Introduction to Quantum Transport in

Low-Dimensional Systems

K J ThomasK J Thomas

Department of Physics

Central University of Kerala

Kasaragod, Kerala

February 22, 2016Harish –Chandra Research Institute, Allahabad

Outline Low-dimensional electron systems

A. Early history

B. Semiconductor technology

C. Two-dimensional electron gases

Quantum wells

High electron mobility transistors

D. Density of free electron states

3D DOS

2D DOS 2D DOS

1D DOS

E. Lowering the electron dimension

1. 3D to 2D

2. 2D to 1D

F. Energy bands in a semiconductor

G. Time and length scales in mesoscopic systems

1. Fermi wavelength λF

2. Mean free path ℓ

3. Phase coherence length ℓФ

4. Thermal length LT

5. Length scales in a magnetic field B

Quantum Wire: Conductance Quantisation

Semiconductor industry: Very large and fast growing. There is a wealth of materials

science, chemistry, engineering and physics that drives the semiconductor industry.

Moore’s law: Over the past few decades it has followed Moore’s law; this is an

empirical observation put forward in 1965 by Gordon Moore of Intel:

that the number of transistors on an integrated circuit for minimum component cost

doubles every 24 months.

Introduction

In the search for smaller feature sizes (physical gate lengths are now < 25 nm),

there lies the prospect that quantum effects may become important, or indeed that

quantum logic could itself be implemented.

At present top down fabrication processes (lithography, etching, etc.) dominate, but

many argue that bottom up fabrication using molecular synthesis and self-assembly

may become important.

Physicists were responsible for the invention of the semiconductor transistor, and

are now involved in understanding mesoscopic physics - the area of physics that lies

between the macroscopic and the microscopic.

Low Dimensionality

The low-dimensionality is achieved by confinement, where the electrons

are restricted from moving freely in all three dimensions.

� Patterning quantum wires, quantum dots etc. from a

semiconductor crystal using lithographic techniques.

� By growing nanowires, nanorods etc by chemical process,

exfoliating graphene sheets from graphite – a new class of 2D exfoliating graphene sheets from graphite – a new class of 2D

materials, and many more such 2D materials being currently

discovered!

First we will look at two-dimensional electron gases (2DEGs).

In the early days these were created in Si devices, but now are commonly

created at the interface of GaAs/AlGaAs heterostructure or similar III – V

semiconductors or II – VI semiconductors.

These artificially engineered systems have very high mobilities (μ ≥ 106

cm2/Vs) and extremely long mean free paths, ℓ ≈ 10-1000 μm.

The dimensionality of a system is not an absolute property, but depends on the

length scale of the physical property being studied.

To quantise electron motion in a particular direction, the confining potential must

be comparable to the electron’s de Broglie wavelength.

Transport properties at low temperatures are determined by the behaviour of the

electrons near the Fermi energy.

Electrons will be considered to be in a low-dimensional system if one of the

Early History

Electrons will be considered to be in a low-dimensional system if one of the

sample dimensions is comparable to the de Broglie wavelength of electrons at

the Fermi level (Fermi wavelength, λF ).

Up to the 1960s quantum confinement could only be observed in natural

structures such as atoms, molecules or crystals.

The first artificial structures were thin metallic films with a thickness W larger

than λF , and because the electron gas could still explore the perpendicular

direction these systems are quasi-two-dimensional rather than two-dimensional.

Semiconductor Technology

�The field effect in semiconductors

is the basis for important electronic

devices and is illustrated in Fig. 1.

�The semiconductor is separated

from a conductive plate (known as

the gate) by an insulator. the gate) by an insulator.

�Suppose that a positive voltage is

applied to the gate and the

underlying semiconductor is n-type.

The positive Vg will attract electrons

to the surface of the semiconductor

and the concentration of carriers in

the vicinity of the surface will be

different from that in the bulk

semiconductor.

FIG. 1 Silicon metal-oxide-semiconductor field-

effect transistors (MOSFET). In the Intel Core

Duo Desktop Processor the channel length

approaches 45 nm.

The MOSFET is fabricated by putting an insulating

thermally oxidised layer of SiO2 onto the surface of

a doped crystalline Si substrate, and then placing a

metallic gate electrode on the top.

When a sufficient number of electrons are attracted, a surface layer with appreciable

conductivity will be formed. This type of layer where extra electrons (remember the

semiconductor is n-type) have been attracted to the surface is known as an

accumulation layer.

If the semiconductor had been p-type, then the electron layer induced at the surface

is known as an inversion layer ; it will be separated from the interior of the

semiconductor by a region from which the carriers have been depleted.

Early research into 2D electron systems used silicon metal-oxide-semiconductor field-

Simple Operation of a MOSFET

Early research into 2D electron systems used silicon metal-oxide-semiconductor field-

effect transistors (Si-MOSFETs), as shown in Fig. 1.

When a voltage is applied between the gate and the source terminals, the generated

electric field penetrates through the oxide and creates an inversion channel in the

substrate at the SiO2-Si interface.

In an n-channel MOSFET, the substrate is p-doped, and the application of positive

gate voltage creates an inversion layer of electrons in the p-type substrate, while in p-

channel MOSFET the substrate is n-doped and the application of negative gate

voltage creates an inversion layer of holes in the n-type substrate. This channel

spreads from the source to the drain and provides conductivity of the transistor.

The advantage of the Si-MOSFET over deposited thin films (and earlier

transistors) is the high-quality SiO2-Si interfaces, which did not generate localised

electron traps (interface states) between the silicon and its oxide layer, leading to

low scattering rates (1/τ) and high mobilities (μ).

A second advantage is the ability to control the carrier concentration by doping or

Advantages and Disadvantages of Si - MOSFET

A second advantage is the ability to control the carrier concentration by doping or

applying an electric field. In this way the carrier density could be made much

lower than in metallic films, with the Fermi wavelength λF comparable to

structures that can be fabricated using lithography.

The physical properties of 2DEG in MOS type structures are determined by the

roughness at the interface between the monocrystalline semiconductor (Si)

and the amorphous oxide (SiO2) which limits the mobility.

Two-dimensional electron gases

In a semiconductor such as GaAs it is possible to control the band gap Eg through

what is known as band gap engineering.

Take for example GaAs, if a fraction x of the Ga atoms are replaced by Al atoms, then

the alloy AlxGa1−x As is produced. If the substitution is achieved randomly, then the

band gap of the resulting alloy E (Al Ga As) is given by:band gap of the resulting alloy Eg(AlxGa1−x As) is given by:

Eg(AlxGa1−x As) = (1.424 + 1.247 x) eV (1.1)

where in practice the fraction x is in the range 0 ≤ x ≤ 0.45.

FIG. 2 Band gap versus lattice

constant for various semi-

conductor alloys. The line

connecting AlAs and GaAs is nearly

vertical, showing that different

alloys of AlxGa1−xAsare lattice

matched.

At 300 K, a (AlxGa1−xAs) = (5.6533 +

0.0078 x) °A.

An Important structural information is shown in Fig. 2, which shows how the lattice

constant a varies in different semiconductor alloy systems. The important feature to

note from this figure is that GaAs will be lattice matched to Al0.4Ga0.6As, and from Eq.

1.1 the two have band gaps that differ by 0.5 eV. The structural similarity of the two

alloys means that one can be easily grown on top of the other, and there will be little

strain at junction between the two (called a heterojunction).

Therefore very few dislocations will be formed at the junction which will be

atomically sharp; the lack of strain and dislocations will mean that there will be few

scattering centres in the vicinity of the heterojunction.

Molecular-beam epitaxy (MBE) allows semiconductor crystals to be grown layer-by-layer

with atomic precision, and with the opportunity to engineer the band structure.

GaAs/AlGaAs heterostructures are grown in an MBE chamber, as shown schematically in

Fig. 3.

The elements that compose the heterostructure: Ga, As, Al, and Si (n-type dopant) are

effused onto a heated GaAs substrate under high vacuum. The layer-by-layer growth is

slow, with a typical growth rate of one monolayer/second (1 μm per hour).

MBE

FIG. 3 Schematic diagram

of an MBE chamber. The

effusion cells on the left

hand side consist of

heated crucibles of Ga,

As, Al, and Si - this creates

molecular beams of each

element that can be

controlled by shutters.

Quantum wells

MBE allows the growth of heterostructures, alternate layers of GaAs and

AlxGa1−xAs, where the layer widths can vary from a few nm up to μm.

The conduction and valence bands are shifted (band offset) in the AlxGa1−xAs

relative to the GaAs, so that quantum wells are formed for both electrons and

holes in the GaAs layers, see Fig. 4.

FIG. 4 (a) Top: a quantum well

consisting of a thin layer

of GaAs, width W, sandwiched

between Al0.3Ga0.7As.

(b)Bottom: The corresponding

band structure, where Eg(GaAs)

is the gap of bulk GaAs. The

lowest confined electronic level,

with wavefunction ψ1, lies Ee1

above the conduction band

edge of bulk GaAs.

The free carriers are now confined along the z -axis and form standing waves

in the potential well; they are still free to move in the x-y plane.

For an infinite potential well of width W, the energies of the electron states are

where p is a quantum number giving the number of half wavelengths

of the electron standing wave in the well.

Because of the different effective masses, different levels will be formed for

heavy and light holes. The energy states for a given quantum number p are

known as subbands.

Technically the quantum well is a

double heterojunction and because

Eg(GaAs) < Eg(AlGaAs)

the well in the conduction band occurs

in the region composed of GaAs.

If the system is doped with electrons,

then the Fermi energy E will lie nearthen the Fermi energy EF will lie near

the bottom of the quantum well, and

only one state will be occupied.

If growth and electron confinement are in the z-direction, then the electron can move

freely in the xy-plane; such a system is known as a two-dimensional

electron gas (2DEG).

The simplest structure that can be fabricated is a quantum well of width 100-150 °A,

with the conduction band profile shown in Fig. 5.

High electron mobility transistors (HEMT)

If a vertical electric field is superimposed on the band structure, then a

narrow quantum well can be formed at an AlGaAs/GaAs interface.

Electrons in this well populate the lowest subband and form a two

dimensional electron gas (2DEG).

These electrons can come from donors in a doped layer which is separated

by a spacer layer. This is called modulation doping and gives rise to high by a spacer layer. This is called modulation doping and gives rise to high

mobilities, as the charged impurities are well removed from the carriers.

Such structures are used in high electron mobility transistors, more

commonly known as HEMTs.

Mobilities of 106-107 cm2/Vs can be routinely achieved, compared to 6000

cm2/Vs in Si-MOSFETs, which gives the possibility of much faster devices.

A more complicated band structure occurs in the simple heterostructure, see Fig. 6,

where it is necessary to understand the band bending which gives rise to a triangular

confinement well.

To increase the purity of the heterojunction it is modulation doped - this means that

the n-type dopant (in this case Si) is introduced into the wide band gap AlGaAs at

some distance from the 2DEG formed at the heterojunction interface. This means

that the positive ionised Si atoms, upon donating their electrons will be situated

further from the 2DEG, leaving the narrow band gap GaAs free from intentional

doping.doping.

FIG. 6 Conduction band profile of a

modulation doped GaAs-AlxGa1−xAs

heterostructure. A triangular

potential well is formed in the

undoped GaAs at the interface with

the n-doped AlGaAs. The spatial

extension of the electron

wavefunction in the growth direction

is typically 100 °A.

The Fermi energy EF is by definition constant throughout the sample.

Figure 6 shows that the bottom of the conduction band in the AlGaAs region is much

higher than EF , and the electrons only populate the narrow gap GaAs.

At temperatures above 100 K the electrons are thermally activated across the

electrostatic barrier and spill over from the n-AlGaAs, leaving behind positively

charged donors.

As shown, this space charge causes band bending, which prevents electrons

How is the triangular confinement achieved?

As shown, this space charge causes band bending, which prevents electrons

returning to their donors.

The electrons transferred into the lower lying conduction band of GaAs are confined

in a triangular potential well, producing a two-dimensional electron gas (2DEG) as the

heterojunction.

The spatial extension of the electron wavefunction in the growth direction is typically

10 nm. The modulation doping technique, introduces a spacer layer of undoped

AlGaAs between the GaAs and the n-AlGaAs. In this way very high carrier

mobilities ∼ 106 cm2/Vs can be achieved, due to the greater spatial separation of the

donor atoms in theAlGaAs layer from the conduction electrons.

This separation reduces the scattering cross section due to the disordered

potential caused by the ionised impurities.

The motion of electrons in a GaAs/AlGaAs 2DEG is therefore limited by small-angle

elastic scattering.

The corresponding mean free path is ℓ = 10-100 μm; therefore it is possible to

fabricate devices in which 2D electrons are fully ballistic.

Because they do not collide often with other particles, the electrons have a long Because they do not collide often with other particles, the electrons have a long

phase coherence length - they can travel coherently for many microns as

quantum mechanical waves with a well defined phase and with the same energy.

Summary

2DEGs can be easily fabricated in GaAs because of the following:

1. AlxGa1−xAs has a tunable band gap with x.

2. AlxGa1−xAs and GaAs are lattice matched.

3. Single-crystal samples can be grown by MBE.3. Single-crystal samples can be grown by MBE.

The 2DEGs are very clean, as measured by the mobility μ = eτ/m.

If a macroscopic gate is evaporated on the sample surface, see Fig. 8, the

electron density n2D can be varied according to

It is easy to derive Eq. 1.3 from the expression for a simple parallel plate

capacitor C = εrε0A/x. Si-MOSFETs, which are the mainstay of modern

electronic circuits, rely on this capacitive effect to open and close the

conducting channel.

FIG. 8 The two-dimensional electron gas (2DEG) is processed in a clean room -

this allows Ohmic contacts (shown in black) to be annealed into the 2DEG, and a

gate to be deposited onto the surface of the device. By applying a voltage Vg to

the gate, the electron density n2D in the 2DEG (which lies a distance x below the

surface) can be tuned according to Eq. 1.3.

It is also necessary to make electrical contact to the 2DEG so that the sample

can be measured, this is achieved by shaping the 2DEG into a mesa and

annealing Ohmic contacts into the device. The finished device, called a Hall bar,

is shown in Fig. 9.

FIG. 9 During processing a Hall bar is

defined by photolithography. A typical

mesa has a width of 100 μm and length mesa has a width of 100 μm and length

of 1000 μm, defining a region of 2DEG

that is connected to external electrical

circuits via Ohmic contacts (shown in

black).

3D → 2D

The potential energy in a quantum well depends only on the coordinate z normal

to the layers. If the Lz < λF , the system can effectively be considered as two-

dimensional and we can write the potential as

V (r) =V (z).

The electron eigen state in a 2DEG is a plane wave for motion in xy-plane and

shows confinement in the z-direction:

The Schr¨odinger equation reduces to its one-dimensional form in the z variable:

where ε is the energy of the 2D subband. The electron energy is given by

where En is the quantised energy corresponding to u(z). Equation 1.18 is a

dispersion relation that gives a parabola for each value of integer n, called a two-

dimensional subband with a band bottom at En when plotted against k. Within

the nth subband there are a continuum of states where the electron motion is

free in the xy-plane. The DOS of a two-dimensional subband is energy

independent:

which is sketched in Fig. 11. In GaAs/AlGaAs systems the energy separation

between the 2D subbands is quite high, E2 − E1 ≈ 25 meV, so that for typical

electron densities n2D ≈ 2 × 1011 cm−2, only one 2D subband is occupied.

The electrons behave two-dimensionally, with a single step-like density of

states. Their eigenstates characterised by a single wavefunction u1(z) for a

motion in z-direction. The density of electrons can be written as

(1.20)

2D → 1D

If we further confine the 2DEG along the y-direction the electrons remain free to

move along x and we may write the electron wavefunction

where ω(y) is the confined wavefunction in the y - direction.

The energy is

where Em is the energy corresponding to ω(y). For each pair of n,m bound

states there is a continuum of free running waves in x-direction. Each value

of Em,n = En+Em now becomes the bottom of a one-dimensional subband.

In practice the tighter the 1D confinement, the higher the 1D subband spacing.

The transverse confinement in a typical quantum point contact is λF /2 ≈ 25 nm,

making the typical subband spacing of the order of 1 meV.

Each subband can be treated as an independent 1D subband, so the total DOS

per unit length is the sum of the density of states for each individual subband:

(1.23)

The 1D DOS has a singularity at E = En at the bottom of each 1D subband,

which in reality is removed by the effects of disorder and temperature.

Energy bands in a semiconductor

In the free electron model, parabolic energy bands

are created. If the electron moves in a periodic crystal, then this periodicity will

be reflected in k-space, such that

In a simple model that takes into account the interaction between electrons and

the ions then gaps will arise at the zone boundaries. In this approximation

the bands are still parabolic but the electron mass is altered to me - the effective

electron mass.

About the gap the conduction and valence bands will have the form:

The latter is equivalent to a negative effective mass.

A hole can be thought of as a positive charge carrier with a positive effective

mass.

It is much easier to confine electrons in a semiconductor structure than in a

metal structure. This is because the Fermi energy EF is much smaller in

semiconductor (meVs rather than eVs), and consequently the Fermi wavelength

λF is much longer (100s of °A, rather than 2-3 °A), see Table II.

Electron-beam lithography can pattern on submicron length scales, whereas

arranging the single atoms in a metal is not easy.

Time and length scales in mesoscopic systems

1. Fermi wavelength λF

At low temperatures only electrons within a few kBT of the Fermi energy

participate in electrical transport.

n2D = g (E) f(E) EF = EF ; EF = ℏ2kF2/2m

λF is a key length scale which determines whether or not an electron can be

confined. In semiconductors, the low electron density (compared to metals)

results in high λF .

A typical carrier density in a GaAs-based 2DEG (with effective mass of 0.067

me) is n2D = 2 × 1011 cm−2 = 2 × 1015 m−2, giving a Fermi wavelength

At this carrier density the Fermi energy is EF = 7.2 meV, and the corresponding

Fermi velocity is vF = 1.85 × 105 ms−1.

2. Mean free path ℓ

The most elementary length scale is the mean free path, ℓ, which is the average

distance the electron travels between successive collisions with impurities,

phonons, defects, etc.

In such a collision the electron’s momentum is changed and the associated time

between collisions is called the momentum relaxation time τ; the distance

covered between each collision is simply given by

where vF is the Fermi velocity.

Note that although the momentum is changed in a collision, the electron’s

energy is usually conserved, that is, the collisions are elastic.

A measure of the scattering time τ in a material can be obtained from its

mobility μ = eτ/m, for a 2DEG of mobility μ = 106 cm2/Vs the scattering time

will be τ = 40 ps.

At this mobility and n2D = 2 × 1011 cm−2, the mean free path is ℓ = 7.4 μm.

Comparing the mean free path ℓ with the characteristic dimension L of the

system, it is possible to discriminate between two distinct transport regimes:

In the diffusive regime, at long times t (where t ≫ τ) the electron will have

undergone many collisions, and will have diffused a distance , where D is

the diffusion constant.

In 2D the diffusion constant is given3 by D = vF2τ/2.

In general the diffusion constant is given by D = vF2τ/d, where

d is the dimensionality.

Although it is expected that v, τ and D all depend on energy, however because

transport takes place in a narrow energy band close to EF , it is possible to

disregard this dependence and allow them to assume their values at EF .

3. Phase coherence length LФ

In a quantum mechanical (or at least semi-classical) description, electrons

carry not only momentum, energy and spin but also a phase; for example, in

a plane wave the phase φ is a well defined quantity (often

chosen to be φ = 0 for convenience). In an interference pattern between two

waves, for example the classic Young’s slit experiment, the interference

pattern

depends on the phase difference φ1 − φ2. If φ1,2 become completely randomized,

then the time-averaged interference will be zero. The phase coherence time τφ is

the average time it takes for the interference term to be suppressed by e−1, and in

general this suppression follows an exponential form exp(−t/τφ) with time t. In

mesoscopic physics, quantum interference effects are important as is τφ.

τφ is sometimes called the phase breaking time.

Elastic impurity scattering from static impurities does not destroy phase

coherence, whereas dynamical scattering, for example by phonons or other

electrons will randomize the phase of an electron wave, as will more

complicated scattering by the spin of an impurity atom (spin-flip processes).

At temperatures less than 1 K the phonons are effectively frozen out, and in pure

samples the dominant decoherence mechanism is scattering of

electrons off the fluctuating electromagnetic fields generated by all other

electrons (called Nyquist scattering).

In ballistic samples, the associated coherence length is given by Lφ = vF τφ,

whereas in diffusive samples

At low temperatures or in a system with large disorder τφ > τ, the diffusive

limit is appropriate and Lφ can be much longer than the mean free path ℓ.

4. Thermal length LT

At finite temperature, only electrons within a few kBT of EF contribute to

transport.

By definition an electron at energy EF will get out of phase by one radian with an

electron at EF + kBT after traveling the thermal length LT (because the two

wavelengths are slightly different).

In a ballistic sample, this length is easily calculated using In a ballistic sample, this length is easily calculated using

[k(EF + kBT ) − k(EF )] × LT = 1 where the dispersion relation is

Assuming EF ≫ kBT , one finds

the corresponding thermal time is , which could have been

determined from the time-domain Heisenberg uncertainty relation

Using ∆E = kBT. With similar sample parameters as before at T = 0.1 K, one obtains

LT ∼ 15 μm, which is larger than the calculated mean free path, rendering the

ballistic approach incorrect. In a diffusive approach, the thermal length is

which gives LT ∼ 7.4 μm at T = 0.1 K. In this case this is equal to the mean free

path, so neither a purely diffusive nor purely ballistic treatment is appropriate

here.

5. Length scales in a magnetic field B

In a magnetic field the electron still undergoes scattering, but rather than

having straight line trajectories between each collision, they become curved having straight line trajectories between each collision, they become curved

by the applied magnetic field. In general the magnetic field tries to make

electrons perform circular orbits in the plane whose normal is parallel to ~B .

A semi-classical length scale associated with this motion is the radius of this

circle; this is called the cyclotron radius lcycl

and is derived by equating the Lorentz force

to the centripetal force as

This length scale depends on the wavevector, which itself depends on the

sheet density as kF ∼ √n2D.

As the magnetic field increases, Eq. 1.27 predicts that the circular motion

becomes more and more localised.

..

becomes more and more localised.

An additional length scale associated with B is the magnetic length,

which corresponds to the spatial extent of the wavefunction in the lowest

Landau level. In contrast to lcycl, the magnetic length is a fundamental length

scale independent of sample properties

(carrier density, mass, etc.).

Density of free electron states

The density of free electron states g(E) = dn/dE in 1D, 2D and 3D is the number of

available states per unit energy per unit volume.

This could also be called the density of available states per unit energy. Whatever

the dimension d (d = 0 − 3) the density of states (DOS) can be written as

The kinetic energy of free electrons is given by E = ћ2k2/2m, therefore

and Eq. 1.4 becomes

The number of states in a cubic box of side L is found by imposing periodic

boundary conditions, for example, in the x-direction this condition is

Therefore eikxL = 1, which means that kx = mx2π/L where mx is an

integer. Similar boundary conditions apply in all three directions, so

the wavevector is:

where the quantum numbers mx, my and mz are integers.

Hence the electron states are uniformly distributed in k-space,

with spacing 2π/L.

3D DOS

In 3D the available states between k → k + dk are contained in a shell of radius k

and thickness dk, see Fig. 10(a), which has a volume 4πk2dk. The electron states

are uniformly distributed in k-space, with spacing 2π/L . Therefore the number

dN of energy states contained in the shell is

where the extra factor of 2 is due to the

spin degeneracy of spin 1/2 electrons.

The volume of the sample is L3,

so the carrier density is defined as n =

FIG. 10 (a) Visualization of three

dimensional k-space. The number of

available states is determined by the

number of points contained in the shell of

radius k and thickness dk. The boundary

between occupied and unoccupied states

is called the Fermi surface.

so the carrier density is defined as n =

N/V = N/L3. Therefore density of states

per unit volume in k-space is

Using Eq. 1.5 the 3D density of free

states is

2D DOS

In 2D the available states between k → k + dk are contained in a ring of radius k and

thickness dk, see Fig. 10(b), which has an area 2πkdk. The electron states are

uniformly distributed in k-space, with spacing 2π/L . Therefore the number dN of

energy states contained in the ring is

where the extra factor of 2 is due to the where the extra factor of 2 is due to the

spin degeneracy of spin 1/2 electrons. The

area of the sample is L2, so the carrier

density is defined as n = N/L2. Therefore

the density of states per unit “volume” in

k-space is:

FIG 10 (b): In two dimensions states

at T = 0 K are filled up to radius kF ,

and those states k > kF are un-

occupied; the boundary between

the two is called the Fermi circle.

Using Eq. 1.5 the 2D density of free states is

1D DOS

In 1D the available states between k → k + dk occur at the two points ±k over a

length dk, see Fig. 10(c), which has length 2dk. The electron states are uniformly

distributed in k-space, with spacing 2π/L . Therefore the number dN of energy

states contained in the two segments is:

where the extra factor of 2 is due to the

spin degeneracy of spin 1/2 electrons.

The length of the sample is L, so the The length of the sample is L, so the

carrier density is defined as n = N/L.

Therefore density of states per unit

“volume” in k-space is

FIG 10(c): Fermi segment in 1D. There is

confinement in the y- direction, which

increases the spacing ky between

adjacent states. When ky > kF , the

condition Ly < λF is satisfied.

Using Eq. 1.5 the 1D density of free states

is

Summary of gd

Table I gives the density of states g in different dimensions d; in general we can see Table I gives the density of states gd in different dimensions d; in general we can see

that (i) as d is reduced, gd becomes more peaky at the bottom of the subband, and,

(ii) the lower the mass, the lower the density of states.

We have not derived the result, but in line with observation

(i) we find that the density of states in zero dimensions is given by a Dirac delta

function, g0D(E) = δ(E).

DOS in 3D, 2D, 1D and 0D