технологический of nanosize structures...
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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.
20
Fig. 1.33. Processes during the MBE growth.
1.3.2. Metalorganic Chemical Vapour Deposition
Fig. 1.16. Schematics of the MO CVD growth.
21
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
23
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.
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