1

Universidade de

Aveiro

Departamento de Física

Nikolai A. Sobolev, Svetlana P. Kobeleva

Physics and technology of nanosize structures

2014/2015

Национальный исследовательский технологический

университет «МИСиС»

National University of Science and Technology

"MISIS"

Education is what remains after all school learning is forgotten. It is less the factual knowledge that is important, than the facility one requires for scientific reasoning.

Max von Laue (The Nobel Prize in Physics 1914)

0. INTRODUCTION

The principle of increasing the emitter efficiency, by using a larger band gap

than in the base (a heterojunction), was mentioned by W.B. Shockley in his

extensive transistor patent (filed 1948, granted 1951) and was discussed

theoretically by I. Gubanov in 1951.

Now, it is almost impossible to imagine the modern solid-state physics

without semiconductor heterostructures (HSs), quantum wells (QWs), superlattices

(SLs) and nowadays also quantum dots (QDs). They are the investigation subjects

of 60 to 70 % of all research groups in the field of semiconductor physics.

The possibility to control the type of conductivity by means of the doping

with various impurities and the idea of the injection of non-equilibrium charge

carriers led to an enormous development and growth of the semiconductor

electronics.

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The advent of the HSs, QWs, SLs and QDs allows to solve a much more

general problem, namely, that of controlling the fundamental parameters of

semiconductor materials and devices: energy gap, effective masses, carrier

mobilities, refractive index, electron energy spectrum, etc.

Our daily life has been changed due to the development of the

semiconductor quantum size structures too. They are used in the optical fibre

telecommunication systems (Internet!), in the high frequency systems like satellite

television. They are used as light detectors, solar cells (the latter especially at the

space stations). Even at home, everybody has a double heterostructure or QW

diode laser inside a CD player.

It is worth noting that the Nobel Prize in Physics for the year 2000 has been

awarded for the invention of heterostructures and other quantum-size

semiconductor components. Here is an excerpt from an official memorandum of

the Nobel Prize Committee:

“Through their inventions this year's Nobel Laureates in physics have laid a

stable foundation for modern information technology. Zhores I. Alferov and

Herbert Kroemer have invented and developed fast opto- and microelectronic

components based on layered semiconductor structures, termed semiconductor

heterostructures. Fast transistors built using heterostructure technology are used in

e.g. radio link satellites and the base stations of mobile telephones. Laser diodes

built with the same technology drive the flow of information in the Internet's fibre-

optical cables. They are also found in CD players, bar-code readers and laser

pointers. With heterostructure technology powerful light-emitting diodes are being

built for use in car brake lights, traffic lights and other warning lights. Electric

bulbs may in the future be replaced by light-emitting diodes.” (Jack S. Kilby, the

third Nobel laureate in Physics of the year 2000, has been rewarded for his part in

the invention and development of the integrated circuit, the chip.)

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1. METAL-SEMICONDUCTOR AND SEMICONDUCTOR-

SEMICONDUCTOR CONTACTS

A contact of two different semiconductors or of a semiconductor and a metal

causes the appearance of potential barriers in the interface layers. The charge

carrier concentration in the layers can be substantially different from those of the

bulk.

1.1. METAL-SEMICONDUCTOR CONTACT

1.1.1. Potential barriers in the contact region

We must use the same energy scale for the quantum states of the metal and

the semiconductor. We choose as a reference the energy of a static electron at

infinity and call this the “vacuum energy level”.

Let us consider an n-type semiconductor:

(A) Before contact (separated metal and semiconductor, Fig. 1-1a)

Fig. 1-1

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A – thermodynamic work function of the electron (NB! The holes cannot leave the

solid!);

ς – chemical potential of the electrons;

χ – electron affinity of the semiconductor (external work function).

(B) Contact before establishing the equilibrium (Fig. 1-1b)

As soon as the contact is established, the electrons from the semiconductor

can pass into the metal by tunnelling and vice versa. Therefore, two currents

appear, namely j1 and j2. (Note that the current directions are opposite to that of the

electron velocities ve1 and ve2.)

If the currents are caused by the thermionic emission, they can be calculated

using the Maxwell-Boltzmann statistics:

* 2 ,

1,2 exp S MeAj A T

kT

, (1)

where 2

* 2 2120 A/cm /K2 32

em k mn nA

me

(2)

is called the Richardson constant.

(a) If AMe > AS, then j1 > j2, and more electrons pass from the

semiconductor into the metal than vice versa.

(b) If AMe < AS, then j1 < j2. NB!! But it is very rarely the case.

The very first theory of the metal-semiconductor contact based upon the

tunnelling phenomenon has been developed as early as 1932 by Joffe and Frenkel.

(C) Contact in equilibrium

The equilibrium condition of two bodies being able to exchange particles (in

an arbitrary way) is

( )FE r const

(3)

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Let us consider the case AMe > AS (j1 > j2):

Neutral region

Evac

++

++

W

Neutral regionSpace charge

(depletion) region

0 x

++

EF E

F

Fi

Ev

Ec

Metal

Evac

++ +++

+ ++ +

eUc

+

xn

E

B

AS

AMe

Built-inel. field

n-Typesemicond.

Fig. 1-2. Energy band scheme of a contact of an n-type semiconductor and a metal with AMe > AS.

Semiconductor is charged positively, metal does negatively. An electric field

emerges that hinders the charge carriers from being further separated. No current

flows anymore. This is a Schottky barrier.

The contact potential difference is

Me Sc | |

A AU

e

(in this case < 0) (4)

and the energy barrier is

b MeA (5)

Uc is also called the built-in voltage (Ubi).

Uc is the potential difference in the bulk of the semiconductor and the contact

(interface) plane.

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W is the thickness of the space charge region. In the case shown above, the

positive charge of the semiconductor is caused by the bound charges of the

ionised donors and the charges of the free holes. As the electron concentration

in the metal is very high, the screening is very strong, and only the surface of

the metal is affected.

The band bending is caused by the mutual dependence between n (p) and EF.

E.g., for the non-degenerated case we have

exp

exp

F CC

V FV

E En N

kTE E

p NkT

(6)

In practice, Schottky barrier heights are quite different from those predicted

by Eq. (5) and shown in Fig. 1-2. Unlike in the simple theory, the Schottky barrier

height is only weakly dependent on AMe. Bardeen developed a model explaining

this phenomenon by the effect of the surface states at the boundary between the SC

and a thin oxide layer that is almost always present at the surface. However,

Bardeen’s model cannot explain many properties of the Schottky barrier diodes.

Other models have been developed, but none of them are completely conclusive.

That is why the effective barrier height is usually determined from the

experimental data (current-voltage characteristics, internal photoemission and

capacitance measurements).

In an ideal contact between a metal and a p-type semiconductor, the

barrier height is

b g MeΦ E A (7)

It follows from Eqs. (5) and (7) that for a given SC and an arbitrary metal the

sum of the barrier heights for the n- and p-type is equal to the band gap:

bn bp gΦ Φ E . (8)

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1.1.2. Space charge region

The width of the space-charge region is found by solving the Poisson

equation: in the depleted region of the semiconductor the charge density is eNd. (e

= |e|).

From 02

2

deN

dx

Vd , (9)

and the boundary conditions for x = W

0,

( ) (0) ( ),

dVdx

e V W V A AMe S

(10)

we deduce that the bending is parabolic and that

SAMeAWdNe

02

22

or

22 ( ) / ( )Me0 dsW A A e N . (11)

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1.1.3. Ohmic contacts

Since the situation AS > AMe is very rare, there is a question how to fabricate

a low-resistance metal-semiconductor contact with a linear IU-characteristic

(“Ohmic contact”).

A practical way to obtain a low resistance Ohmic contact is to increase the

doping near the Me-SC interface to a very high value so that the depletion layer

caused by the Schottky barrier becomes very thin, and the current transport through

the barrier is enhanced by tunneling (Fig. 1-3).

Fig. 1-3. A real Ohmic contact

(a) in equilibrium;

(b) under forward bias.

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1.2. CLASSICAL HETEROSTRUCTURES

1.2.1. Semiconductor heterojunctions

Let us consider the contact of two different semiconductors with Eg1 > Eg2

(bandgaps of some III–V ternary and quaternary alloys are given in Fig. 1-4).

(A) A1 > A2.

Contact potential difference 02121

e

AAcUcUcU .

Conduction band offset 12 cE .

Valence band offset 212121 gEgEcEgEgEvE .

Fig. 1-4. Bandgap versus lattice parameter for elementary and compound semiconductors.

In a heterojunction, the heights of the potential barriers are different for the

electrons and the holes (see two examples in Fig. 1-5). This difference is one of the

main advantages of the heterojunctions as it allows to vary the ratio of the electron

and the hole currents.

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Fig. 1-5

The possible conduction and valence band offsets for different relative

values of A and χ are shown in Fig. 1-6.

Fig. 1-6

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The types of heterojunctions are classified according to the relative position

of the band edges. The possible arrangements, corresponding band bendings and

multilayer structures are shown in Fig. 1-7.

Fig. 1-7

1.2.2. Two-dimensional electron gas in a heterojunction

If e.g. a doped AlGaAs layer is grown on top of an undoped GaAs substrate,

a 2D electron gas is formed near the interface due to a difference in the electron

affinities and workfunctions.

The electron motion is confined and therefore quantized perpendicularly to

the junction plane because the de Broglie wavelength is greater that the potential

well width. That means the electrons are confined within a quantum well that in

this case has a triangular form.

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Fig. 1-8

NB!! In the xy-plane the electron motion is not confined and therefore not

quantized, therefore instead of sharp energy levels we have energy subbands.

Example: GaAs, T = 300 K → λ = 26 nm and it increases with decreasing

temperature.

This quantization effect is well known in the inversion layers on Si,

however, in AlGaAs/GaAs heterostructures it is more pronounced because of the

smaller effective mass of electrons in GaAs (m*(Si) = 0.26m0, m*(GaAs) =

0.067m0).

The underlying idea is that, at equilibrium, charge transfer occurs across the

heterojunction to equalize the Fermi level on both sides.

The electrons are transferred from the doped AlGaAs to the undoped GaAs.

As a result, conduction electrons appear in a high-purity, high-mobility

semiconductor without introduction of mobility-limiting donor impurities. It is

called the modulation doping of heterostructures.

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There are many high-performance devices based on this phenomenon:

HEMT (high electron mobility transistor), TEGFET (two-dimensional electron gas

FET), SDHT (selectively doped heterostructure transistor), MODFET

(modulation-doped FET), etc.

The Fermi level throughout the structure, and therefore the charge transfer,

have to be calculated. The qualitative result of simple calculations for the AlGaAs–

GaAs system is shown in Fig. 1-9.

Fig. 1-9

To achieve a high mobility in GaAs, the latter is usually nominally undoped

to avoid the charge carrier scattering by impurities. As a rule, it is a weakly doped

p-type with an unintentional acceptor concentration of ~ 1014 cm–3 which

corresponds to a surface acceptor density QB/e = 41010 cm–2. This is a small value

as compared to the typical surface electron densities in the 2D gas, Ns ~ 1011 – 1012

cm–2 (see Fig. 1-10).

All calculations rely on the assumption that the wavefunction of the GaAs

electrons has a negligible penetration into the barrier. In fact, the maximum

penetration measured in the experiment does not exceed ~ 20 Å << W. However,

an undoped AlGaAs “spacer” layer of thickness Wsp is usually used to further

separate the ionized donor atoms from the channel electrons: increasing this spacer

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layer diminishes the Coulomb interaction between the ionized donors and the

electrons, resulting in an increased mobility. But increasing Wsp tends do decrease

the channel electron density Ns.

The experimental determination of the density of transferred electrons as a

function of the doping level ND2 and the spacer thickness Wsp is shown in Fig. 1-10.

Fig. 1-10

As was mentioned above and can be seen from Fig. 1-10, in single

GaAs/AlGaAs heterostructures Ns do not exceed 1012 cm–2. Inclusion of an

undoped spacer layer additionally diminishes the charge transfer. In order to obtain

higher carrier densities in the channel, modulation-doped multiple quantum well

(MQW) structures are used.

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A historical view of the electron mobility ZnO and GaAs heterostructures

showing the steps leading to its improvement is given in Fig. 1-11.

Fig. 1-11. A historical view of the mobility of electrons in ever-cleaner ZnO and GaAs heterostructures and the steps leading to this improvement. (a) The mobility progress achieved for ZnO up to the record mobility of 180 000 cm2/(V·s). The highest mobility reported for a bulk single crystal of ZnO is also shown for comparison. (b) The mobility progress achieved for GaAs over the last three decades up to the present mobility record of 36 000 000 cm2/(V·s). The curve labelled ‘bulk’ is for a GaAs single crystal doped with the same concentration of electrons as the 2DEGs. MBE, molecular-beam epitaxy; UHV, ultra-high vacuum; LN2, liquid nitrogen; ‘undoped setback’, an undoped layer prior to the modulation doping to further separate the ionized impurities from the 2DEG.

1.2.3. Heterojunction bipolar transistor

The heterojunction bipolar transistor (HBT) is a bipolar junction transistor

(BJT) where differing semiconductor materials are used for the emitter-base

junction and the base-collector junction, creating heterojunctions. The effect is to

limit the injection of holes from the base into the emitter region, since the potential

barrier in the valence band is higher than in the conduction band. Unlike BJT

technology, this allows a high doping density to be used in the base, reducing the

base resistance while maintaining gain.

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The HBT improves on the BJT in that it can handle signals of very high

frequencies, up to several hundred GHz. It is commonly used in modern ultrafast

circuits, mostly radio-frequency (RF) systems, and in applications requiring a high

power efficiency, such as RF power amplifiers in cellular phones.

Fig. 1.12. Energy band diagram of a HBT.

1.2.3. High-electron-mobility transistor

A high-electron-mobility transistor (HEMT), also known as heterostructure

FET (HFET) or modulation-doped FET (MODFET), is a modification of the field-

effect transistor using a heterojunction with a 2DEG as the channel instead of a

doped region. The classical material combination is again GaAs with AlGaAs. In

recent years, gallium nitride HEMTs have attracted attention due to their high-

power performance. HEMT transistors are able to operate at higher frequencies

than ordinary transistors, up to millimeter wave frequencies.

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Fig. 1.13. Band diagram of GaAs/AlGaAs heterojunction-based HEMT, at equilibrium.

The Fermi level of the gate metal is matched to the pinning point, which is

1.2 eV below the conduction band (Fig. 1.13). With the reduced AlGaAs layer

thickness, the electrons supplied by donors in the AlGaAs layer are insufficient to

pin the layer. As a result, band bending is moving upward and the 2DEG does not

appear. When a positive bias greater than the threshold voltage is applied to the

gate, electrons accumulate at the interface and form a 2DEG.

The HEMTs find applications in microwave and millimeter wave

communications (e.g., cell phones, satellite television receivers), imaging, radar,

and radio astronomy – any application where high gain and low noise at high

frequencies are required. HEMTs have shown current gain to frequencies greater

than 600 GHz and power gain to frequencies greater than 1 THz. Furthermore,

gallium nitride HEMTs on silicon substrates are used as power switching

transistors for voltage converter applications. Compared to silicon power

transistors gallium nitride HEMTs feature low on-state resistances, and low

switching losses due to the wide bandgap properties.

A traditional HEMT structure is conductive at zero gate bias voltage, due to

the polarization-induced charge at the barrier/channel interface. Consequently,

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depletion-mode (D-mode) transistors are well-studied and have been further

developed for several years.

The properties of GaN and AlN and their heterostructures have encouraged

the research of AlGaN/GaN based transistors for various applications.

Consequently, outstanding results have been reported for the D-mode HEMTs.

However, for several applications enhancement mode (E-mode) devices are

essential.

1.2.4. Quantum Hall effect

Fig. 1.14. (a) Drawing of a typical Hall bar, fabricated from a two-dimensional electron gas formed in a GaAs/AlGaAs heterostructure.

(b) At a temperature of T = 1.3 K, when the perpendicular magnetic field B is increased, the Hall (transverse) resistance RH, defined as the ratio of the transverse voltage VH to the current I, shows plateaus at values RK/i, where i is an integer and RK theoretically equals h/e2. Simultaneously, the longitudinal resistance (Rxx = Vxx/I, where Vxx is the longitudinal voltage) drops to zero, reflecting the absence of dissipation in the 2DEG. The value of the Hall resistance on the plateaus is a very reproducible (deviations of < 10–10) resistance reference. ns, carrier density; μ, carrier mobility.

The integer quantum Hall effect relies on the charge carriers in the system

occupying a series of discrete energy levels known as Landau levels (LLs), which

correspond to the quantization of the cyclotron motion of charge carriers in the

magnetic field. The value at which RH is quantized depends on the number of filled

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LLs, described by the filling factor ν = nsh/eB, where ns is the carrier density (Fig.

1.14(b)).

The quantum resistance standard is based on a Hall bar usually fabricated

from a 2DEG formed in a GaAs/AlGaAs semiconductor structure (Fig. 1.14(a)).

When operated at low temperature (T ≤ 1.5 K) and high magnetic field (B ≈ 10 T),

with measurement currents of a few tens of microamperes on the RH plateau at

RK/2 (ν = 2), such a Hall bar allows the calibration of a wire resistor in terms of a

conventional value of RK (i.e., 25.812807 kΩ) with a typical accuracy of 10–9.

1.3. GROWTH OF HETEROSTRUCTURES

1.3.1. Molecular beam epitaxy

Fig. 1.15. Schematics of the MBE growth of GaAs-AlGaAs heterostructures.

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Fig. 1.33. Processes during the MBE growth.

1.3.2. Metalorganic Chemical Vapour Deposition

Fig. 1.16. Schematics of the MO CVD growth.

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1.3.3. Atomic Layer Deposition

Fig. 1.17. Schematic representation of an ALD process.

1.3.4. Pulsed Laser Deposition

Fig. 1.18. Scheme of a typical PLD setup for large-area film growth. The main functional parts are designated on the left. The fundamental processes during (I) target ablation, (II) plasma expansion, and (III) growth are shortly described on the right.

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1.4. CARRIER INJECTION IN A HETEROSTRUCTURE

Fig. 1.19A.

Refractiveindex

Photondensity

Activeregion

n ~ 5%

2 eV

Holes in VB

Electrons in CB

AlGaAsAlGaAs

1.4 eV

Ec

Ev

Ec

Ev

(a)

(b)

pn p

Ec

(a) A doubleheterostructure diode hastwo junctions which arebetween two differentbandgap semiconductors(GaAs and AlGaAs).

2 eV

(b) Simplified energyband diagram under alarge forward bias.Lasing recombinationtakes place in the p-GaAs layer, theactive layer

(~0.1 m)

(c) Higher bandgapmaterials have alower refractiveindex

(d) AlGaAs layersprovide lateral opticalconfinement.

(c)

(d)

GaAs

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Fig. 1.19B. (a) An NpP AlGaAs/GaAs/AlGaAs double heterostructure (DHS; the capital letters represent the larger band gap semiconductors) in equilibrium.

(b) The DHS has been forward biased causing an injection of electrons and holes into the device, the depletion region is reduced, and the bands of the N-type AlGaAs shift upwards. When the voltage is sufficient, the quasi-Fermi level for the N-type material is at the same energy and the electrons can overcome the potential barrier DEc and flow into the p-GaAs region where they are confined by the lower band gap material. Similarly, holes flow in from the P-type AlGaAs to the p-GaAs valence band. The electrons and holes are confined where they can recombine radiatively.

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1.5. SUMMARY OF THE PROPERTIES OF CLASSICAL

HETEROSTRUCTURES

1.5.1. Basic physical phenomena

‘‘The recombination, light-emitting, and population inversion zones coincide and are

concentrated in the middle layer. Due to potential barriers at the boundaries of

semiconductors having forbidden bands of different width, the through currents of electrons

and holes are completely absent, even under strong forward voltages, and there is no

recombination in the emitters (in contrast to p-i-n, p-n-n+, n-p-p+ homostructures, in which

the recombination plays the dominant role) … . Because of a considerable difference

between the permittivities, the light is completely concentrated in the middle layer, which

acts as a high-grade waveguide, and thus there are no light losses in the passive regions

(emitters)’’

(Zh. I. Alferov, Nobel Laureate in physics 2000, written in 1966)

Fig. 1-20. Main physical phenomena in classical heterostructures:

(a) One-side injection and superinjection;

(b) diffusion in built-in quasielectric field;

(c) electron and optical confinement;

(d) wide-gap window effect;

(e) diagonal tunneling through a heterostructure interface.

a) one-side injection and super-injection (concentration of injected carriers in the

base region can exceed the carrier concentration in the emitter by several

orders of magnitude);

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b) diffusion in the built-in electric field;

c) electronic and optical confinement;

d) the wide gap window effect;

e) diagonal tunneling through the interface.

1.5.2. Important practical applications

a) low-threshold semiconductor lasers exhibiting a CW regime at RT;

distributed feedback (DFB) lasers;

lasers with distributed Bragg reflectors (DBR);

surface-emitting lasers;

IR lasers based on type-II heterostructures;

b) solar cells and photodetectors using a wide band gap window;

c) semiconductor integrated optics based on DFB and DBR lasers;

d) heterobipolar transistors with a wide band gap emitter;

e) transistors, thyristors and dynistors with optical signal transfer;

f) power diodes and thyristors;

g) IR-to-VIS light transformers;

h) efficient cold cathodes.

1.5.3. Important technological peculiarities

a) inherent necessity of a good lattice parameter match;

b) use of ternary and quaternary alloys to achieve the lattice parameter match;

c) inherent necessity of epitaxial growth technologies.