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

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2.8 Problems 63

2.7 Summary

1. In the framework of classical physics it was impossible to explain the dependence of

the spectral density on the radiation frequency in the spectrum of blackbody radiation.

2. Planck’s formula for the spectral density of blackbody radiation fully describes this

phenomenon over the entire frequency range. The derivation of this formula is based

on the assumption that electromagnetic radiation is emitted by a heated body in the

form of single portions of energy (quanta).

3. The quantum properties of electromagnetic radiation become apparent not only during

its emission, but also upon its absorption by a substance as well as during the propaga-

tion of radiation. Electromagnetic radiation is a ﬂux of photons – particles with mass

at rest equal to zero, energy E

= h

ω, and momentum p

= h

k. These relationships

connect the particle and wave properties of radiation.

4. Any particle with momentum p is related to a wave process, whose wavelength is

deﬁned by the relationship λ

= 2π h

/ p. The wave properties of particles become

apparent when the linear size of the region within which the motion of the particles

takes place is comparable to the de Broglie wavelength, λ

5. The product of the uncertainties of two observables whose product has the dimension-

ality of an action, [J s], cannot be less than Planck’s constant, h, i.e., it cannot be less

than the quantum of action, h. The main uncertainty relationships in quantum physics

are r

p

≥ h and E t ≥ h.

6. The concept of wave–particle duality applied to a substance and a ﬁeld allows a

complete description of many physical phenomena.

2.8 Problems

Problem 2.1. Using Planck’s formula, ﬁnd the most probable frequency, ω

prob

and mean frequency, ω

mean

, in the blackbody radiation spectrum at T = 2000 K.

(Answer: ω

prob

≈ 7.4 × 10

−1

and ω

mean

≈ 10

−1

Problem 2.2. Monochromatic radiation of wavelength λ = 0.55 µm is incident

on the photosensitive surface of the photoelectric cell and it liberates N

= 10

electrons from the cell. The sensitivity of the photoelectric cell, which is deﬁned

as γ = N

e/(N

ω), is equal to γ = 10 mA W

−1

. Estimate the number of

photons, N

, incident on the surface of the photoelectric cell.

Problem 2.3. An isolated small iron ball of radius r = 1 mm is exposed to

monochromatic light with wavelength λ = 0.2 µm. What is the maximum

potential, ϕ

, of the ball’s charging and how many electrons, N

, will it emit?

The work function of iron is equal to A

= 4.36 eV. (Answer: ϕ

= 1.85 V

and N

≈ 1.3 × 10

Problem 2.4. The cut-off wavelength for a given metal is λ

= 0.62 µm. How

much larger is the highest kinetic energy, K

max

, of photoelectrons emitted by

the metal surface after its exposure to light of wavelength λ = 0.35 µm than

64 Wave–particle duality and its manifestation

the mean thermal electron energy, K

. Assume room-temperature conditions.

(Answer: K

max

≈ 40.)

Problem 2.5. A laser beam with wavelength λ = 0.35 µm, power N = 5 W, and

cross-sectional diameter d = 200 µm is perpendicularly incident on a cube face

with side length a = 100 µm. Find the number of incident photons per unit time,

n, and the pressure, P, and force, F, that they create. (Answer: n ≈ 2.8 ×10

P ≈ 1.1 Pa, and F ≈ 1.1 ×10

−8

N.)

Problem 2.6. In a homogeneous magnetic ﬁeld with magnetic ﬂux density

B = 0.1 T an electron gyrates in a circle with a radius 100 times greater than

the electron de Broglie wavelength. Calculate the radius, r , and the electron de

Broglie wavelength, λ

.(Answer:r ≈ 0.8 µm and λ

≈ 8nm.)

Problem 2.7. A parallel array of electrons is perpendicularly incident on a

narrow slit of width b = 5 µm. The electrons are accelerated by a voltage

U = 100 V. Find the distance, X , between two maxima of the ﬁrst order in the

diffraction picture. The screen is located at a distance L = 20 cm from the slit.

(Answer: X ≈ 14.8 µm.)

Problem 2.8. The wavefunction of a particle moving along the x-axis has

the form of Eq. (2.149). Considering all parameters of this function as real

parameters, ﬁnd the dependence on coordinate of the real part of this function,

Re{}, and the square of its modulus, ||

,attimet = t

Chapter 3

Layered nanostructures as the simplest

systems to study electron behavior in a

one-dimensional potential

The description of an electron’s behavior is very simple if the corresponding

probability current density, j, is constant. In this case, as we already found in

Section 2.6.3, the electron wavefunction, (r, t), is the product of a wavefunction

ψ(r) that is the solution of the stationary Schr

odinger equation and an exponential

factor, exp(iEt/h

), which depends on time (see Eq. (2.191)). Depending on the

conﬁguration of the potential within which the electron motion takes place, the

solution of the time-independent Schr

odinger equation (Eq. (2.194))



−

∇

+U (r)



ψ(r) = Eψ(r) (3.1)

deﬁnes the allowed energy states and the corresponding wavefunctions. Equa-

tion (3.1) is easier to solve than the time-dependent Schr

odinger equation (2.182).

In this chapter we will consider several important examples of electron behav-

ior in one-dimensional potential wells and demonstrate that the solutions of the

time-independent Schr

odinger equation (3.1) explain the main peculiarities of

the behavior of conﬁned electrons. The main advantage of the examples consid-

ered is that we can ﬁnd numerical values for electron energy levels (the so-called

electron spectrum) in potentials of different one-dimensional proﬁles. The dis-

crete character of the electron spectrum, which was discovered experimentally

for various atoms, demonstrates that the classical concepts are irrelevant for the

description of microscopic particle behavior on smaller scales. The solutions of

the time-independent Schr

odinger equation also allow us to explain the proba-

bility of electron penetration into the region of the potential barrier as well as

into the region behind the barrier, which is a purely quantum phenomenon.

As an example of one-dimensional potentials we will use layered nanostruc-

tures, which we have brieﬂy considered in Section 1.3, and whose fabrication

we will consider in Section 8.1. With the help of a modern technology called

molecular-beam epitaxy (MBE) it became possible to fabricate structures with

a precision of up to one atomic layer. The solutions of the Schr

odinger equation

for such structures were available in quantum-mechanical textbooks a long time

before the layered structures were actually grown. This set of toy problems from

66 Layered nanostructures

the quantum-mechanical tool box became applicable to real structures and now-

adays they are a hot topic of contemporary nanoelectronics. By combining layers

of semiconductor materials with different forbidden energy gaps, for example

materials such as GaAs and Al

1−x

As, where x is the fraction of Al in the

alloy, we can limit the space within which an electron can freely move. As you

seeinFig.1.4(a), electrons can be conﬁned within the GaAs material, which

has a smaller forbidden energy gap, E

, than Al

1−x

As. Electrons can move

freely only in the y- and z-directions, whereas in the x-direction their motion is

restricted. If we allow the percentage of Al in the Al

1−x

As alloy, which serves

as a barrier for electrons in the GaAs layer, to vary with respect to coordinate, we

can create potentials of various proﬁles, such as those which will be considered

in Section 4.3.2.

The simplest form of space in which electron motion is restricted only in one

of the directions is the so-called quantum well. It can be made in the form of a

sandwich: a thin layer of GaAs is placed between two thick layers of AlGaAs

alloy (see Fig. 1.4). To analyze the peculiarities of electron motion in such

layered (sandwiched) structures we will ﬁrst consider the quantum-mechanical

description of free electron motion in a vacuum. After that we will consider

electron motion in a quantum well with inﬁnite barriers, which is a simpliﬁed

model of a quantum well with real barriers. In such a well an electron cannot

escape from the well or penetrate under the barrier, but it can freely move

in the plane of the well, where its motion is not restricted. Then we will consider

the case when one of the barriers is ﬁnite and the other is inﬁnite, and after that

the quantum well with both barriers being ﬁnite, which corresponds to a real

quantum well of a layered structure.

In the last sections of Chapter 3 we will consider the case of electron transfer

through a barrier (the so-called electron-tunneling phenomenon). Such a barrier

can easily be fabricated in the form of a layered structure consisting of three

layers, whose middle layer creates a potential barrier for electrons instead of a

well.

3.1 The motion of a free electron in vacuum

The simplest example that is described by the Schr

odinger equation (3.1)isthat

of a free electron moving in an unbounded region. The potential energy of such

an electron does not depend on coordinates, i.e., it is constant, and therefore as

a reference point the condition U (r) = U

= 0 can be chosen. From classical

mechanics it is well known that in such a potential the electron momentum, p,

and energy, E, are conserved. Since U (r) = 0 the time-independent Schr

odinger

equation (3.1) for a free electron with mass m

has the form

−



∂

∂x

∂

∂y

∂

∂z



ψ(x, y, z) = Eψ(x, y, z). (3.2)

3.1 The motion of a free electron in vacuum 67

Since the electron’s motion along each of the coordinate axes can be considered

independent, the total wavefunction ψ(x, y, z) can be written as a product of sep-

arate wavefunctions ψ

(x), ψ

(y), and ψ

(z) (i.e., we have separated variables),

ψ(x, y, z) = ψ

(x )ψ

(y)ψ

(z), (3.3)

and the total energy E as

E = E

+ E

. (3.4)

On substituting Eqs. (3.3) and (3.4) into the Schr

odinger equation (3.2) and

dividing it by ψ

(x)ψ

(y)ψ

(z) we obtain the following expression:

(x )



∂

(x )

∂x

+ E

(x )



(y)



∂

(y)

∂y

+ E

(y)



(z)



∂

(z)

∂z

+ E

(z)



= 0. (3.5)

Since each of the three terms in Eq. (3.5) depends only on one of the independent

variables x, y, and z, each of these terms should be equated to zero to satisfy

Eq. (3.5). Thus, Eq. (3.5) can be rewritten as a system of three independent

second-order differential equations:

−

(x )

= E

(x ), (3.6)

−

(y)

= E

(y), (3.7)

−

(z)

= E

(z). (3.8)

Note that Eqs. (3.6), (3.7), and (3.8) have the same form, and therefore it suf-

ﬁces to ﬁnd a solution for one of them, for example the solution of Eq. (3.6) only,

and then the solutions of the other two equations can be written by substituting

for the variable x the variables y and z, correspondingly.

Since the electron’s energy E

is deﬁned through the wavenumber k

, (3.9)

where α = x, y, z, let us rewrite Eq. (3.6) in the following form:

(x )

+ k

(x ) = 0, (3.10)

where

. (3.11)

There are two types of solutions for the one-dimensional equation (3.10). One

is for an electron moving in the positive direction along the x-axis, and has the

68 Layered nanostructures

following form:

(x ) = Ae

. (3.12)

The second is for an electron moving in the negative direction along the x-axis:

(x ) = Be

−ik

. (3.13)

The general solution of Eq. (3.10)isthesumofEqs.(3.12) and (3.13):

(x ) = Ae

+ Be

−ik

. (3.14)

Analogously for the y- and z-directions,

(y) = De

+ Ee

−ik

, (3.15)

(z) = Fe

+ Ge

−ik

. (3.16)

Considering an electron moving in the positive direction along the x-axis,

we can ﬁnd the time-dependent wavefunction 

(x, t) of a free particle in a

stationary state if we take into account Eqs. (3.12) and (2.191):



(x , t) = ψ

(x )e

−iE

t/h

= Ae

i(k

x−E

t/h

)

. (3.17)

The solutions in the other two directions have analogous forms and the total

wavefunction of an electron moving in an arbitrary direction is

(r, t) = Ce

i(k·r−Et/h

)

, (3.18)

where k ·r = k

x + k

y + k

z and E = E

+ E

. The wavefunction

(3.18) deﬁnes the probability amplitude of ﬁnding the particle at any arbitrary

point in space as well as at any instant in time. Since this function is complex,

only the square of its modulus has physical meaning. It deﬁnes the corresponding

probability density, ρ(r, t):

ρ(r, t ) =|(r, t)|

= 

∗

(r, t)(r, t ). (3.19)

On substituting Eq. (3.18) into Eq. (3.19) we ﬁnd that the probability density of

a quantum state of a free electron does not depend on time:

ρ(r, t) =|C|

. (3.20)

Example 3.1. Find the probability current density for a free electron that is in a

stationary state with momentum p and energy E.

Reasoning. Let us choose the positive direction of the x-axis along the direction

of the vector p. Then the vector of the probability density ﬂux will have only one

component along the x-axis, which, according to Eq. (2.190), has the form





∂

∗

∂x

− 

∗

∂

∂x



. (3.21)

Here the wavefunction, 

(x, t), of the free electron depends only on one spatial

coordinate x and is deﬁned by Eq. (3.17). The complex-conjugate function

3.2 An electron in a potential well with inﬁnite barriers 69



∗

(x, t) is deﬁned as



∗

(x , t) = A

∗

−i(k

x−E

t/h

)

. (3.22)

Let us calculate the corresponding derivatives from Eq. (3.21):

∂

∂x

= ik



∂

∗

∂x

=−ik



∗

. (3.23)

By substituting these expressions into Eq. (3.21) we obtain



−ik



∗

− ik



∗







∗



∗

. (3.24)

Taking into account Eq. (3.19) for the probability density 



∗

=|A|

= ρ(x),

we obtain

ρ(x) = v

ρ(x), (3.25)

where v

is the electron’s velocity. The result obtained for a free electron is

consistent with the classical expression for the ﬂux density since it is proportional

to the electron’s velocity, v

, and the electron density, ρ.

3.2 An electron in a potential well with inﬁnite barriers

Let us study the energy spectrum of an electron conﬁned in a one-dimensional

rectangular quantum well with impenetrable walls. This case corresponds to an

electron being in a thin (several nanometers thick or even thinner) semiconductor

ﬁlm. In this case the potential energy of the electron depends only on one

coordinate (in the example considered here, on the x-coordinate) and can be

written as

U (x ) =







∞, x ≤ 0,

0, 0 < x < L,

∞, x ≥ L.

(3.26)

The time-independent Schr

odinger equation (3.1) for this one-dimensional

potential well can be written as



−



∂

∂x

∂

∂y

∂

∂z



+U (x)



ψ(x, y, z) = Eψ(x, y, z). (3.27)

As in the case of a free electron, the wavefunction ψ(x, y, z) can be taken as a

product of three separate wavefunctions ψ

(x), ψ

(y), and ψ

(z) (see Eq. (3.3)).

On repeating the same procedure as for Eq. (3.2) we ﬁnd that the Schr

odinger

equation in the y- and z-directions is the same as for a free electron (see Eqs. (3.7)

and (3.8)). Thus, the electron motion in the y- and z-directions remains free

and the solutions to the Schr

odinger equation, ψ(y) and ψ(z), are deﬁned by

70 Layered nanostructures

Eqs. (3.15) and (3.16), respectively. The values of the kinetic energy for free

motion in the y- and z-directions are deﬁned by Eq. (3.9), i.e.,

(3.28)

and

. (3.29)

The Schr

odinger equation for the x-coordinate differs from (3.6) since it contains

a term with potential energy U(x):



−

+U (x)



(x ) = E

(x ). (3.30)

Equation (3.30) is a one-dimensional differential equation and, to avoid unnec-

essary use of subscripts, we will henceforth omit the subscript x from the eigen-

function, ψ

, and eigenvalue, E



−

+U (x)



ψ(x) = Eψ(x). (3.31)

For the region 0 < x < L where U (x) = 0 and where the electron is conﬁned,

the Schr

odinger equation (3.31) takes the form

+ k

ψ = 0, (3.32)

where

k =

√

Here, for simplicity, we omitted the subscript x from the wavenumber k

.The

solutions to differential equations similar to (3.32) can be sought in the form of a

combination either of harmonic functions or of exponential functions as we have

already done in Eq. (3.14):

ψ(x) = A sin(kx) + B cos(kx), (3.33)

ψ(x) = A



ikx

+ B



−ikx

. (3.34)

Let us seek solutions for the case of a potential well with inﬁnite walls in the

form of Eq. (3.33):

ψ(x) = A sin(kx) + B cos(kx) = C sin(kx + φ). (3.35)

Here we introduced two unknown parameters, C and φ. To ﬁnd these parameters

we should take into account the boundary conditions imposed by the potential of

the well. Since the potential barriers at the boundaries of the well are inﬁnite (see

Eq. (3.26)), the electron cannot penetrate into the regions outside of the well.

3.2 An electron in a potential well with inﬁnite barriers 71

Thus, we can write

ψ(0) = 0, (3.36)

ψ(L) = 0. (3.37)

By substituting the wavefunction (3.35) into Eqs. (3.36) and (3.37) we obtain a

system of two equations for three unknowns, C,φ, and k:

ψ(0) = C sin φ = 0 (3.38)

and

ψ(L) = C sin(kL + φ) = 0. (3.39)

Since the parameter C = 0(C = 0 corresponds to the trivial solution ψ(x) ≡ 0),

from Eq. (3.38) it follows that

φ = 0. (3.40)

Equations (3.35) and (3.39) can be rewritten as

ψ(x) = C sin(kx), (3.41)

sin(kL) = 0. (3.42)

From the last equation we obtain the following allowed values of the wavenumber,

k, for the electron’s wavefunction in the potential well being considered:

=±

πn

, where n = 1, 2, 3,... (3.43)

Taking into account the relation of this parameter with the electron’s energy,

we come to the important conclusion that the electron energy in the potential

well cannot change continuously – it can have only certain discrete values of

allowed energies (see Fig. 3.1):

, (3.44)

where n = 1, 2, 3,... The electron energy, E

, in the potential well is thus

quantized and the parameter n, which deﬁnes the electron energy state, is called

the principal quantum number.Forn = 1 the electron’s energy is minimal and

equal to

0.3737

eV. (3.45)

This state, which has the lowest energy, is often called the ground state.The

energy E

is inversely proportional to L

and its numerical value in electron-volts

for an electron with free electron mass m

can be calculated with the coefﬁcient

0.3737 when the width of the well, L, is measured in nanometers. The quantum

number n cannot be equal to zero because this would mean that the wavenumber

k = k

= 0 and there would be no electron in the well because ψ(x) = 0atk = 0

72 Layered nanostructures

U(0)= ∞

U(L)=∞

Figure 3.1 Quantization of

an electron’s energy in a

potential well with

impenetrable walls of

U(0) =∞and U(L) =∞.

The two lowest energy

levels, E

and E

,andthe

corresponding

wavefunctions, ψ

and ψ

are shown.

(see Eq. (3.41)). This leads us to the important conclusion that a conﬁned electron

can never be in a state of rest. Even in its ground state the electron has non-zero

momentum, i.e., from the point of view of classical mechanics the electron is in a

state of constant motion. This is a direct consequence of Heisenberg’s uncertainty

principle and wave–particle duality.

For the type of potential well considered here the positions of the energy states

are not equidistant. The distance between adjacent energy states increases with

increasing quantum number, n, according to the law

E = E

n+1

− E

= (2n + 1)E

. (3.46)

If the width of the potential well, L, within which the electron is conﬁned

(for example the width of the above-mentioned semiconductor ﬁlm) increases,

then the energy of the ground state E

decreases and therefore the distance

between the energy states, E, decreases and tends to zero at L →∞.The

energy spectrum transforms from a discrete spectrum into a quasicontinuous or

practically continuous spectrum and the electron’s behavior becomes classical.

The unknown coefﬁcient in the expression for the wavefunction (3.41) can be

found from the normalization condition (2.140):



∞

−∞

|ψ(x)|

dx = C



sin

x)dx = 1. (3.47)

Calculation of this integral gives us



sin

x)dx =

[

−sin(k

L)cos(k

L) + k

]

. (3.48)