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

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3.2 An electron in a potential well with inﬁnite barriers 73

U(0) = ∞

L/2

U(L) =

∞

Figure 3.2 The probability

density distributions,

|ψ

(x)|

and |ψ

(x)|

,for

the two lowest energy

levels, E

and E

,inawell

with impenetrable walls.

Taking into account Eq. (3.42), this integral reduces to



sin

x)dx =

L. (3.49)

The parameter C is

C =



. (3.50)

Finally, the wavefunction of the electron in the potential well for the nth energy

state can be written as

(x ) =



sin



nπ x



. (3.51)

The graphs of the wavefunctions ψ

(x) and ψ

(x)forn = 1 and n = 2, are

shown in Fig. 3.1. The distribution of the probability density, |ψ

(x)|

, of ﬁnding

an electron in the region of the potential well whose number of maxima is

determined by the quantum number n is shown in Fig. 3.2. We see that in the

ground state (n = 1) the probability of ﬁnding an electron has a maximum at the

center of the potential well. In the ﬁrst excited state (n = 2) the probability of

ﬁnding an electron at the center of the well is equal to zero, whereas the probability

of ﬁnding an electron has maxima at the points x = L/4 and x = 3L/4. This

behavior of the electron differs drastically from the behavior of a classical particle

in a potential well, for which the probability of ﬁnding a particle is the same at

any point in the well, for any value of the particle’s energy (see Appendix A,

Section A.2.6).

74 Layered nanostructures

Note that there are similarities between the mathematical description of elec-

tron motion in a potential well with inﬁnite walls and the oscillations of a

string with ﬁxed ends. In both cases the boundary conditions are such that

the length of the system (in our case it is the width of the potential well or

the distance between ﬁxed ends in the case of the string) must be equal to an

integer multiple of half of the wavelength, which is described by the wavefunc-

tion (in the case of an electron) or by the transverse wave (in the case of a

string).

Example 3.2. An electron is located in a rectangular potential well with inﬁnite

walls. The width of the well is L = 10 nm. Find the energies of stationary

states of the electron for which at the well’s boundary x = L the derivatives of

the wavefunction are negative and equal to −τ

(here τ

are arbitrary positive

constants).

Reasoning. The wavefunction, ψ

,forthenth stationary state in the potential

well with inﬁnite walls is given by Eq. (3.51). Let us ﬁnd its derivative:

dψ



nπ

cos



nπ x



. (3.52)

For the point x = L,Eq.(3.52) gives the relation between the derivative of the

wavefunction and its quantum number, n:



nπ

cos(nπ) =−τ

. (3.53)

Since the quantum number n is an integer and positive, and because the left-hand

side of Eq. (3.53) must be negative, the quantum number n must be an odd

number:

cos(nπ) =−1, n = 1, 3, 5,... (3.54)

On substituting Eq. (3.54) into Eq. (3.53) we obtain the expression for n for a

given τ

n =



. (3.55)

On substituting Eq. (3.55) into the formula for the energy of the nth state,

Eq. (3.44), we obtain





Lτ

. (3.56)

On substituting the value of L = 10 nm into Eq. (3.56) we obtain

= 3.1 × 10

−47

J = 1.91 × 10

−28

eV. (3.57)

3.3 An electron in a potential well with ﬁnite barriers 75

In reality we never deal with the case of given τ

, and rather relation (3.55) has

to be rewritten in the form

nπ



, (3.58)

i.e., Eq. (3.58) deﬁnes the value of the derivative of the wavefunction at the

boundary of the potential well and the most important conclusion from Eq. (3.58)

is that the derivative τ

is a quantized quantity. Thus, from Eq. (3.58) we obtain

≈ 4.4 × 10

n m

−3/2

, (3.59)

where n = 1, 3, 5,... By substituting the ﬁrst two values of τ

into Eq. (3.57)

we obtain for the magnitudes of the ﬁrst two odd energy levels E

= 3.7meV

and E

= 33 meV. Note that at boundaries with inﬁnite walls the condition of

continuity of the derivative of the wavefunction is violated while the continuity

of the wavefunction is preserved since each wavefunction becomes zero at the

boundaries.

3.3 An electron in a potential well with ﬁnite barriers

The technology of fabrication of planar layered structures with given physical

properties allows the experimental realization of the conﬁnement of electrons

inside potential wells of various proﬁles. Most important for practical realizations

are one-dimensional potential proﬁles of rectangular wells, (1) with one of the

potential barriers being ﬁnite and the other inﬁnite, and (2) with both potential

barriers ﬁnite.

3.3.1 The potential well with one of the barriers ﬁnite

Let the dependence of the potential energy of an electron on the coordinate x

have the following form:

U (x ) =







∞, x ≤ 0,

0, 0 < x < L ,

, x ≥ L .

(3.60)

Such a potential proﬁle is shown in Fig. 3.3. The ﬁnite value of the potential at

the right boundary, x = L, considerably changes the solutions of the Schr

odinger

equation and correspondingly the behavior of an electron. It is now necessary

to distinguish between two different types of electron motion in the well. The

ﬁrst type corresponds to E ≤ U

, and the electron in the potential well can have

bound states, which correspond to the discrete set of energy levels. The second

type of motion corresponds to E > U

, and the electron does not have bound

states and can be found at any point with coordinate x > 0. This type of motion

corresponds to a continuous energy spectrum. In this section we will consider

only the case when E ≤ U

and the electron is conﬁned in the potential well.

76 Layered nanostructures

xL0

E,U

U =

(x)

E >U

E U

Figure 3.3 The ﬁrst two

energy states, E

and E

and the corresponding

wavefunctions, ψ

and ψ

foranelectronina

potential well with the

right-hand-side potential

barrier ﬁnite.

A classical particle cannot move out of the well because its kinetic energy

would be negative when it is outside of the well. On reﬂecting from the walls

of the well the particle would move only inside the well. The behavior of the

electron described by the solution of the corresponding Schr

odinger equation is

fundamentally different from the behavior of a classical particle.

Because the potential proﬁle U (x) is a step-like function, let us divide the

electron’s accessible region x > 0 into regions 1 and 2 with potential ener-

gies U

= 0 and U

= U

. In each of these regions we will obtain solutions of

the Schr

odinger equation. Then, in correspondence to the boundary conditions

applied to the wavefunctions, we will join these solutions in such a way that these

wavefunctions and their derivatives are continuous at the boundaries.

Let us write the Schr

odinger equation for each of these regions, 1 and 2:

(x )

+ k

(x ) = 0, 0 ≤ x ≤ L , (3.61)

(x )

− κ

(x ) = 0, x > L. (3.62)

Here we introduced for the wavenumbers of the electron wave in the potential

well

k =



, (3.63)

and outside of the potential well

κ =



− E)

. (3.64)

3.3 An electron in a potential well with ﬁnite barriers 77

The solutions of Eqs. (3.61) and (3.62) can be written as follows:

(x ) = A

ikx

+ B

−ikx

, (3.65)

(x ) = A

κx

+ B

−κx

, (3.66)

where A

, A

, B

, and B

are constants, which can be found from the bound-

ary conditions. In our case the boundary conditions are the continuity of the

wavefunction and its derivative at the boundaries of the potential well. From the

condition of continuity of the wavefunction at x = 0 we obtain the equation for

unknowns A

and B

+ B

= 0, (3.67)

since in the region x ≤ 0 the function ψ(x) = 0. Taking into account the last

equation, Eq. (3.65) transforms into

(x ) = 2iA

sin(kx). (3.68)

The wavefunction must be ﬁnite everywhere, in region 2 as well. For this reason

= 0 because in region 2 the term A

κx

increases inﬁnitely with increasing

x. At the boundary x = L the following conditions for the continuity of the

wavefunction and its derivative must be satisﬁed:

(L) = ψ

(L), (3.69)

dψ



x=L

dψ



x=L

. (3.70)

After substitution of the corresponding expressions for ψ

and ψ

, the boundary

conditions (3.69) and (3.70) can be rewritten in the form

2iA

sin(kL) = B

−κ L

, (3.71)

2ikA

cos(kL) =−κ B

−κ L

. (3.72)

Let us divide Eq. (3.71)byEq.(3.72). As a result we obtain the following

equation:

tan(kL) =−

, (3.73)

which deﬁnes the energy spectrum of the electron in the potential well. This is

a transcendental equation, whose solution cannot be obtained analytically. For

the analysis of such equations very often graphical methods of solution are used,

together with numerical methods.

Let us transform Eq. (3.73) in order to carry out the graphical analysis. Taking

into account the trigonometric identity

sin ϕ =±

tan ϕ



1 +tan

, (3.74)

78 Layered nanostructures

−1

f=−

f= kL

f=sin(kL)

Figure 3.4 Graphical

solution of the

transcendental equation

(3.77).

we can transform Eq. (3.73) into the form

sin(kL) =±

√

+ κ

. (3.75)

According to Eqs. (3.63) and (3.64), for the terms under the square root in

Eq. (3.75) we obtain

+ κ

. (3.76)

By substituting Eq. (3.76) into Eq. (3.75) we obtain

sin(kL) =±γ kL, (3.77)

where we introduced the parameter

γ =



, (3.78)

which deﬁnes the slope f (kL) =±γ kL.

Let us plot graphs of the right-hand and left-hand sides of Eq. (3.77)as

functions of the parameter kL (see Fig. 3.4). The points of intersection of the

sine function with the lines f =±γ kL deﬁne the roots of Eq. (3.77). From all

the roots of Eq. (3.77) we separate the roots which are the roots of Eq. (3.73).

The chosen roots must satisfy the condition tan(kL) < 0. The tangent function

is negative only in the following intervals of values of kL where the roots can

exist:

+ nπ ≤ kL ≤ π + nπ. (3.79)

Thus, we have to choose the roots of Eq. (3.77) which exist in the interval (3.79).

In Fig. 3.4 intervals of kL (Eq. (3.79)) are shown by thick lines. Note that from the

expressions derived we can identify wells for which the solutions of Eq. (3.73)do

3.3 An electron in a potential well with ﬁnite barriers 79

not exist. So there would be no roots and no bound energy states of the electron in

those potential wells. We show by the dashed line the maximal slope of f = γ kL

for the parameter γ = 2/π when the roots for Eq. (3.73) exist. For higher values

of the slope, γ , there are no bound states. Thus, the condition for which there is

at least one energy state with E ≤ U

reduces to the inequality

γ ≤ 2/π. (3.80)

Taking into account relation (3.78), inequality (3.80) takes the form

√

≤

. (3.81)

This condition can be rewritten in more convenient form:

≥

, (3.82)

where the left-hand side of the inequality is deﬁned by the potential-well param-

eters only, and the right-hand side by the type of the particle conﬁned in the

potential well. In relatively shallow and/or narrow potential wells, there may be

no bound states for an electron, i.e., the electron cannot be captured by such a well

and the electron has energy E > U

, which is greater than U

. That corresponds

to the case of an unbound particle, which will be considered in Section 3.4.1.

The second energy level in the well can exist at γ ≤ 2/(3π ), i.e.,

≥

9π

. (3.83)

As the parameter U

increases from the value at which equality is achieved

in Eq. (3.80) to the value at which equality is achieved in Eq. (3.83), the ﬁrst

level appears at E

= U

and goes down to E

= 0.3 × U

, and the second level

appears at E

= U

. Further increase of U

brings both levels, E

and E

down and, at γ ≤ 2/(5π), the third energy level appears. Thus, the spectrum of

eigenvalues of the electron energy is discrete.

Let us now write expressions for the wavefunctions in each of the two regions

(Eqs. (3.65) and (3.66)), taking into account the relations between the coefﬁcients

which we have obtained:

= A sin(kx), (3.84)

= A sin(kL)e

κ L

−κx

, (3.85)

where we introduced the constant A = 2iA

. From these two expressions it

follows that inside the potential well the wavefunction is an oscillatory function,

as it was in the case of the potential well with inﬁnite barriers. Outside of the

well, as the electron moves away from the potential barrier, its wavefunction

decreases exponentially. This means that in a bound state there is a non-zero

probability of the electron being outside of the potential well where its energy

80 Layered nanostructures

is E < U

and therefore the electron’s kinetic energy, K , would be negative.

This can be explained as follows: in quantum mechanics the total energy, E ,

cannot be presented as a sum of kinetic and potential energy because, according

to Heisenberg’s uncertainty principle, kinetic and potential energies (as well as

momentum and coordinate) cannot be both exactly measured simultaneously.

The uncertainty relations Eq. (2.117) relate the particle’s coordinates and its

corresponding momentum projections, which deﬁne potential (coordinates) and

kinetic (momenta) energies. Thus, the situation with E < U

is possible for a

quantized particle, and it is very common in various theoretical and applied

problems, as we will see in later chapters. The qualitative dependence of the

wavefunctions for the ﬁrst two energy states on the coordinate x is shown in

Fig. 3.3.

Example 3.3. Find the ratio E

for an electron in a well with the poten-

tial proﬁle described by Eq. (3.60) when only one bound state exists, which

corresponds to the parameter kL = 3π/4.

Reasoning. The parameter k is deﬁned by Eq. (3.63),

k =



Then, from the relation kL = 3π/4 we obtain for the energy, E

,thefollowing

equation:

9π

32m

. (3.86)

The parameter γ , which deﬁnes the slope of the function f = γ kL,asshownin

Fig. 3.4, can be found from Eq. (3.77):

γ =

sin(kL)

√

2/2

3π/4

√

3π

. (3.87)

Using the deﬁnition of the parameter γ (Eq. (3.78)), we obtain the expression

for the height of the potential barrier U

9π

16m

. (3.88)

Taking into account Eq. (3.86), we obtain

= 0.5.

Example 3.4. Find the normalization constant A for the wavefunction described

by Eqs. (3.84) and (3.85).

Reasoning. Since the electron’s behavior in regions 1 and 2 is described by its

own wavefunction, the normalization condition must be written in the form



∞

|ψ|

dx =



|ψ

dx +



∞

|ψ

dx = 1. (3.89)

3.3 An electron in a potential well with ﬁnite barriers 81

Let us substitute into this condition the expressions for the wavefunctions ψ

and

|A|





sin

(kx)dx + sin

(kL)e

2κ L



∞

−2κx



= 1. (3.90)

By carrying out the integration we obtain

|A|



−

sin(2kL)

sin

(kL)

2κ



= 1. (3.91)

Let us assume that the constant A is real. Then

A =



L − [sin(2kL)]/(2k) + [sin

(kL)]/κ

. (3.92)

At U

→∞the parameters κ →∞and kL → nπ . Then

A =



which coincides with the normalization constant from Eq. (3.50).

3.3.2 A symmetric potential well with ﬁnite

potential barriers

The dependence of the electron potential energy in a symmetric potential well

with ﬁnite potential barriers is deﬁned as

U (x ) =







, x ≤ 0,

0 , 0 < x < L,

, x ≥ L .

(3.93)

Such a potential well is shown in Fig. 3.5. Two cases, E ≤ U

and E > U

,as

in Section 3.3.1 are possible. Let us consider here one of these cases, in particular

the case with E ≤ U

. By analogy to Eqs. (3.65), (3.66), and (3.68) the solutions

of the Schr

odinger equation for the three regions 1, 2, and 3 (see Fig. 3.5), taking

into account that wavefunctions at x →±∞must be ﬁnite, can be written in the

following form:

(x ) = C

κx

, x ≤ 0,

(x ) = C

sin(kx + φ), 0 < x < L ,

(x ) = C

−κx

, x ≥ L.

(3.94)

In Eqs. (3.63) and (3.64) we have already introduced the parameters k =

√

E/h

and κ =

√

− E)/h

. The boundary conditions of continu-

ity of the wavefunctions and their derivatives at points x = 0 and x = L give us

a system of four equations to ﬁnd the constants C

, C

, and φ as well as the

82 Layered nanostructures

xL0

123

E > U

E U

Figure 3.5 A symmetric

potential well with

potential barriers of ﬁnite

height. The ﬁrst two

lowest energy levels, E

and E

,andtheir

corresponding

wavefunctions, ψ

(x)and

(x), are shown.

relationships between the parameters, which replace Eq. (3.73):

tan φ =

, (3.95)

tan(kL +φ) =−

. (3.96)

These equations can be reduced to

sin φ =

, (3.97)

sin(kL +φ) =−

, (3.98)

where the parameter b =

√

. If we eliminate from these equations the

parameter φ we obtain an equation that deﬁnes the energy spectrum of an electron

in the symmetric potential well with ﬁnite barrier heights:

kL = nπ − 2arcsin





, (3.99)

where the quantum number n = 1, 2, 3,... deﬁnes the electron energy-level

number in the well. Since the positive values of the argument of the arcsine

function cannot exceed unity, i.e., k/b ≤ 1,

max

= b =

√

. (3.100)

The energy of an electron in the potential well is quantized in accordance with

Eq. (3.99), i.e., its energy spectrum is discrete. Let us show this graphically.

Figure 3.6 shows the graphs of f (k) = kL and f

(k) = nπ − 2arcsin(k/b),

which correspond to the left-hand and right-hand sides of Eq. (3.99). The index

n of the function f

(k) coincides with the quantum number n. The points of

intersection of the line f (k) with the curves f

(k) deﬁne the roots of Eq. (3.99).

We see from Fig. 3.6 that the spectrum of values of k and their related values