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

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8.3 Selected examples of nanodevices and systems 283

GaAs

(a)

-GaAs

(b)

Quantum well

Potential barrier

Figure 8.15 (a) The

schematic structure of a

GaAs/Al

1−x

resonant-tunneling diode

and (b) its potential

proﬁle. U

is the height of

the potential barriers, L

the thickness of the

barriers, and L

the width

of the quantum well.

(a)

(b)

(c)

Figure 8.16 The system of

energy levels in a

resonant-tunneling diode:

(a) under equilibrium

conditions, (b) at

resonance under an

applied voltage, and

applied voltage. The

electron potential energy

and population of the

energy band in contact

-regions are sketched

in the left and right parts.

denotes the bottom of

the conduction band.

8.3.1 Resonant-tunneling diodes

A nanoscale phenomenon called resonant tunneling can be described as fol-

lows. The current through a nanostructure consisting of two potential barriers

with a quantum well between them (a so-called double-barrier structure), sharply

increases when the Fermi energy of the electrode that supplies electrons becomes

equal to the energy level in the quantum well. Such a double-barrier structure

with two contacts is called a resonant-tunneling diode (RTD). The scheme of

an RTD and its potential proﬁle are shown in Fig. 8.15. The system of energy

levels of such a simple structure is shown in Fig. 8.16. This structure consists

of two barriers divided by a region with low potential energy (a quantum well

with quantized electron levels). Let us assume that the quantum well has only

284 Nanostructures and their applications

(c)

(b)

(d)

(a)

Figure 8.17 An energy

diagram and

current–voltage

characteristic of the

resonant-tunneling diode:

the portions labeled (a),

(b), and (c) correspond to

the physical situations of

Fig. 8.16. Part (d) shows

what happens with the

current when the second

electron state enters into

resonance with the Fermi

level, E

a single discrete energy level. The thickness of the barriers and width of the

quantum well can be several nanometers. The regions in the right-hand and left-

hand sides from the barriers serve as electron reservoirs, and these regions can be

considered as the contacts to the structure. Such a double-barrier structure has one

distinctive feature: its tunneling probability has a resonant character. Even if the

transmission through each barrier is low, this does not mean that the probability of

electron tunneling through the double-barrier structure is small. When the energy

of electrons incident on the barrier is equal to the energy of one of the discrete

levels of the quantum well between the two barriers, the tunneling probability for

the double-barrier structure increases sharply. The effect of resonant tunneling

can be understood in terms of interference. Indeed, due to the interference of

the transmitted and reﬂected electron waves the wave that is reﬂected from the

double barrier returns to the quantum well. Therefore, the wave incident on the

double-barrier structure from the left has a high probability of being transmitted

to the right.

Let us describe how the RTD works. The current that ﬂows through the double-

barrier structure depends on the applied voltage. Note that the voltage is mostly

applied to the double-barrier region, since the regions from the left-hand and

from the right-hand sides of the barriers have higher conductivity. If the applied

voltage is small and the energy of electrons incident on the barrier from the

left is lower than the energy of the discrete energy level in the quantum well,

then the transmission through the barrier and, consequently, the current through

the double-barrier structure will be small (see part (a) of the curve in Fig. 8.17

and Fig. 8.16(a)). The current reaches its maximum at the voltage at which the

energy of electrons becomes equal to the energy of the discrete energy level

(see part (b) of the curve in Fig. 8.17 and Fig. 8.16(b)). At higher voltages the

energy of incident electrons becomes higher than the energy of a discrete energy

level (see part (c) of the curve in Fig. 8.17 and Fig. 8.16(c)). As a result, the

8.3 Selected examples of nanodevices and systems 285

tunneling probability of the double-barrier structure decreases and consequently

the current decreases. Thus, there appears a maximum of the current–voltage

characteristic, which is followed by a region with negative-differential resistance

(NDR), i.e., by a region where the current decreases with increasing voltage.

With the further increase of the applied voltage, the current begins to increase

again. If the potential well between two barriers has several discrete energy levels,

then the current–voltage characteristic will have several maxima, each of them

at a voltage at which the corresponding energy level comes into resonance with

the Fermi energy, E

, of the left-hand-side region (see part (d) of the curve in

Fig. 8.17).

Having these regions, the RTD may be used in electronic circuits not only

as a rectiﬁer but also for performing various other functions. The existence

of such an NDR region (or NDR regions) is promising for the development

of multilevel logical elements, memory elements, and solid-state ultra-high-

frequency generators. The addition of a control electrode in the central region

of an RTD transforms it into a resonant-tunneling transistor and increases the

number of its applications. Such transistors have turn-off and turn-on frequencies

of about 10

Hz. This is several orders of magnitude higher than the frequencies

of the best silicon transistors in modern integrated circuits. Therefore, such

transistors can be used in the next generation of integrated circuits.

8.3.2 Single-electron transistor s

The operation of single-electron devices is based on the effect of tunneling of

a single electron. Various structures and devices based on this effect have been

realized experimentally. In the simplest version, this effect can be observed in

a single-electron tunnel junction. The single-electron tunnel junction may be

created in the form of two plates (we consider here the simplest case, in which

both plates are made of the same metal) of small cross-sectional area, S, and a thin

layer of dielectric of thickness d between them (see Fig. 8.18). The capacitance

of this system is given by

C = 

, (8.15)

where  is the dielectric permittivity of the dielectric separating the metallic

plates. The potential difference between the plates is determined by the charge

accumulated on these plates:

U =

. (8.16)

If the charge, Q, is reduced by the charge of one electron, e, then the potential

difference becomes

U =

Q − e

. (8.17)

286 Nanostructures and their applications

Metal

Dielectric

Tunnel junction

Metal

+Q−Q

Figure 8.18 A capacitor

with an induced charge

and a single-electron

tunnel junction.

As a result of transmission of a single electron from one electrode through

the dielectric layer to the opposite electrode, the voltage at the tunnel junction

changes by the following magnitude:

U =

. (8.18)

The energy stored in the capacitor is deﬁned as

E =

. (8.19)

The change of the energy stored in the capacitor as a result of transfer of a single

electron is equal to

E =

−

(Q − e)

e(Q − e/2)

. (8.20)

Electron tunneling occurs only if E > 0, i.e., if energy is decreased due to the

transfer of a single electron. Thus, from Eq. (8.20) it follows that, for electron

tunneling to occur, the initial charge on a metallic plate must satisfy the following

condition:

Q ≥

. (8.21)

This corresponds to the voltage between plates, U, being equal to

U ≥

. (8.22)

Thus, by applying a voltage higher than e/(2C), we can make current ﬂow. Since

the current cannot ﬂow if

U <

, (8.23)

the effect is known as Coulomb blockade. The current–voltage characteristic for

a device with Coulomb blockade is shown in Fig. 8.19.

8.3 Selected examples of nanodevices and systems 287

0 U

Figure 8.19 The

current–voltage

characteristic for a device

with Coulomb blockade.

Because of the nanometer width of the tunnel junction, d, and small capaci-

tance, C, each electron passing through the junction causes a substantial change

in the voltage U (see Eq. (8.18)). If the condition (8.22) is not satisﬁed, the elec-

tron tunneling is blocked by a large blocking voltage, caused by the tunneling of

a single electron. The energy which is necessary to charge the capacitor in order

to satisfy the condition (8.22) is equal to

= eU =

. (8.24)

This is the charging energy required to add a single electron with charge e to the

capacitor. Note that for Q = 0 the expression (8.20) is reduced to Eq. (8.24).

Let us estimate the capacitance, C, and the energy, E

, required to transfer

a single electron in nanoscale capacitors. For example, for a capacitor with

S = 100 nm

, d = 10 nm, and  = 1 (we assume that the dielectric is vacuum)

we get C ≈ 9 × 10

−18

F and E

≈ 0.9 eV. The energy needed to transfer a single

electron is equal to almost 1 eV! Let us compare the capacitance and charging

energy of a nanoscale capacitor with those for a macroscopic capacitor with

S = 100 mm

, d = 1 mm, and  = 1. For these parameters we get C ≈ 9 ×10

−13

F and E

≈ 9 × 10

−8

eV. We see that the difference between the energy needed

to transfer a single electron, E

, in the case of a nanoscale capacitor and that for

a macroscopic capacitor is enormous. In the case of a macroscopic capacitor, E

is almost negligible. The charging energy of the macroscopic capacitor (E

9 × 10

−8

eV) is lower by six orders of magnitude than the thermal energy of

an electron at room temperature (E

= 2.5 × 10

−2

eV)! Therefore, Coulomb

blockade cannot be observed in macroscopic objects.

Let us now consider the Coulomb-blockade effect in a quantum-dot single-

electron transistor, as shown in Fig. 8.20. Just like any conventional transistor,

it has three electrodes: source, drain, and gate. A quantum dot is placed in

288 Nanostructures and their applications

Gate

Source

Drain

Dielectric

Tunnel

junctions

Figure 8.20 A schematic

representation of a

quantum-dot

single-electron transistor:

the quantum dot (QD) is

isolated from the source,

drain, and gate by

dielectric.

vac

(a)

vac

(b)

Source

Drain

Figure 8.21 Band

diagrams of a

quantum-dot

single-electron transistor

with source–drain voltage,

, equal to zero: (a)

gate voltage U

= 0and

(b) gate voltage U

> 0.

Here, E

= e

/ (2C).

the channel, i.e., in the region between the source and the drain. The quantum

dot, which is often called the island, is either a metallic or a semiconductor

nanoparticle, which is isolated from all three electrodes by a dielectric. Electron

motion can take place only through the dielectric, under certain conditions.

Here we give you a very simpliﬁed version of the operational principles of

a quantum-dot single-electron transistor. The band diagrams of a quantum-dot

single-electron transistor without and with an applied positive gate voltage, U

are shown in Fig. 8.21 (the voltage between source and drain, U

, is equal to

zero). The quantum dot (the island) in the channel of a single-electron transistor

is a nanoparticle, which is made from the same metal as the source and the drain.

AsshowninFig.8.21(a), the electrons in the source and the drain have higher

energy than the electrons have in a quantum dot. Therefore, it is energetically

8.3 Selected examples of nanodevices and systems 289

favorable for an electron from the source or the drain to tunnel to the quantum

dot. However, because of Coulomb blockade, the electron must have energy

greater than the charging energy E

. At zero gate voltage (U

= 0), as shown in

Fig. 8.21(a), the electron in the source has energy lower than E

and therefore

tunneling of the electron from the source to the quantum dot cannot happen. If

a positive gate voltage (U

> 0) is applied to the gate, as shown in Fig. 8.21(b),

the energy in the dot is decreased by E

and the electron from the source will

be able to tunnel to the quantum dot and then to the drain. Thus, by controlling

the potential on the gate we can control the transmission of a single electron

through the dielectric barrier and therefore control the current in a circuit. This

is the essential feature of a transistor. If the electron that tunneled from the source

stayed in the dot rather than going into the drain, then the energy in the dot would

increase by E

and tunneling of the next electron would be blocked.

In digital integrated circuits based on single-electron transistors one bit of

information (two possible states, 0 or 1) may be presented as the presence or

absence of a single electron in the quantum dot of the channel. A whole memory

block with a capacity of 10

bits (which is 100 times higher than that which

modern chips have) could be placed on a crystal with a surface area of 1 cm

The practical realization of such devices is the goal of researchers all over the

world.

8.3.3 Quantum-optoelectronic devices

Optoelectronics is a ﬁeld of physics and technology that is related to the con-

version of optical radiation into electrical signals and vice versa. The number of

optoelectronic devices is large. The operational principles of these devices are

based on quantum mechanics. We can distinguish two main types of optoelec-

tronic devices: (1) light-emitting devices and (2) light-absorbing or photodetect-

ing devices.

Light-emitting devices

These are the devices that convert electric current into light. They can be

incandescent lamps, electroluminescent indicators, semiconductor light-emitting

diodes (LEDs), and lasers. The most widely used devices developed with the help

of nanotechnology are LEDs and lasers. We will talk about lasers – sources of

coherent radiation – later. Now we will discuss the operational principles of the

LEDs with a p–n-junction.

Light-emitting diodes are sources of non-coherent radiation. They emit light

only when the current through the p–n-junction ﬂows in the forward direction.

The forward direction for the p–n-junction corresponds to the case when a

negative potential is applied to the n-region. Electrons come into the junction

region from the n-region and holes come from the p-region. After recombination,

holes and electrons disappear and their energy in the form of a light quantum

290 Nanostructures and their applications

- electron

- hole

Light

(a)

p-type

(b)

electrons

holes

n-type

Forward direction

Interface between

n- and p-type regions

Figure 8.22 A schematic

representation and band

diagram of a

semiconductor

light-emitting diode.

is emitted, as shown in Fig. 8.22. The light intensity should be proportional

to the current ﬂowing through the LED. However, when the current increases

the thermal heating of the LED also increases, leading to an increase of the

non-radiative losses in the LED.

For a long time mass production of LEDs was limited to LEDs emitting only

in the red and infrared regions of the electromagnetic spectrum. Green LEDs

were made ﬁrst from gallium phosphide (GaP) and blue LEDs from silicon

carbide (SiC). However, these materials either emitted low-intensity radiation or

quickly became overheated due to their low efﬁciency. Nitrides, materials based

on elements from the third and ﬁfth groups of the Periodic Table of the elements

(AlN, GaN, and InN), have become practical for the fabrication of LEDs in the

visible range of the electromagnetic spectrum. These materials emit throughout

the entire range of the visible and ultraviolet regions of the electromagnetic

spectrum from 240 to 640 nm. During the 1970s scientists from IBM fabricated

violet and blue diodes based on epitaxial ﬁlms of GaN. However, these diodes

quickly overheated and thus became non-operational.

The development of nanotechnology has allowed the growth of heterostruc-

tures constructed on the basis of GaN, AlN, and InN that can be used for the

fabrication of heterojunction LEDs. The junction is called a heterojunction when

the interface is formed by two different materials. The junction is called a homo-

junction if both parts of the junction are made from the same material with

different doping. The most frequently encountered type of homojunction is the

p–n-junction, which has the same material on both sides of the junction, doped

with donors (in the n-region) and acceptors (in the p-region) (the p–n-junction

showninFig.8.22 is a homojunction). A GaInN/GaAlN heterojunction is shown

in Fig. 8.23.

8.3 Selected examples of nanodevices and systems 291

(b)

p-type

- electron

- hole

Light

(a)

n-type

GaAlN

GaInN

Heterointerface

Figure 8.23 A schematic

representation and band

diagram of an GaInN/

GaAlN light-emitting

diode.

The advantage of a heterojunction lies in the possibility of controlling the

doping in n- and p-regions independently of the size of the region where electrons

recombine: in the case of the heterostructure shown in Fig. 8.23 the recombination

region is controlled by the thickness of the GaInN layer. Doping is the addition

of impurity atoms to a semiconductor, which deﬁnes its type of conductivity,

p or n. The resistance of p- and n-regions signiﬁcantly decreases when they

are heavily doped. Therefore, when current ﬂows through a more highly doped

heterojunction there is less heat production in the highly doped n- and p-regions

and lower currents may be used if the recombination region is made narrower

than in the case of a homojunction. The concentration of electrons and holes in

the narrow recombination region of the LED will be signiﬁcantly higher than in

the case of a homojunction and, therefore, the intensity of radiation will also be

higher. The materials that can be grown by means of modern technology allow

one to cover the entirety of the visible spectrum.

Lasers on quantum wells

A laser, which is the acronym of light ampliﬁcation by stimulated emission

of radiation, is an optical quantum generator, i.e., it is a source of coherent

electromagnetic radiation. Here, we will discuss lasers for the visible or close-

to-visible range of the electromagnetic spectrum. Note that modern lasers cover

all frequencies from X-rays (wavelength ∼10 nm) up to 1 THz (wavelength

∼300 µm). The operation of the laser is based on the utilization of the stimulated

emission (or induced emission) of atoms, molecules, and crystalline structures.

The coherence of laser radiation is due to the fact that light waves are emitted in

a coordinated fashion: the emitted light waves have the same frequency, wave-

length, and direction of propagation. The most compact, efﬁcient, and reliable

292 Nanostructures and their applications

lasers are semiconductor lasers. Because of these properties they are used in

CD and DVD players, laser printers, and computers. Telephony, the Internet, and

optical and other types of cable communications got their second lease of life

due to a wide use of semiconductor lasers.

The operation of any type of laser is based on two important quantum phe-

nomena: (1) the existence of states with population inversion and (2) stimulated

emission. A medium containing a system of quantum particles (atoms, molecules,

or ions), whose quantum state with higher energy (excited state) can be populated

signiﬁcantly more than the state with lower energy, is called an active medium.

The state of the active medium in which the majority of all particles of the system

is in the excited state is called the state with population inversion. The excita-

tion of a quantum system of particles is done by the pump, i.e., a constant or

pulsed action on the active medium to maintain the state of population inversion

of carriers. As an example, we consider here an active medium that consists of

atoms rather than molecules or more complicated objects. In the active medium

with population inversion photons, which either are injected into the medium or

are created as a result of transitions of atoms from the excited to the non-excited

state, are scattered more often by excited atoms than by non-excited atoms.

The emission radiated during the spontaneous transition of an electron in

the excited atom from a higher energy level to the lower energy level is called

spontaneous emission. Spontaneous emission, which occurs from a large number

of atoms in the active medium, happens incoherently (i.e., in non-coordinated

fashion), since quantum transitions of electrons with the emission of photons

occur in each atom independently. However, the transition of an electron from a

higher energy level, E

, to a lower energy level, E

, not only may be spontaneous,

but also can happen under the inﬂuence of an external electromagnetic ﬁeld, for

example as a result of an atom’s interaction with the photons which already exist

in the system.

For such a type of transition the photon frequency, ω, must be equal to the

frequency ω

ω = ω

− E

. (8.25)

Such radiation is called stimulated or induced. As a result of such an interaction

of the excited atom with a photon whose frequency is equal to the frequency of

the electron quantum transition from state E

to state E

, two photons absolutely

identical in energy, direction of propagation phase, and polarization occur (see

Fig. 8.24). Thus, the electrons from the energy levels with population inversion

emitting induced radiation amplify the original light beam, forming a light beam

that is coherent, narrow, and polarized.

For the operation of any type of laser it is necessary to have an optical

resonator, which is usually a system of two parallel mirrors. The role of the

resonator is to conﬁne electromagnetic radiation within the active medium and