlled bands can have elements with arbitrary valency.

Thus, for beryllium, Be, with electron conﬁguration 1s

, the energy bands which

correspond to atomic levels with n = 2andl = 0 and with n = 2andl = 1overlap,

forming one wide band. Therefore, beryllium, Be, is a metal.

The difference in the form of ﬁlling of the highest allowed band is the basis for crystals’

classiﬁcation according to their conducting properties. The highest of the completely ﬁlled

bands is usually called the valence band. The maximum energy in this band is denoted

as E

and is called the top of the valence band. The lowest partially ﬁlled or completely

unﬁlled band is called the conduction band. The minimum energy in this band is denoted

as E

andiscalledthebottom of the conduction band. Metals are crystals whose valence

band is ﬁlled completely and whose conduction band is not completely ﬁlled. Dielectrics

are crystals whose valence band is completely ﬁlled and whose conduction band is empty.

The region separating the valence and conduction bands, called the bandgap,has,for

good dielectrics a width, E

, of several electron-volts (see Fig. C.11).

C.5.4 Effective mass

The solutions of the Schr

odinger equation (C.81) describe the behavior of electrons in

a crystal in the absence of external ﬁelds. However, the response of an electron sys-

tem to applied external ﬁelds is of practical interest. For analysis of electron behavior

in a crystal in an external ﬁeld an approximation called the effective mass method is

widely used. The effective mass itself can be introduced using the dispersion relation.

C.5 The energy spectrum of an electron in a crystal 407

Let us expand the energy E

(k) in a series near one of the extremum points k

in the ﬁrst

Brillouin zone:

(k) = E

) +





∂

∂k



0α

(

− k

0α

)

. (C.93)

To describe the motion of an electron in a periodic ﬁeld of a crystal in the same way as

we describe the motion of a free electron, let us introduce the tensor of inverse effective

mass, the components of which on the main axes have the form

∗



∂

∂k



0α

. (C.94)

Then, the electron dispersion relation in a crystal (Eq. (C.93)) takes the same form as that

of a freely moving particle whose inertial properties are different along each of the axes:

(k) = E

) +



− k

0α

)

∗

qα

. (C.95)

If all three components of the tensor 1/m

∗

are the same in crystals with a cubic lattice, we

can use the scalar effective mass m

∗

. In this case the dispersion relation can be presented

in the form

(k) = E

) +

(k − k

)

∗

. (C.96)

Near the bottom of the band, where the function E

(k) has its minimum, the effective

mass of the electron m

∗

is positive, since at the point of the minimum



∂

∂k



k=k

> 0. (C.97)

Near the top of the band the function E

(k) has its maximum, and the second derivative

(C.97)isnegativeandm

∗

< 0. The effective mass, just like an ordinary mass, relates a

force applied to the particle to the acceleration produced. If m

∗

< 0 then the direction

of electron acceleration is opposite to the direction of the applied force. This must not

be a surprise, since the effective mass of the electron takes into account the inﬂuence of

the periodic crystalline potential on the electron. The inﬂuence of a periodic ﬁeld and an

external force on the electron, which has wave properties, can lead to such an effect.

Let us write the one-electron Schr

odinger equation for an electron in a crystal in the

presence of external forces:

−

∇

(r) + [E

−U

eff

(r) + U (r)]ψ

(r) = 0. (C.98)

Here, the inner-crystal potential U

eff

(r) is a periodic function, which is usually unknown.

The function U (r) describes a non-periodic ﬁeld, which affects the electron through the

408 Appendix C. Crystals as atomic lattices

defects of crystalline structure and external electric and magnetic ﬁelds. Equation (C.98)

in the approximation of the effective mass for a cubic lattice can be reduced to the

following equation:

−

∗

∇

(r) +[E

−U (r)]ϕ

(r) = 0, (C.99)

where the unknown function U

eff

(r) is absent and, instead of the mass of a free electron,

, we introduce the electron effective mass, m

∗

. The new wavefunction ϕ

(r) is related

to Bloch’s modulation factor, G

(r) which is a property of the electron wavefunction,

(C.77)by,

(r) = G

(r)ϕ

(r). (C.100)

Equation (C.99) has the simplest form in the absence of external ﬁelds in an ideal (i.e.,

without defects) crystalline structure:

−

∗

∇

(r) = E

(r). (C.101)

Equation (C.101) describes the motion of an electron in a crystal as the motion of a free

particle with mass equal to the effective mass, m

∗

, and energy, E

, which deﬁnes a sta-

tionary energy state of an electron in one of the allowed bands. The electron wavefunction

in this case has the form of a de Broglie wave:

(r, t) = Ae

i(k·r−E

(k)t/h

)

, (C.102)

where the dependence of the electron energy, E

, on the wavevector, k,isdeﬁnedby

Eq. (C.86). Knowledge of the form of the dispersion relation, E

(k), and the wave-

function, ϕ

(k, t ), allows us to ﬁnd the most important characteristics related to the

behavior of the electron subsystem in general.

C.5.5 Quasiparticles – electrons and holes

In the expression for the wavefunction (C.102) the energy E

(k) is the total electron

energy in the energy band deﬁned by the set of quantum numbers {q}. It contains the

electron kinetic energy and energy of interaction with all other electrons and nuclei. Let

us expand the expression (C.86) in series over the wavenumbers near the extremum point

= 0:

(k) = E

−





− 6 A

+ A

= E

−

∗

, (C.103)

wherewehaveintroducedp = h

k and

−

= E

−





− 6 A

, (C.104)

∗

E

, (C.105)

C.5 The energy spectrum of an electron in a crystal 409

where E

is the width of the allowed band. From the equations obtained above it follows

that the magnitude of the electron effective mass, m

∗

, is inversely proportional to the

width of the band, E

. For the inner bands the magnitudes of A

and E

are small

and therefore the electron effective mass there is large. The consequence of this is larger

inertia of electrons of inner bands and their weak response to external inﬂuences. Only in

the conduction band is the magnitude A

such that the effective mass of an electron, m

∗

is similar to the mass of a free electron, m

. Thus, in the approximation considered here,

we can present the total electron energy in a crystal in the form of the kinetic energy of

some pseudo-electron with mass m

∗

= E

(k) − E

−

∗

. (C.106)

The electron in a crystal is called a pseudo-particle because its “momentum” is a

quasimomentum, i.e., it is a periodic quantity in momentum space and all its physically

different quantum states can be reduced to the ﬁrst Brillouin zone:

−

≤ p

≤

. (C.107)

Moreover, the mass of such an “electron” has not much in common with the mass of a

free electron and, as we have already noted, the effective mass of an electron at the top of a

band may even be negative. Therefore, we named it a “pseudo-electron,” which reﬂects the

unusual properties of this particle. Thus, with the help of the formal method considered

here, we can reduce the complex problem of the motion of electrons interacting with each

other and with nuclei to the problem of the motion of free pseudo-electrons.

Let us now discuss the question of how to present the dispersion relation near the

maximum of the E

(k) dependence, i.e., near the top of a band near the points k

0α

=±π/a.

In this case the expression for the electron energy can be written as

(k) = E

−





+ 6 A

− A

(k −k

)

. (C.108)

If we make the substitution k − k

= k



and shift the origin of the energy to the top of the

band, then the dispersion relation can be presented as

= E

(k) − E

=−A



∗

, (C.109)

where we have introduced p = h



and

= E

−





+ 6 A

, (C.110)

∗

=−

. (C.111)

Therefore, the pseudo-electrons near the top of the band are characterized by a negative

effective mass and a negative kinetic energy, which is not usual for the behavior of free

electrons. We can introduce near the top of the band instead of pseudo-electrons with

negative effective mass quasiparticles called holes, whose mass is positive. Since these

quasiparticles are related to vacancies of quantum states near the top of the band, their

410 Appendix C. Crystals as atomic lattices

charge is positive and equal to the charge of the electron. It is commonly accepted that

parameters of quasiparticles – electrons and holes – may be written with corresponding

indexes “n” and “p” that deﬁne their charge (negative and positive). Further, introducing

these indexes will allow us to omit the index “q”, which deﬁnes the set of quantum

numbers of an electron in a band.

In order to describe a hole in terms of its usual physical characteristics, its kinetic

energy is counted down from the top of the allowed and not completely ﬁlled band

(usually this is a valence band). The energy of a hole is positive since

=−

> 0, (C.112)

where p

=−p

and m

=−m

> 0. Here, p

and p

are the quasimomenta of the

electron and hole, which are related to the group velocities of the corresponding particles:

, (C.113)

−p

−m

= v

. (C.114)

Thus, the group velocities of these two quasiparticles are the same, i.e., v

= v

.For

quasiwavevectors there are the following relationships:

=−

=−k

. (C.115)

Thus, near the top of the energy band, we can consider the system of interacting electrons

in a crystal from two points of view. First, this system is an array of free quasielectrons

with negative charge having negative mass. Second, it is an array of free holes with positive

charge and positive mass, which are moving in the opposite direction to the electrons. The

description of the physical processes in a crystal which take place with the participation

of electrons near the top of the valence band with the help of the notion of holes is more

suitable and commonly accepted.

Example C.5. Under the conditions of the Hall effect the magnitude of the magnetic ﬂux

is B = 0.1 T and the current density along the x-axis in a semiconductor sample of n-type

is equal to j

= 10

−2

. Estimate the Hall voltage, ϕ

, and Hall constant, R

,ifthe

concentration of carriers is equal to n = 10

−3

and the length of the sample is equal

to L = 5 ×10

−3

m (see Fig. C.12).

Reasoning. The current density of charge carriers is related to their concentration and

velocity by

= nqv

. (C.116)

From the last expression we can ﬁnd the carrier velocity:

. (C.117)

C.6 Electrons in semiconductors 411

Figure C.12 The Hall

effect.

The Hall effect is the occurrence of a transverse (with respect to the current direction)

voltage in a sample placed in perpendicular electric and magnetic ﬁelds (i.e., B ⊥ E). This

voltage can be found from the condition that, when the current ﬂows along the x-direction,

the electric force, F

= qE

, and magnetic force, F

= qv ×B = j

B/n, which affect

the carriers counterbalance each other, i.e.,

, (C.118)

where E

is the electric ﬁeld intensity of the transverse Hall ﬁeld. Taking into account

that E

= ϕ

/L, we obtain

ϕ

= R

B, (C.119)

where the Hall constant is equal to R

= 1/(nq). For the semiconductors with their

different types of carriers we have

=±

, (C.120)

where e is the modulus of the electron charge. Using the values of n and e, we obtain

= 6.25 × 10

−3

−1

and ϕ

= 3.1 × 10

−3

V. By ﬁnding the sign of the Hall

constant, i.e., the direction of the Hall voltage, we can ﬁnd the type of current carrier in

semiconductors. Note that Fig. C.12 corresponds to charge of carriers q = e > 0.

C.6 Electrons in semiconductors

C.6.1 Intrinsic and extrinsic conductivity

Semiconductors are a special group of crystals. These materials have such a name because

their resistivity is intermediate between those of dielectrics and metals. Thus, at room

temperature the resistivity of metals is ρ ≈ 10

−8

 m; for dielectrics the magnitude of

this quantity is in the interval 10

−10

 m; whereas for semiconductors this quantity

412 Appendix C. Crystals as atomic lattices

is in the interval 10

−4

−10

 m (pure germanium, Ge, has ρ ≈ 0.5  m). The materials

which have semiconductor properties can be divided into two groups. The ﬁrst group

includes the so-called “elemental” semiconductors, which consist of atoms of only one

element. The second group includes semiconductor compounds, which consist of atoms

of two or more types.

The main representatives of the ﬁrst group are semiconductors such as silicon, Si,

whose atomic number is Z = 14, and germanium, Ge, whose atomic number is Z = 32.

Both elements have four valence electrons on their outermost shell. The crystalline lat-

tices of these semiconductors consist of atoms surrounded by four neighboring atoms,

each of which is bound to a particular atom by covalent two-electron bonds. Therefore,

in contrast to metals, whose valence electrons in a crystalline lattice become free, in

semiconductors at T = 0 K there are no free carriers. This makes a semiconductor have

the properties of a dielectric. However, there is a signiﬁcant difference between semi-

conductors and dielectrics. The energy which is necessary to break the electron bond is

deﬁned by the width of the bandgap, E

, which for different types of semiconductors

ranges from 0.01 eV to 2 eV, is signiﬁcantly smaller than that in dielectrics. For com-

parison, the width of the bandgap in Si is equal to E

= 1.1eVandinGeE

= 0.7 eV,

whereas good dielectrics have E

of about several electron-volts (for example, diamond has

= 5.4 eV).

The second group of semiconductor materials is large. We can mention ﬁrst of all

binary compounds of elements of the third and ﬁfth groups of the Periodic Table of the

elements (for example, GaAs and InSb). These compounds are usually denoted as A

and their properties are very similar to those of “elemental” semiconductors from the ﬁrst

group. Binary semiconductors also include A

compounds (for example, ZnS), metal

oxides (for example, Cu

O), and organic compounds.

External inﬂuences on such semiconductors (for example, heating or exposure to

radiation) may detach some of the electrons from atoms and free them. This means that

these electrons from the valence band move to the conduction band, where they occupy the

lowest levels. Being in the conduction band, electrons in an external electric ﬁeld behave

as the electrons in a metal would, i.e., they participate in the directed transfer of charge.

Simultaneously with the appearance of electrons in the conduction band, vacancies are

formed in the valence band, which are called holes, i.e., not completely occupied levels.

These levels may, because of external inﬂuences, be occupied by the electrons residing

on the lower energy levels of a valence band. The direction of a displacement of these

electrons in a valence band under the inﬂuence of an external electric ﬁeld is equivalent

to the direction of a displacement of positively charged holes. Thus, the existence of

holes makes additional input to the electrical conductivity of a semiconductor. Its total

conductivity is composed from electron and hole conductivities and, taking into account

that the concentration of electrons, n

, in the conduction band and that of holes, n

,inthe

valence band are equal to each other, the conductivity can be written as

σ = q

, (C.121)

where q

,µ

and q

,µ

are the charge and mobility of electrons and holes, respectively.

The electron and hole mobilities are similar in magnitude despite the motion of a hole being

a peculiar “relay-race” motion of many electrons. Therefore, we can assume that the total

conductivity is approximately twice the electron conductivity. This type of conductivity is