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

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7.5 Double-quantum-dot structures (artiﬁcial molecules) 223

(1)

(0)

U = U =

(2)

Figure 7.15 The

dependences of the

energies of the

symmetric, E

(1)

,and

antisymmetric, E

(2)

, states

of the double-

quantum-dot structure on

the width of the barrier, d,

separating individual

quantum dots.

in the structure with two coupled quantum dots are in the state of lowest energy,

then the wavefunction must be symmetric. The electrons in quantum dots in this

state are considered to be in phase. If the electrons are in the state with higher

energy, then the wavefunction must be antisymmetric. The electrons in quantum

dots in such a state are considered to be out of phase.

7.5.2 Double-quantum-dot structures: exact solution

According to Fig. 7.13, for each of the three regions of the symmetric structure

we can write the Schr¨odinger equation and its solution. Here, to simplify the

calculations, we assume that the electron effective mass, m

∗

, in the potential

wells and that in the barriers are the same. We will use this assumption for all

subsequent calculations. For the regions 1 and 3, where d ≤|x|≤d + L

,we

have

−

∗

(x )

= E

(x ), (7.98)

(x ) = A

ikx

+ B

−ikx

, (7.99)

where j = 1, 3 and

k =



∗

. (7.100)

Let us assume that for the region |x|≤d with j = 2 the electron energy, E

is smaller than the height of the potential barrier, U

. Then, the Schr¨odinger

equation and its solution can be written in the form

−

∗

(x )

(x ) = E

(x ), (7.101)

(x ) = A

κx

+ B

−κx

, (7.102)

224 Quantization in nanostructures

where we introduced the parameter κ:

κ =



∗

− E

)

. (7.103)

The boundary conditions for this structure are

(−d − L

) = ψ

(d + L

) = 0, (7.104)

(−d) = ψ

(−d), (7.105)

(d) = ψ

(d), (7.106)



dψ

−

dψ





x=−d

= 0, (7.107)



dψ

−

dψ





x=d

= 0. (7.108)

Equation (7.104) reﬂects the inability of the electron to escape the considered

region and Eqs. (7.105)–(7.108) reﬂect the continuity of the wavefunction and

its derivative at the boundaries of the potential barrier U

. On substituting the

expressions for the wavefunction (7.99) and (7.102) into Eqs. (7.104)–(7.108)

we obtain two dispersion relations:

tan(kL

) +

coth(κd) = 0, (7.109)

tan(kL

) +

tanh(κd) = 0, (7.110)

where we introduced the hyperbolic functions tanh and coth:

tanh α =

coth α

− e

−α

+ e

−α

. (7.111)

The dispersion relation (7.109) corresponds to the symmetric states and the

dispersion relation (7.110) to the antisymmetric states. For sufﬁciently large

distances between the quantum dots, i.e., at κd  1, dispersion relations are

reduced to one, namely to

tan(kL

) ≈−

=−



− E

. (7.112)

For the ﬁrst two deepest levels at E

 U

we obtain from Eq. (7.112)

≈ π (7.113)

and, on substituting Eq. (7.100) into Eq. (7.113), we get

(1,2)

≈

∗

= E

(0)

. (7.114)

Therefore, when the potential wells are positioned far away from each other,

i.e., when d →∞, the levels that correspond to the symmetric wavefunction,

, and the antisymmetric wavefunction, ψ

, merge, as shown in Fig. 7.15.

7.5 Double-quantum-dot structures (artiﬁcial molecules) 225

U = U =

, E

, U

−d − L

−ddd+Lx

(2)

(1)

> U

Figure 7.16 The

wavefunctions of a

double-quantum-dot

structure, ψ

and ψ

when the quantum dots

are close to each other.

In the case of a narrow barrier (see Fig. 7.16), i.e., at κd  1, the dispersion

equations (7.109) and (7.110) become

tan(kL

) =−

, (7.115)

tan(kL

) =−kd. (7.116)

Equation (7.115) corresponds to the symmetric state and Eq. (7.116)tothe

antisymmetric state. Let us rewrite Eq. (7.116)as

= arctan(−kd). (7.117)

In the case of small d (i.e., at kd  1) we can obtain from Eq. (7.117)the

dependence of the energy of the antisymmetric state on the distance between the

quantum dots, d. We can expand the right-hand side of Eq. (7.117)inaTaylor

series taking into account only its ﬁrst two terms. Using this approximation,

arctan t can be written as

arctan t =π+ t. (7.118)

Taking into account Eq. (7.118), we can rewrite Eq. (7.117)as

= π − kd. (7.119)

By substituting into Eq. (7.119) the expression for k from Eq. (7.100), we ﬁnd

(2)

≈

∗

(1 +d/L

)

. (7.120)

We see that at d = 0 the energy of the antisymmetric state becomes equal to

the energy of the ﬁrst excited level of an individual quantum dot with width 2L

and the electron wavefunction becomes an antisymmetric function, ψ

,ofthe

226 Quantization in nanostructures

individual quantum dot of width 2L

(see Fig. 7.14):

(2)

≈

∗

= E

(0)

. (7.121)

As the distance between quantum dots, d, increases the level E

(2)

slightly

decreases, but with further increase of d the level E

(2)

tends to E

(0)

according to

Eq. (7.114). The dependence of E

(1)

, which corresponds to the symmetric state,

on d isshowninFig.7.15, but it can be obtained only by numerical solution of

Eq. (7.109). In the limiting case when d = 0, from Eq. (7.115) it follows that

kL =

(7.122)

and

(1)

(0)

. (7.123)

Let us consider now the case when the electron energy of motion along the x-

axis in the double-quantum-dot structure is greater than the height of the potential

barrier, U

, which separates the potential wells (E

> U

). The solutions of the

Schr¨odinger equation for regions 1 and 3 have the same form (7.99), and the

solution in region 2 can be written as

(x ) = A

+ B

−ik

, (7.124)

where



∗

−U

)

. (7.125)

Compare Eqs. (7.124) and (7.125) with the corresponding Eqs. (7.102) and

(7.103). After substitution of the wavefunctions ψ

(x) into the boundary con-

ditions (7.104)–(7.108) we arrive at the following dispersion relations, which

deﬁne the energy spectrum of an electron in the quantum-dot structure:

tan(kL

) −

cot(k

d) = 0, (7.126)

tan(kL

) +

tan(k

d) = 0, (7.127)

where Eq. (7.126) corresponds to the allowed symmetric states and Eq. (7.127)

to the allowed antisymmetric states. Compare Eqs. (7.126) and (7.127) with the

corresponding Eqs. (7.109) and (7.110).

Finding the general analytical solutions of Eqs. (7.126) and (7.127) is impos-

sible because these are transcendental equations. Therefore, let us consider one

of the possible particular solutions of Eqs. (7.126) and (7.127). Let us assume

that in the region of the quantum dots, 1 and 3, as well as in the region of the

barrier, 2, the electron waves are de Broglie standing waves. For the formation

7.5 Double-quantum-dot structures (artiﬁcial molecules) 227

of these waves the following conditions must be satisﬁed:

d = mπ and kL



n +



π (7.128)

for the symmetric states and

d = mπ and kL

= nπ (7.129)

for the antisymmetric states. Therefore, if condition (7.128) is satisﬁed the energy

of the stationary symmetric state must be equal to

∗



n +



= U

∗

. (7.130)

Here, we took into account that the equation for kinetic energy is deﬁned as

E =

∗

We see that m = 1 corresponds to the lowest above-barrier energy state. This

state may exist if the height of the potential barrier is equal to

∗



(n + 1/2)

−



. (7.131)

For the energy of antisymmetric states, from condition (7.129) we get

∗

= U

∗

. (7.132)

The lowest energy value corresponds to m = 1. This state can be realized if the

barrier height is equal to

∗



−



. (7.133)

Thus, different barrier heights correspond to symmetric and antisymmet-

ric above-barrier states (see Eqs. (7.131) and (7.133)). If the conditions of

Eqs. (7.131) and (7.133) are satisﬁed only one of these states can be realized.

From the discussion above it follows that by controlling the height and width

of the barrier we can shift the allowed energy levels and change the distance

between them.

From Eqs. (7.130) and (7.132) it follows that for a particular set of nanos-

tructure parameters the realization of the above-barrier regime with symmetric

or antisymmetric states, deﬁned by quantum numbers n and m, is possible only

if one of the following conditions is satisﬁed:

d >

n + 1/2

, or d >

. (7.134)

Example 7.6. Apart from under conditions (7.128) and (7.129), the station-

ary above-barrier states in a double-quantum-dot structure are possible also at

228 Quantization in nanostructures

d = (m + 1/2)π . Find the lowest energy for the above-barrier symmetric and

antisymmetric states.

Reasoning. For symmetric states, according to Eq. (7.129), the condition kL

nπ must be satisﬁed. Then, we have

∗

= U

∗



m +



. (7.135)

For antisymmetric states the condition kL

= (n + 1/2)π must be satisﬁed.

Then, we have

∗



n +



= U

∗



m +



. (7.136)

From Eqs. (7.135) and (7.136) it follows that in the case considered here the

realization of the above-barrier regime is possible only if one of the following

conditions is satisﬁed:

d >

m + 1/2

or d >

m + 1/2

n + 1/2

(7.137)

for symmetric and antisymmetric states, respectively. The lowest possible energy

of the symmetric states corresponds to m = 0 and n = 1, and that of antisymmet-

ric states corresponds to m = 0. The above-barrier symmetric state has energy

greater than that of the antisymmetric state.

Example 7.7. Write the expressions for the wavefunction and the electron energy

for double-quantum-wire and double-quantum-well structures.

Reasoning. From the nanostructures of two coupled quantum dots considered

above we can remove the restriction along one of the directions of quantization.

For example, along the y-direction we can remove the restriction for electron

motion by increasing inﬁnitely the length L

. Then, the electron motion in this

direction will be free. As a result we will obtain a nanostructure consisting of

two coupled quantum wires. For an electron in such a structure the wavefunction

and energy can be written as

ψ(x, y, z) =



sin



πn



(x ), (7.138)

E = E

∗

, (7.139)

where ψ

(x) and E

are deﬁned by the solutions of the one-dimensional

Schr¨odinger equation for the potential U(x) which we have already obtained.

If we remove one more restriction for electron motion along the z-direction by

increasing inﬁnitely the length L

, then we will obtain a nanostructure consisting

of two coupled quantum wells. The electron wavefunction and the energy for

such a nanostructure have the forms

ψ(x, y, z) = ψ

(x )e

i(k

y+k

, (7.140)

E = E

∗



+ k



. (7.141)

7.6 An electron in a periodic one-dimensional potential 229

The electron motion remains quantized only along the x-axis and electron motion

in the yz-plane is free. The forms of the one-dimensional wavefunction ψ

(x)

and energy E

are completely deﬁned by the form of the potential U(x). For the

dependence U (x) deﬁned by Eq. (7.92) the wavefunction ψ

(x) and energy E

have already been found earlier in this section.

7.6 An electron in a periodic one-dimensional potential

The most important structures which are suitable for practical applications whilst

still being simple enough for theoretical analysis are periodic structures consist-

ing of coupled nanoobjects such as quantum dots, quantum wires, or quan-

tum wells. There is a conventional name for periodic structures of repeat-

ing quantum wells. They are called superlattices. The origin of this name

is that each of the quantum wells which are the elements of the periodic

structure has its own lattice and in the superlattice the ordering of quantum

wells is superimposed on the lattice of quantum wells. Therefore it is not

simply a lattice but a superlattice. We can generalize this notion and under-

stand under the term “superlattice” those periodic structures which include not

only quantum wells but, instead of quantum wells, other nanoobjects such as

nanowires or quantum dots. From the point of view of ﬁnding solutions of the

Schr¨odinger equation, one-dimensional superlattices, which can be formed by

placing nanoobjects regularly along one of the directions, are the simplest. In

this section we will consider the general properties of electron behavior in a one-

dimensional superlattice, which do not depend on the particular form of a periodic

potential.

7.6.1 Bloch’s theorem

Let us consider the general form of the wavefunction of an electron moving in a

periodic potential (of period D) of a one-dimensional superlattice. The condition

of periodicity of the potential, which has the form

U (x + nD) = U (x), (7.142)

leads to the condition of translational invariance for the electron wavefunction

in the superlattice:

|ψ(x + nD)|

=|ψ (x)|

. (7.143)

Here n is an arbitrary integer number. The simplest example of a step-like one-

dimensional periodic potential is shown in Fig. 7.17. The last expression shows

that the probability density of ﬁnding an electron at the point with the coordinate

x is a periodic function with the period D.

230 Quantization in nanostructures

AABBB

U(x)

−d d + L

−d − L

Figure 7.17 The simplest

example of a

one-dimensional step-like

periodic potential. A

denotes the material of

the potential wells and B

the material of the

barriers which separate

potential wells. D is the

superlattice period, U

the

height of the barriers, d

the width of the barriers,

and L

the width of the

quantum wells.

The wavefunctions ψ(x) and ψ(x + nD) satisfy the same Schr¨odinger equa-

tion with the potential (7.142). Therefore, these functions must be linearly depen-

dent:

ψ(x + nD) = C (nD)ψ(x). (7.144)

From condition (7.143) it follows that

|C(nD)|

= 1, (7.145)

and from Eq. (7.145)wehave

C(nD) = e

, (7.146)

where K

is a real parameter, which has the dimension of the wavenumber.

Thus, the shift of the wavefunction’s argument, x,bynD changes the electron

wavefunction as follows:

ψ(x + nD) = e

ψ(x). (7.147)

Let us make the following transformations in Eq. (7.147):

ψ(x) = e

−iK

ψ(x + nD) = e

−iK

(x +nD)

ψ(x + nD)e

= ϕ(x + nD)e

= ϕ(x)e

. (7.148)

Here we introduced the periodic function

ϕ(x + nD) = e

−iK

(x +nD)

ψ(x + nD), (7.149)

which satisﬁes the following condition:

ϕ(x + nD) = ϕ(x). (7.150)

Note that, according to Eq. (7.150), ϕ(x) is a periodic function, whereas only

the square of the modulus of the function ψ(x), i.e., |ψ (x)|

, is periodic

7.6 An electron in a periodic one-dimensional potential 231

(see Eq. (7.143)). The wavefunction ψ (x) itself is a product of a periodic function

ϕ(x) and a free-electron wavefunction, e

Thus, the electron wavefunction, ψ(x), in the periodic potential of a super-

lattice always can be presented as a wavefunction of a free electron, e

modulated by a periodic function, ϕ(x), i.e.,

ψ(x) = ϕ(x)e

, (7.151)

where the parameter K

is an effective wavenumber (and in the general case it is

an effective wavevector) of an electron in a superlattice. This statement constitutes

Bloch’s theorem. The wavefunction (7.151) is called Bloch’s wavefunction.The

wavefunction (7.151) must satisfy the following cyclic boundary conditions:

ψ(x + L

) = ψ(x), (7.152)

where L

is the superlattice length in the x-direction. The superlattice length,

, contains a sufﬁciently large number of superlattice periods, D.Fromthis

condition, taking into account Eq. (7.151), it follows that the wavenumber, K

must have discrete values:

2π

l, l = 0, ±1, ±2,... (7.153)

The momentum of the particle p is related to the effective wavevector K as

p = h

K, and in the case of one-dimensional motion we have

= h

2π h

l. (7.154)

7.6.2 Quasimomentum

For free electron motion the magnitudes k and p are well deﬁned, in contrast

to the case for electron motion in a superlattice, for which k and p are not

unambiguously deﬁned, which is a consequence of the periodicity of U (x). Let

us change the wavenumber K by K



= K + 2πl/D, where l is an integer number.

Here we have omitted the subscript x from the wavevector K

for convenience.

Let us translate the wavefunction, ψ(x)bynD according to Eq. (7.147):

ψ(x + nD) = e

iKnD

ψ(x) = e

[



−2πl/D

]

ψ(x) = e



ψ(x)e

−i2πln

= e



ψ(x),

(7.155)

where we took into account that the product ln is an integer number and

−i2πln

= 1. (7.156)

From a comparison of Eqs. (7.155) and (7.147) it follows that the electron

quantum states with wavenumbers K and K



= K + 2πl/D in the superlattice

correspond to the same physical state. Because of this equivalence for the electron

in the superlattice the magnitude K is called the quasiwavevector rather than just

232 Quantization in nanostructures

the wavevector. We can also talk about the physical equivalence of the states with

magnitudes of momentum in the superlattice p and p + 2πl/D. Therefore, p is

called the quasimomentum.

In Chapter 2 it was shown that for a free particle the principle of conservation

of momentum p holds. This is due to the translational homogeneity of free space

(of vacuum). In the superlattice the homogeneity of space is broken because

the potential energy of the electron becomes a periodic function. The existence

of a periodic potential in the superlattice leads to the conservation not of the

momentum of the electron but of its quasimomentum. Thus, for the electron’s

quasimomentum Newton’s second law is valid:

= F

ext

, (7.157)

where the force F

ext

is related to the external ﬁelds and does not include the

periodic force F

related to the superlattice potential, U(r), by

=−∇U(r). (7.158)

The interval of quasimomenta which contains all physically non-equivalent states

is symmetric relative to the origin of the coordinates, i.e.,

−

π h

≤ p ≤

π h

. (7.159)

This interval is called the ﬁrst Brillouin zone. The second zone is located sym-

metrically with respect to the ﬁrst zone with respect to the origin of coordinates.

It includes two intervals:



−

2π h

, −

π h



and



π h

2π h



. (7.160)

Then, the third Brillouin zone is located similarly and so on. The width of the

interval which corresponds to any Brillouin zone is always equal to 2π h

/D.

Let us now discuss what values the quasiwavevector K = p/h

, which deﬁnes

the boundaries of the Brillouin zone, may have. Let us assume that an electron

with wavevector K is moving along the x-direction and that the period of the

superlattice in this direction is equal to D. Let us assume that the interaction

of the electron with an individual nanoobject of the superlattice is weak.This

means that during propagation of the electron wave in the periodic potential of the

superlattice the amplitude of the reﬂected wave from an individual nanoobject

is much smaller than the amplitude of the incident electron wave. Since the

superlattice contains many nanoobjects, the phase relationships between the

reﬂected waves play a signiﬁcant role. If the electron de Broglie wavelength

exceeds the distance between nanoobjects, i.e., exceeds the period D, then the

phase difference between the waves reﬂected from neighboring objects is small.