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

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Chapter 7

Quantization in nanostructures

In Chapters 3 and 4 we have discussed electron behavior in potential wells of

various proﬁles and dimensionalities. We have established that localization of

electrons in such potential wells, regardless of their form, leads to the discretiza-

tion of the electron energy spectrum whereby the distance between energy levels

substantially depends on the geometrical size of the potential wells. If this size

is macroscopic then the distance between the energy levels is so small that we

can consider the energy spectrum to be practically continuous (or quasicontinu-

ous). Electrons in metallic samples of macroscopic sizes have this kind of energy

spectrum. Another limiting case is that of small clusters consisting of just a few

atoms, where the distance between energy levels is of the order of electron-volts.

Gradual decrease of one or several geometrical dimensions of the potential well

from macroscopic to about 1 µm practically does not change the form of the

electron energy spectrum. Very often macroscopic materials (or macroscopic

crystals) are referred to as bulk materials or bulk crystals. Changes happen

only when the size of structures is of the order of or less than 100 nm. Such

structures are called nanostructures. The change of the electron spectrum from

quasicontinuous to discrete implies changes in most of the physical properties

of nanostructures compared with those in bulk crystals. In this chapter we will

consider the main peculiarities of the electron energy spectrum in nanostructures

of various dimensionalities.

7.1 The number and density of quantum states

Electric, optical, thermal, and other properties of macroscopic crystalline mater-

ials signiﬁcantly depend on the energy states of electrons. Let us deﬁne two

quantities, which are important for describing the features of the electron energy

spectrum in these materials. The ﬁrst quantity, N (E), is the number of quantum

states corresponding to the energy interval (0, E). The second quantity, g(E),

called the density of states, deﬁnes the number of quantum states, N , correspond-

ing to the unit energy interval in the vicinity of E. In order to calculate these

quantities we consider ﬁrst a three-dimensional potential well, which represents

193

194 Quantization in nanostructures

bulk crystal. Such a structure is called three-dimensional because of the num-

ber of dimensions along which the electron can freely move. In such a case the

electron momentum space is also three-dimensional. For the analysis of elec-

tron behavior in a crystal in an external ﬁeld an approximate method, which is

called the effective-mass approximation, is widely used. In this approximation the

Schr¨odinger equation for an electron with free electron mass, m

, in the periodic

inner-crystal potential is reduced to the Schr¨odinger equation which describes

the behavior of free electrons with mass equal to the so-called effective mass, m

∗

We will show in Section 7.6.3 how to deﬁne the effective mass of an electron.

To ﬁnd how electrons are distributed over the allowed quantum states in a

crystal with linear dimensions L

, L

, and L

, where they can freely move, we will

assume the boundary conditions (4.4): ψ (0, y, z) = ψ(x, 0, z) = ψ(x, y, 0) =

ψ(L

, y, z) = ψ (x, L

, z) = ψ(x, y, L

) = 0. The components of momentum

according to Eq. (4.15) can take the following values:

π h

, p

π h

, p

π h

, (7.1)

where n

=1, 2, 3, . . . and α =x, y, and z. We will consider ﬁrst the case of large

, L

, and L

. Thus, the spacing between components of the momentum, p

π h

, is small and we can consider the momentum, p,tobequasicontinuous.

Therefore, in order to calculate the number of states we can use integration over

p instead of summation.

Let us choose an inﬁnitesimally small volume of momentum space, dV

around some momentum p:

= d p

d p

. (7.2)

It is easy to carry out calculations for a macroscopic crystal, i.e., for macroscopic

values of L

, L

, and L

. In this case the change of momentum projections, p

, and p

, is practically continuous and the electron motion can be considered

as a classical motion. Taking into account Eq. (7.1), we obtain the following

expression for the quantum numbers n

, which deﬁne the number of quantum

states:

π h

, (7.3)

where α = x, y, and z. On differentiating Eq. (7.3), we ﬁnd the number of

quantum states dn

in the interval dp

π h

d p

. (7.4)

The number of quantum states is equal to the number of sets of n

, n

, and

in this three-dimensional interval of numbers dn = dn

.Wehaveto

take into account the fact that each value of n

corresponds to the two values

of p

with the same absolute value, but with opposite signs. Therefore, the

total number of states in the three-dimensional interval of momentum space is

7.1 The number and density of quantum states 195

equal to

dn =

(2π h

)

d p

. (7.5)

In this expression the physical quantity V = L

represents the volume in

real space which is available for an electron, and the quantity dV

= d p

d p

represents the differential volume element in the momentum space available for

an electron. In the case of a free electron the momentum space is isotropic

since all directions are equivalent. Therefore, for the differential volume of the

momentum space, we can use spherical coordinates and write

= d p

d p

= 4π p

d p. (7.6)

Taking into account all the above, the expression (7.5) can be rewritten as

dn(p) =

2π

d p. (7.7)

The total number of states in the momentum space of radius p is deﬁned as

follows:

N ( p) = 2



dn(p) =



d p =

3π

. (7.8)

Note that we put a factor of 2 in front of the integral in Eq. (7.8). This is because

the set of quantum numbers n

, n

, and n

does not completely deﬁne the state

of an electron. Because the electron has its own internal angular momentum

(spin) (see Section 6.3), the quantum state of the electron has to be deﬁned by an

additional quantum number m

=±

. (7.9)

According to the Pauli exclusion principle, two electrons with opposite spins can

be in a state with the same momentum p. Therefore, the number of quantum

states is twice as large as it otherwise would be.

The number of states that corresponds to the unit interval of momentum space

is given by the density of states, g(p):

g(p) =

dN ( p)

d p

. (7.10)

We see that with increasing momentum, p, the number of states, N (p), increases

proportionally to the momentum raised to the third power, and that the density

of states, g( p), increases proportionally to the square of the momentum.

In many cases it is more convenient to consider energy, E, rather than momen-

tum, p, as a variable. In order to do this, we have to perform a transformation

of the equations from the momentum space to the energy space. Taking into

account that the energy of a free electron has a quadratic dependence on the

momentum, the electron surface of equal energy in the momentum space, p,isa

196 Quantization in nanostructures

sphere of radius p:

E =

∗

, (7.11)

dE =

∗

d p, (7.12)

and

d p =



∗

dE . (7.13)

By substituting Eqs. (7.11) and (7.13) into Eq. (7.7) we obtain the following

expression for the number of states in the energy interval dE:

dn(E) =

√

2Vm

∗

3/2

2π

√

E dE. (7.14)

The total number of quantum states in the interval (0, E) is deﬁned by the integral

N (E) = 2



dn(E) = 2

√

2Vm

∗

3/2

3π

3/2

. (7.15)

For the density of states, g(E), i.e., the number of states that corresponds to the

unit energy interval, we obtain

g(E) =

dN (E)

√

2Vm

∗

3/2

√

E. (7.16)

We see that the number of states, N (E), for the electron in a three-dimensional

space of macroscopic volume V increases with increasing energy proportionally

to E

3/2

, and that the density of states, g(E), increases proportionally to

√

E.Both

quantities depend linearly on the volume, V , and, therefore, they can be deﬁned

per unit volume. In this case, the volume V in Eqs. (7.15) and (7.16) is canceled

out. Figure 7.1 shows the electron’s surface of equal energy in the momentum

space, E(p), and Fig. 7.2 shows the dependences of the number of states, N , and

the density of states, g, on energy, E, for the electron’s three-dimensional motion

in a macroscopic crystal.

Note that Eqs. (7.2)–(7.16) can be used only in the case of crystals with

macroscopic volume V . All three dimensions, L

, L

, and L

, are so large that

the differences between different values of electron momentum and energy are

small, and thus the magnitudes p and E can be considered as continuous (or

quasicontinuous) variables.

If there are N electrons in the volume V , then at zero temperature they occupy

the lowest energy states and on each level there are two electrons with opposite

spins deﬁned by Eq. (4.18).

The distribution of electrons among the energy levels at an arbitrary temper-

ature, T , is described by Fermi–Dirac quantum statistics for particles with spin

=±1/2. The probability of ﬁnding an electron in the state with energy E is

7.1 The number and density of quantum states 197

E(p)

Figure 7.1 The surface of

equal energy in the

momentum space for the

electron’s

three-dimensional

motion. Here,

|p|=



+ p

, g

g(E) ~ E

1/2

N(E) ~ E

3/2

Figure 7.2 The

dependences of the

number, N(E), and the

density of states, g(E), on

energy, E, in the case of

an electron’s

three-dimensional motion

in a crystal.

given by the Fermi–Dirac distribution function, f (E):

f (E) =

(E−E

)/(k

T )

+ 1

, (7.17)

where E

is the Fermi energy and k

is Boltzmann’s constant. From Eq. (7.17)

it follows that at T = 0 K the probability of occupation of all the states with the

energy E ≤ E

is equal to unity, and the probability of occupation of the states

with energy E > E

is equal to zero (see Fig. 7.3). Thus, the Fermi energy, E

deﬁnes the energy up to which all the energy states at T = 0 K are occupied.

Energy states higher than E

at T = 0 K are unoccupied.

Given the importance of the notion of the Fermi energy, let us determine E

at T = 0 K. Let us assume that there are N electrons in the volume V . Since

198 Quantization in nanostructures

f (E)

T = 0 K

0.5

> T

Figure 7.3 The

Fermi–Dirac distribution

function, f (E), for various

temperatures. The

distribution functions at

temperatures higher than

0Kareshownbydashed

and dash–dotted lines.

Here, T

> T

two electrons with opposite spins occupy each available state, the number of

electrons, N , that can be placed in all states from E = 0toE = E

is equal to

the number of states N (E

). According to Eq. (7.15) this number is related to

the Fermi energy, E

,by

N = N (E

) = 2

√

2Vm

∗

3/2

3π

3/2

. (7.18)

From the above equation we obtain the following expression for the Fermi energy:

∗



3π



2/3

. (7.19)

Let us estimate E

for the conduction electrons in a metal, for example in

copper. For copper the electron effective mass is close to the free electron mass:

∗

≈ m

. The electron density for copper is equal to ρ = N/V ≈ 10

−3

Thus, the Fermi energy is about several electron-volts and the distance between

energy levels is inﬁnitesimally small compared with E

. Therefore, the energy

spectrum of the electron gas in metals can be considered to be quasicontinuous.

Example 7.1. An electron is conﬁned in a cubic box with impenetrable walls

and with the edge lengths L = 1 µm. Find the distance between the levels E

113

and E

122

, and compare this energy with the electron thermal energy, E

,atroom

temperature:

T ≈ 38.4meV. (7.20)

Determine whether the quantum-mechanical approach is required or the clas-

sical approach with a quasicontinuous spectrum is applicable. Assume that

the mass of the electron in the box is equal to the electron mass in vacuum,

= 9.1 × 10

−31

kg.

7.2 Dimensional quantization and low-dimensional structures 199

Reasoning. According to Eq. (4.18), the energy level E

122

has the same mag-

nitude as the other two states with the sets of quantum numbers (2, 1, 2) and

(2,2,1):

122

= E

212

= E

221

+ 2

). (7.21)

The energy level E

113

has the same magnitude as the other two states with the

sets of quantum numbers (1, 3, 1) and (3, 1, 1):

113

= E

131

= E

311

+ 1

+ 3

). (7.22)

The distance between the energy levels E

113

and E

122

113

− E

122

= 8.75 × 10

−7

eV = 8.75 × 10

−4

meV. (7.23)

According to Eq. (7.23), in the cubic box of dimensions 1 µm × 1 µm × 1 µm,

the distance between energy levels is so small in comparison with the thermal

energy at room temperature of 38.4 meV (see Eq. (7.20)) that we can consider

the electron energy spectrum to be quasicontinuous. It follows from the above

expression that in order to increase the distance between the adjacent energy

levels it is necessary to decrease the size of the region within which the electron

is conﬁned.

7.2 Dimensional quantization and

low-dimensional structures

As we have shown previously, the quantum properties of an electron become

apparent when the electron de Broglie wavelength becomes comparable to the

size of the region within which the electron motion takes place. For estimates

of this region we have to take into account that electron motion takes place not

in vacuum but in a certain medium at a certain temperature. Therefore, we have

to replace the mass of the electron in vacuum, m

, by its effective mass in a

medium, m

∗

. In Section 7.6.3 we will present a detailed discussion about the

effective mass of an electron, but for now we note that the electron motion in a

crystalline material is not completely free since it takes place under the inﬂuence

of the inner-crystal potential of atoms. An electron effective mass is introduced

to take into account the crystal potential and, in many practical cases, to describe

the motion of an electron in a medium, under the inﬂuence of external ﬁelds,

using the conventional equations in which the free-electron mass, m

, is replaced

by the effective mass, m

∗

, of an electron in the particular medium.

We will use the effective de Broglie wavelength, λ

∗

, of an electron which is

connected with its effective mass, m

∗

, in a crystalline material, as a criterion for

200 Quantization in nanostructures

(a)

(b) (c)

Figure 7.4 Different types

of nanostructures: (a) a

quantum well, (b) a

quantum wire, and (c) a

quantum dot. A and B

denote the materials

which constitute the

nanostructures, with A

being the material of the

nanostructure itself and B

being the material of the

surrounding matrix. The

nanostructure is referred

to as free-standing if the

surrounding matrix, B, is

vacuum.

a space quantization:

∗

2π h

∗

2π h

√

∗

. (7.24)

In the case when electron motion along only one direction is limited (for example,

along the x-direction) (see Fig. 7.4(a)), i.e., when the condition

≤ λ

∗

(7.25)

is satisﬁed, the electron energy states corresponding to the motion in this direction

are quantized. At the same time the electron motion in the other two directions

(y- and z-directions) stays free, with the continuous energy spectrum. Such a

structure is called a quantum well (QW).

In the case when the electron motion is restricted along two directions (for

example, along the x- and y-directions), i.e., when the conditions

≤ λ

∗

and L

≤ λ

∗

(7.26)

are satisﬁed, the energy which corresponds to these two directions of motion is

quantized. Such a structure is called a quantum wire (QWR) (see Fig. 7.4(b)).

In the case when the electron motion is restricted along all three directions,

i.e., when the conditions

≤ λ

∗

, L

≤ λ

∗

, and L

≤ λ

∗

(7.27)

are satisﬁed, the energy spectrum of such a structure, which is called a quantum

dot (QD) (see Fig. 7.4(c)), is totally quantized. Quantum dots can take the form

of a cube, a short cylinder, a disk, or a sphere of nanometer size, and they have

the potential to become in the future the main building blocks of nanoelectron-

ics. For example, from a set of quantum dots we can make artiﬁcial crystals

with desired parameters. In semiconductor crystals, which are used for building

nanostructures, the ratio of the electron mass in vacuum, m

, to its effective

mass in a semiconductor, m

∗

, is greater than unity. For example, for GaAs,

7.2 Dimensional quantization and low-dimensional structures 201

which is one of the most widely used semiconductors in nanoelectronics, this

ratio is approximately equal to 10: m

∗

≈ 10. Therefore, at room tempera-

ture the de Broglie wavelength of an electron in GaAs is of the order of 10 nm

according to Eq. (7.24). This deﬁnes in accordance with Eqs. (7.25)–(7.27)the

size of structures at which the quantization of the electron spectrum becomes

noticeable. Note that on lowering the temperature by a factor of 100, i.e., at

temperatures of about 3 K, λ

∗

increases by one order of magnitude. Therefore,

at 3 K the quantization is important in structures with sizes of the order of

100 nm.

The simplest nanostructure with the quantum limitation along one of the

directions, the quantum well, is a thin semiconductor ﬁlm with a thickness of

about L

∼ λ

∗

. The electrons are mostly localized in the ﬁlm and cannot escape

the ﬁlm because the energy of an electron outside of the ﬁlm is higher than the

energy of an electron within the ﬁlm, i.e., there is a potential barrier for the

electrons outside of the ﬁlm. The height of the potential barrier that forms a

quantum well for electrons in the ﬁlm is equal to the work function, A

, which

for most semiconductors is about 3–5 eV. This energy is two orders of magnitude

higher than the energy of thermal motion of free electrons, E

= 3k

T/2. At

room temperature, E

is about 4 ×10

−2

eV and E

decreases linearly with

decreasing temperature, T .

We can estimate the ground-state energy of an electron in a quantum well

using the uncertainty relationships. In a direction of restricted motion the electron

momentum projection is equal to p

∼ p

∼ h/L

. Therefore, for the energy

of the ground state we get

∼

∗

∼

∗

. (7.28)

In a GaAs quantum well with L

= 20 nm and with the effective mass of

the electron equal to m

∗

= 0.067m

the energy of the ground state is equal to

≈ 0.1 eV. The distances between the adjacent lower energy levels are of the

same order of magnitude. The potential well formed in the x-direction does not

affect the electron motion in the yz-plane of the semiconductor ﬁlm. The electron

energy spectrum corresponding to this motion is continuous and quadratic with

respect to momentum, as it is in the case of free electron motion. The total energy

of an electron in such a thin semiconductor ﬁlm has a mixed discrete–continuous

character:

E = E

+ p

∗

, (7.29)

where E

is the energy of the nth level of the quantized motion in the x-direction.

Because of the continuity of the spectrum along the other two directions (the y-

and z-directions) electrons that belong to the nth quantum state may have total

energy, E, equal to any value within the interval E

≤ E < ∞. Such a set of

202 Quantization in nanostructures

quantum states for the given n is usually called the subband of dimensional

quantization.

In order to observe the quantization of the energy spectrum experimentally,

the distance between the adjacent energy levels has to be sufﬁciently large. First

of all it must be larger than the thermal energy of electrons in a quantum well,

i.e.,

n+1

− E

> k

T. (7.30)

This condition must be satisﬁed in order to exclude the possible thermal tran-

sitions of the electron between quantized energy levels which will hinder the

observation of the quantum-dimensional effects (QDEs). Second, for observa-

tion of QDEs the mean free path of the electron, l

, in the medium must be

substantially larger than the size of the region of quantized motion. This is

because the quantization takes place only if the electron wavefunction has the

form of a standing wave in the region of electron motion. Such a regularity

of the wavefunction is possible only in the case of weak electron scattering on

vibrations of atoms of the ﬁlms as well as on defects, which are always present in

real structures and destroy the coherent character of the electron motion. To have

such a regularity, an electron in the quantum state with momentum p

must have

a chance to make several ﬂights from one wall of the quantum well (quantum

wire or quantum dot) to another during its time of free ﬂight, i.e., during the time

between two collisions:

, (7.31)

where v

is the characteristic velocity that corresponds to momentum p

given by

∗

This may be possible only in the case of l

 L, where L is the characteristic

size of a QW, QWR, or QD. Let us note that criterion (7.30) can be applied to

semiconductors in which the electron concentration is low and a small number

of levels E

is ﬁlled. Metallic nanostructures are not suitable for the observation

of QDEs because, due to the large number of electrons, the energy of conduction

electrons in typical metals is several electron-volts, which is much larger than

the energy distance between the levels of dimensional quantization. Therefore,

nowadays semiconductor thin-ﬁlm structures are used for the observation of

QDEs. It is worth noting that the ﬁrst experiments in which the quantization of

the electron spectrum was observed were carried out with semimetallic ﬁlms of

bismuth (ﬁrst of all the effective mass of electrons in bismuth, Bi, is very small

and second, it is easy to grow thin ﬁlms of bismuth by laser ablation).

Alongside structures containing quantum wells, structures containing a poten-

tial barrier with a width of about λ

∗

are also of great interest. An electron incident