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

Подождите немного. Документ загружается.

3.3 An electron in a potential well with ﬁnite barriers 83

max

2π

3π

, f

4π

f = kL

Figure 3.6 Graphical

solution of Eq. (3.99).

Here, f

= nπ −

2 arcsin(k/ b).

of electron energy are discrete. For the case shown in Fig. 3.6 there exist only

three solutions of Eq. (3.99) and correspondingly three energy levels in the

potential well, since the line f (k) = kL and curves f

(k) have only three points

of intersection, n = 1, 2, and 3, in the region of allowed values of k: k ≤ k

max

The larger the width of the well, L, the steeper the line f (k) = kL is, and therefore

there are more roots of Eq. (3.99) and correspondingly more energy levels exist

in the well.

As U

→∞, the parameter b tends to inﬁnity. Thus, in this limiting case the

roots of Eq. (3.99)arekL = nπ as was calculated earlier for the well with inﬁnite

potential barriers (compare with Eq. (3.43)). For the given value of the width of

the potential well, L, as the depth, U

, of the well decreases the magnitude k

max

also decreases, which leads to a decrease in the number of energy levels in the

well. In the case k

max

<π/L, i.e., at

, (3.101)

only one energy level can exist in the well. In contrast to the non-symmetric well

with one of the walls being of ﬁnite height (Fig. 3.3), for which the bound state

of an electron can exist only if condition (3.82) is satisﬁed, in the symmetric well

of ﬁnite depth there is always at least one bound state for an electron.

To plot the wavefunctions (3.94), let us write the expressions that relate the

constants C

, C

, and C

√

, (3.102)

=−e

κ L

√

. (3.103)

84 Layered nanostructures

As has already been mentioned, relations (3.102) and (3.103) were obtained from

the conditions of continuity of the wavefunction at the boundaries of the well.

The constant C

can be found from the condition of normalization, which in our

case will have the form



−∞

|ψ

dx +



|ψ

dx +



∞

|ψ

dx = 1. (3.104)

Figure 3.5 shows the ﬁrst two energy levels, E

and E

, in the potential well with

barriers of ﬁnite height and corresponding wavefunctions, ψ

and ψ

.Wesee

that for E ≤ U

there is a symmetric probability of ﬁnding an electron outside

the well and inside the barrier. The higher the energy levels, the higher is this

probability.

Example 3.5. Find approximately the energy difference of two levels with energy

E  U

for an electron in a symmetric potential well of width L and barrier

height U

Reasoning. Since the energy of the electron E  U

, the ratio k/b =

√

E/U

a very small number. Therefore, let us substitute the arcsine function by the ﬁrst

term of its expansion in series, arcsin(k/b) ≈ k/b; then, from Eq. (3.99), we get

kL ≈ nπ −

. (3.105)

On solving this equation with respect to the wavenumber, k, we obtain

≈

nπ

L + h



2/(m

)

. (3.106)

Let us ﬁnd from this expression the energy of stationary states of an electron in

the potential well. Since E

= h

/(2m

≈



L + h



2/(m

)



. (3.107)

The values of k

and E

decrease compared with the case of the well with

impenetrable barriers (compare Eqs. (3.106) and (3.107) with Eqs. (3.43) and

(3.44)). At U

→∞Eq. (3.107) coincides with the exact expression for the

energy levels (3.44). The energy distance between the second and ﬁrst energy

levels is

− E

≈

3π



L + h



2/(m

)



. (3.108)

3.4 Propagation of an electron above the potential well

In the previous section we considered restricted motion of an electron when its

energy does not exceed the height of the potential barriers of the quantum well.

3.4 Propagation of an electron above the potential well 85

The conﬁnement of an electron in the potential well leads to the quantization

of its energy, which corresponds to the so-called quantum-conﬁnement effect or

quantum-size effect. Now let us consider the peculiarities of the motion when the

energy of an electron, E, is greater than the height of the potential barrier, U

3.4.1 Reﬂection from an asymmetric well with one inﬁnite

potential barrier

Suppose that in the region of the well with one of the barriers having inﬁnite

height (Fig. 3.3) an electron is moving along the x-axis in the negative direction.

The Schr

odinger equation for each of the regions, 1 and 2, has to be written as

follows:

(x )

+ k

(x ) = 0, k =



, (3.109)

(x )

+ κ

(x ) = 0,κ=



(E − U

)

. (3.110)

We see that, in contrast to the case with E ≤ U

, only the equation for the second

region (Eq. (3.110)) has changed. Equation (3.110) that corresponds to region 2

now has oscillatory solutions as the solutions in region 1. Therefore, the solutions

for the above-mentioned equations can be sought in the form

(x ) = A

ikx

+ B

−ikx

, (3.111)

(x ) = A

iκx

+ B

−iκx

. (3.112)

Taking into account the boundary condition ψ

(0) = 0, we obtain as before the

relation between the constants, B

=−A

, and the same expression (3.68)for

the wavefunction ψ

(x). By applying the boundary conditions (3.69) and (3.70)

at the point x = L, we obtain the expressions for the coefﬁcients A

and B

= e

−iκ L



i sin(kL) +

cos(kL)



, (3.113)

= e

iκ L



i sin(kL) −

cos(kL)



. (3.114)

Since there are no restrictions for the parameters k and κ, the energy E can have

any value greater than U

, i.e., in this case the electron has a continuous energy

spectrum.

In each of the regions 1 and 2 the wavefunctions ψ

(x) and ψ

(x)arethe

sums of two de Broglie waves propagating in opposite directions. The wave that

approaches the well (the part of the total wavefunction proportional to e

−ikx

)

undergoes a partial reﬂection at the boundary x = L, and then the reﬂected wave

86 Layered nanostructures

propagates along the x-axis in the positive direction. The wave that penetrated

into region 1 is totally reﬂected from the boundary x = 0. Then, part of this wave

penetrates the boundary x = L and also propagates along the x-axis inﬁnitely.

The reﬂection coefﬁcient of the wave incident on the well is deﬁned by the

expression

R =

. (3.115)

It follows from expressions (3.113) and (3.114) for the coefﬁcients A

and B

that the reﬂection coefﬁcient R ≡ 1.

The probability density for all values of x ≥ 0 is an oscillating function, which

has different periods of oscillations λ

and λ

in regions 1 and 2,

π h

√

, (3.116)

π h



(E − U

)

. (3.117)

The oscillating character of the probability density is the result of a superposition

of the incident and reﬂected waves. As a result, the wavefunction has the form

of a standing wave similar to that for the reﬂection of an electromagnetic wave

from an ideal conductor (see Section B.3.4 in Appendix B).

3.4.2 Reﬂection from a symmetric well with ﬁnite

potential barriers

Let us consider now the propagation of an electron over the symmetric potential

well with proﬁle deﬁned by Eq. (3.93) (see Fig. 3.5). Suppose that the electron

is moving towards the walls of the well in the positive direction along the x-axis

and its energy is E > U

. The Schr

odinger equation for each of the three regions

1, 2, and 3 can be written as

(x )

+ κ

(x ) = 0, (3.118)

(x )

+ k

(x ) = 0, (3.119)

(x )

+ κ

(x ) = 0, (3.120)

where the parameters k and κ are deﬁned by the same expressions as in

Eqs. (3.109) and (3.110). The solutions for the three regions 1, 2, and 3 can

be written as

(x ) = A

iκx

+ B

−iκx

, x ≤ 0, (3.121)

(x ) = A

ikx

+ B

−ikx

, 0 < x < L , (3.122)

(x ) = A

iκx

+ B

−iκx

, x ≥ L. (3.123)

3.4 Propagation of an electron above the potential well 87

Let us assume that the electron propagates to the well from the negative part

of the x-axis, then in the region x > L only the outgoing wave proportional to

ikx

exists. Since there is no electron coming back from x →∞into the region

of interest, B

= 0. Using the standard boundary conditions such as (3.69) and

(3.70) for the wavefunctions ψ

, ψ

, and ψ

, we arrive at the following expression

for the amplitude of the outgoing wave:

2κke

−iκ L

2κk cos(kL) + i(k

+ κ

)sin(kL)

. (3.124)

According to Eq. (2.190), the probability density currents of the incident wave,

, and the wave transmitted through the well, j

,havetheforms

= e

, (3.125)

= e

. (3.126)

Taking into account these expressions, the transmission coefﬁcient, D, which

characterizes the probability of transmission of an electron over the well, can be

written as

D =

. (3.127)

Taking into account Eq. (3.124) for the transmission coefﬁcient, D, we get the

following expression:

D =



1 +



− κ

2kκ



sin

(kL)



−1

. (3.128)

Taking into account Eqs. (3.109) and (3.110), Eq. (3.128) can be rewritten in the

form

D =



1 +

4E(E − U

)

sin



√



−1

. (3.129)

The reﬂection coefﬁcient, R, which deﬁnes the probability of reﬂection of the

electron from the potential well when it propagates over this well, can be written

R =

= 1 − D. (3.130)

Using Eq. (3.128), we can write the expression for the reﬂection coefﬁcient, R,

R =



1 +

(2kκ)

− κ

)

sin

(kL)



−1



1 +

4E(E − U

)

sin



√

EL/h





−1

. (3.131)

Thus, the transmission and reﬂection coefﬁcients for the particle propagating

over the potential well strongly depend on the ratio of the energy of the electron,

E, and the height of the potential barrier, U

. When the energy, E, is slightly

88 Layered nanostructures

greater than the potential barrier, U

, the transmission coefﬁcient D  1, and

the reﬂection coefﬁcient is close to unity. As the ratio E/U

increases, the

transmission coefﬁcient, D, rapidly increases and at E/U

= 2 its maximal

value is D ≈ 0.9. The electron’s reﬂection from the potential well at E > U

is a purely quantum effect, which is caused by the wave properties of the

electron.

With the increase of the electron energy, E, both coefﬁcients, D and R,

oscillate. This can be explained by invoking the interference of the de Broglie

waves reﬂected from the potential barrier at the boundaries of the well at x = 0

and x = L. Since, at kL = nπ , the electron does not undergo reﬂections, R = 0

and D = 1. The energy of such an electron has to be equal to

, (3.132)

where n is an integer number from n = n

min

to n →∞, where n

min

is deﬁned

from the condition E

≥ U

with E

found from Eq. (3.132). The absence

of electron reﬂection from the potential well is analogous to the effect of an

antireﬂecting coating of a lens. This effect is connected with the destructive

interference of waves reﬂected from the two boundaries of the potential well,

which takes place at

L = n

, (3.133)

where n = 1, 2, 3,... and λ

is the de Broglie wavelength that is deﬁned by

Eq. (2.88): λ

= 2π/k. The opposite situation, i.e., the minima of the transmis-

sion coefﬁcient and, correspondingly, the maxima of the reﬂection coefﬁcient,

occurs if the following condition is satisﬁed:

kL = (2n − 1)

, (3.134)

which can be rewritten as

L =

2n − 1

. (3.135)

Example 3.6. An electron travels over a symmetric rectangular well of width

L and barrier height U

. Find the energy of the electron, E, that corresponds

to equal values for the coefﬁcients of transmission, D, and reﬂection, R.The

well should be considered sufﬁciently narrow, i.e., the condition kL  1must

be satisﬁed.

Reasoning. From the given condition we have that R = D = 1/2 and, taking

into account Eqs. (3.129) and (3.131), we obtain

sin

(kL) = 4



− 1



. (3.136)

3.5 Tunneling 89

Taking into account the smallness of kL, we can write sin(kL) ≈ kL:

= 4



− 1



. (3.137)

Since E = h

/(2m

), on substituting k

by 2m

E/h

in Eq. (3.137) we get the

following quadratic equation for the ratio E/U





−

. (3.138)

On solving this equation and leaving only the positive root (which is the only

one that has physical meaning) we get

= 1 +

. (3.139)

Therefore, the energy which corresponds to equal values for the reﬂection and

transmission coefﬁcients is deﬁned as

E = U



1 +



. (3.140)

3.5 Tunneling: propagation of an electron in the

region of a potential barrier

In Section 3.3 we studied the behavior of an electron in a rectangular potential

well with one barrier of ﬁnite height. This potential barrier has an inﬁnite width,

see Fig. 3.3. In this case an electron with energy, E, less than the potential-barrier

height, U

, cannot move very far from the well since the wavefunction decreases

exponentially (see Eq. (3.85)), i.e., an electron can be found only in the vicinity

of the well in the barrier region. The wavefunctions shown in Fig. 3.3 as well

as Eqs. (3.89) and (3.90) clearly demonstrate the ﬁnite probability of ﬁnding the

electron in the barrier region. If the well has potential barriers of ﬁnite height and

width, then the electron can escape the potential well even when E < U

, i.e.,

the electron can penetrate through the potential barrier. The lower the barrier

height, U

, and the smaller its width, L, the higher is the probability that the

electron can escape the well.

To understand the peculiarities of electron transmission (or, as it is often called,

tunneling) through a barrier of ﬁnite width, let us consider a free (at x ≈−∞)

electron, which is incident on a so-called rectangular potential barrier (see

Fig. 3.7). The potential barrier is a region of space where the potential energy

of a particle is greater than in the surrounding area. For a one-dimensional

rectangular potential barrier, as is shown in Fig. 3.7, the potential energy of an

90 Layered nanostructures

E,U

123

0 L

E < U

E > U

Incident

electron

Incident

electron

Figure 3.7 Electron

transmission through the

rectangular potential

barrier.

electron in different regions can be deﬁned as follows:

U (x ) =







0, x < 0,

, 0 ≤ x ≤ L,

0, x > L .

(3.141)

Let us consider further two possible situations, which correspond to under-barrier

(E < U

) and over-barrier (E > U

) electron motion.

3.5.1 Electron tunneling under the barrier

Consider an electron with energy E < U

incident from the negative side of the

x-axis on the rectangular potential barrier. The electron can be found on both

sides of the barrier. For the three regions 1, 2, and 3 (see Fig. 3.7) the Schr

odinger

equation can be written as

(x )/dx

+ k

(x ) = 0,

(x )/dx

− κ

(x ) = 0,

(x )/dx

+ k

(x ) = 0,

(3.142)

where k and κ are deﬁned by Eqs. (3.63) and (3.64): k =

√

E/h

and κ =

√

− E)/h

. The wavefunctions which are the solutions of the above-

mentioned equations can be written as

(x ) = A

ikx

+ B

−ikx

(x ) = A

κx

+ B

−κx

(x ) = A

ikx

+ B

−ikx

(3.143)

Since with the chosen direction of the electron’s incidence only an outgoing

wave can propagate outside of the barrier, B

≡ 0 as in the case of Eq. (3.123).

From the continuity conditions for the total wavefunction and its derivative at the

3.5 Tunneling 91

boundaries of the barrier such as (3.69) and (3.70), we obtain a system of four

equations, from which we can ﬁnd the relation between the coefﬁcients in the

solutions (3.143):

+ B

= A

+ B

ik(A

− B

) = κ(A

− B

κ L

+ B

κ L

= A

ikL

κ(A

κ L

− B

κ L

) = ikA

ikL

(3.144)

Most interesting for us are the coefﬁcients of reﬂection and transmission through

the barrier, R and D:

R =

, (3.145)

D =

. (3.146)

To ﬁnd R and D, we determine the coefﬁcients B

and A

from the system of

equations (3.144):

+ κ

)sinh(κ L)

− κ

)sinh(κ L) +2iκk cosh(κ L)

, (3.147)

2iκke

−ikL

− κ

)sinh(κ L) +2iκk cosh(κ L)

, (3.148)

where we introduce the hyperbolic sine and cosine,

sinh x =

− e

−x

, cosh x =

+ e

−x

By substituting Eq. (3.147) into Eq. (3.145) we ﬁnd the expression for the

reﬂection coefﬁcient, R:

R =



1 +

+ κ

)

sinh

(κ L)



−1

. (3.149)

The transmission coefﬁcient can be obtained analogously or from the expression

R + D = 1.

As a result this coefﬁcient has the form

D =



1 +

+ κ

)

sinh

(κ L)



−1

. (3.150)

Using relations (3.109) and (3.110), the coefﬁcients R and D can be rewritten in

terms of energy:

+ κ

)

E(U

− E)

(3.151)

92 Layered nanostructures

and

R =



1 +

4E(U

− E)

sinh

(κ L)



−1

, (3.152)

D =



1 +

4E(U

− E)

sinh

(κ L)



−1

. (3.153)

Thus, in the case of a barrier with ﬁnite L and U

, the electron has a ﬁnite

probability of emerging on the other side of the barrier, which from the classical-

physics point of view is impossible. This effect was called tunneling and the

phenomenon of transmission of a particle through the barrier was called the

tunneling phenomenon. This phenomenon is inherent to any quantum particle

regardless of whether the particle has an electric charge.

Suppose that the parameters of the barrier and the energy of the particle satisfy

the condition κ L  1, which can be rewritten in the following form:

− E)L



. (3.154)

In this case

sinh(κ L) ≈

, (3.155)

then the transmission coefﬁcient can be written as

D = D

−2L

√

−E)/h

, (3.156)

where

= 16

E(U

− E)

. (3.157)

From these expressions it follows that the transmission coefﬁcient rapidly

(exponentially) decreases with increasing barrier width, particle mass, and energy

difference U

− E.

In the case of a sufﬁciently narrow barrier, for which the condition κ L 

1or(U

− E)L

 h

/(2m

) is valid, the expressions for the reﬂection and

transmission coefﬁcients, R and D, take the forms

R =

1 +G

, (3.158)

D =

1 +G

. (3.159)

Here we introduced the parameter

G =

(LU

)

, (3.160)

where the product L × U

is often called “the surface” of the potential barrier.

At ﬁnite width, L, and height of the potential barrier U

→∞, or ﬁnite U

and width L →∞, the transmission coefﬁcient, D, for passage of the electron