CHAPTER

1

SEMICONDUCTORS AND
HETEROSTRUCTURES

1 .1

THE MECHANICS OF WAVES

De Broglie (see reference [4]) stated that a particle of momentum p has an associated
wave of wavelength A given by the following

A=�

(1.1)

p

Thus, an electron in a vacuum at a position r and away from the influence of any
electromagnetic potentials, could be described by a state function which is of the
form of a wave, i.e.

(1.2)
where t is the time,
given by:

w

the angular frequency and the modulus of the wave vector is

k

= Ikl = 211'
A

Quantum Wells. Wires and Dots. Second Edition. P. Harrison
@2005 John Wiley & Sons, Ltd.

(1.3)

1


2

SEMICONDUCTORS AND HETEROSTRUCTURES

The quantum mechanical momentum has been deduced to be a linear operator [12]
acting upon the wavefunction 'I/J, with the momentum p arising as an eigenvalue, i.e.

where

-iI1Sl'I/J = p'I/J

(1.4)

a,
a,
a,

\7 = -i+ -j +-k
ay
az

(1.5)

ox

which when operating on the electron vacuum wave function in equation (1.2) would
give the following:

_ili\7ei(kor-wt) = ei(kor-wt)
p

(1.6)

and therefore

:. -iii (ikxi + ikyj + ikji:) ei(k",x+kyy+k. z-wt) = pei( or-w )
k

t

(1. 8)

Thus the eigenvalue:

(1.9)
which not surprisingly can be simply manipulated (p = lik = (h/27r)(27r/>") to
reproduce de Broglie's relationship in equation (1.1).

Following on from this, classical mechanics gives the kinetic energy of a particle
of mass m as

p2
(mv)2
1
=T = -mv2 =
2m
2m
2

--

(LlO)

Therefore it may be expected that the quantum mechanical analogy can also be rep­
resented by an eigenvalue equation with an operator:

1
. )2 'I/J = T'I/J
- ( -zli\7
2m

(Lll )

i.e.
(Ll2)
where T is the kinetic energy eigenvalue, and given the form of \7 in equation (1.5)
then:


\72 =

02
02
+
ax2 ay2

+

02
az 2

(Ll3)

When acting upon the electron vacuum wave function, i.e.
(1.14)

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THE MECHANICS OF WAVES
then
_

� (i2k2 + i2k2y + i2k2) ei (k.r-wt) = Tei(k.r-wt)
2m

z

x


3

(1.15)

Thus the kinetic energy eigenvalue is given by:
T=

h2k2

(1.16)

2m

-

For an electron in a vacuum away from the influence of electromagnetic fields, then
the total energy E is just the kinetic energy T. Thus the dispersion or energy versus
momentum (which is proportional to the wave vector k) curves are parabolic, just as
for classical free particles, as illustrated in Fig. 1.1.

E

k
Figure 1.1

The energy versus wave vector (proportional to momentum) curve for an electron

in a vacuum
In summary, the equation describing the total energy of a particle in this wave

description is called the time-independent Schrooinger equation and for this case
with only a kinetic energy contribution can be summarised as follows:

_�'V21{J = E1{J
2m

(1.17)

A corresponding equation also exists which includes the time-dependency explicitly;
this is obtained by operating on the wave function by the linear operator ino/ at, i.e.

(1.18)
i.e.
(1.19)
Clearly this eigenvalue ru.,; is also the total energy but in a form usually associated
with waves, e.g. a photon. These two operations on the wave function represent
the two complimentary descriptions associated with wave-particle duality. Thus the
second, i.e., time-dependent, SchrOdinger equation is given by the following:
(1.20)

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4

SEMICONDUCTORS AND HETEROSTRUCTURES

1 .2

CRYSTAL STRUCTURE


The vast majority of the mainstream semiconductors have a face-centred cubic Bravais
lattice, as illustrated in Fig 1.2. The lattice points are defined in terms of linear
combinations of a set of primitive lattice vectors, one choice for which is:
(1.21)
The lattice vectors then follow as the set of vectors:
(1.22)

Figure 1.2 The face-centred cubic Bravais lattice
The complete crystal structure is obtained by placing the atomic basis at each
Bravais lattice point. For materials such as Si, Ge, GaAs, AlAs, loP, etc., this consists
of two atoms, one at
and the second at
in units of Ao .
For the group IV materials, such as Si and Ge, as the atoms within the basis are the
same then the crystal structure is equivalent to diamond (see Fig. 1.3 (left». For III-V
and II-VI compound semiconductors such as GaAs, AlAs, loP, HgTe and CdTe, the
cation sits on the
site and the anion on (+
this type of crystal
is called the zinc blende structure, after ZnS, see Fig 1.3 (right). The only exception to
this rule is GaN, and its important lnxGa l-xN alloys, which have risen to prominence
in recent years due to their use in green and blue light emitting diodes and lasers (see
for example [13]) these materials have the wurtzite structure (see reference [2] p. 47).

(�,�,�)

(-�,-�,-�),

(-�,-�,-�)


�,+�,+k);

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CRYSTAL STRUCTURE

Figure 1.3

Figure 1.4

5

The diamond (left) and zinc blende (right) crystal structures

Schematic illustration of the ionic core component of the crystal potential across

the {OO I} planes-a three-dimensional array of spherically symmetric potentials

From an electrostatics viewpoint, the crystal potential consists of a three-dimen­
sional lattice of spherically symmetric ionic core potentials screened by the inner shell
electrons (see Fig. 1.4), which are further surrounded by the covalent bond charge
distributions which hold everything together.

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6


SEMICONDUCTORS AND HETEROSTRUCTURES

1 .3

THE EFFECTIVE MASS APPROXIMATION

Therefore the crystal potential is complex; however using the principle of simplicity

*

imagine that it can be approximated by a constant! Then the Schrooinger equation
derived for an electron in a vacuum would be applicable. Clearly though, a crystal
isn't a vacuum so allow the introduction of an empirical fitting parameter called the
effective mass,

m *.

Thus the time-independent Schrodinger equation becomes:

( 1.23)
and the energy solutions follow as:

( 1.24)
This is known as the

effective mass approximation

and has been found to be very

suitable for relatively low electron momenta as occur with low electric fields. Indeed,

it is the most widely used parameterisation in semiconductor physics (see any good
solid state physics book, e.g.

[ 1,2,7]). Experimental measurements of the effective

mass have revealed it to be anisotropic-as might be expected since the crystal po­
tential along say the

[001] axis is different than along the [1 1 1] axis. Adachi [14]

collates reported values for GaAs and its alloys; the effective mass in other materials
can be found in Landolt and Bomstein

[ 15].

In GaAs, the reported effective mass is around
mass of an electron. Fig.

0.067 m o, where mo is the rest
1.5 plots the dispersion curve for this effective mass, in

comparison with that of an electron in a vacuum.

E

GaAs

,

,

,
,
,

,
,

!

,

k
Figure 1.5 The energy versus wave vector (proportional to momentum) curves for an electron
in

•

GaAs compared to that in a vacuum

Choose the simplest thing first; if it works use it, and if it doesn't, then try the next simplest!

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BAND THEORY
1 .4

7

BAND THEORY


It has also been found from experiment that there are two distinct energy bands within

semiconductors. The lower band is almost full of electrons and can conduct by the

movement of the empty states. This band originates from the valence electron states
which constitute the covalent bonds holding the atoms together in the crystal. In
many ways, electric charge in a solid resembles a fluid, and the analogy for this band,
labelled the

valence band is that
holes.

the empty states behave like bubbles within the

fluid-hence their name

! conduction band
V

k
valence band

Figure 1.6 The energy versus wave vector curves for an electron in the conduction band and
a hole in the valence band of GaAs

In particular, the holes rise to the uppennost point of the valence band and just as
it is possible to consider the release of carbon dioxide through the motion of beer in
a glass, it is actually easier to study the motion of the bubble (the absence of beer),
or in this case the motion of the hole.


In a semiconductor,

the upper band is almost devoid of electrons. It represents

excited electron states which are occupied by electrons promoted from localised
covalent bonds into extended states in the body of the crystal. Such electrons are
readily accelerated by an applied electric field and contribute to current flow. This
band is therefore known as the
Fig.

conduction band.

1.6 illustrates these two bands. Notice how the valence band is inverted-this

is a reflection of the fact that the 'bubbles' rise to the top, i.e. their lowest energy states
are at the top of the band. The energy difference between the two bands is known as

the

bandgap,

labelled as

Egap

on the figure. The particular curvatures used in both

bands are indicative of those measured experimentally for GaAs, namely effective
masses of around 0.067


mo for an electron in the conduction band, and 0.6 mo

for a

(heavy-)hole in the valence band. The convention is to put the zero of the energy at

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SEMICONDUCTORS AND HETEROSTRUCTURES

8

the top of the valence band. Note the extra qualifier 'heavy'. In fact, there is more
than one valence band, and they are distinguished by their different effective masses.
Chapter

1 1 will discuss band structure in more detail; this will be in the context of

a microscopic model of the crystal potential which goes beyond the simple ideas
introduced here.

1 .5

HETEROJUNCTIONS

Figure

1.7

Two dissimilar semiconductors with different bandgaps joined to form a
heterojunction; the curves represent the unrestricted motion parallel to the interface

The effective mass approximation is for a bulk crystal, which means the crystal
is so large with respect to the scale of an electron wave function that it is effectively
infinite. Within the effective mass approximation, the Schr6dinger equation has been
found to be as follows:

1i2

{)2

---1jJ(z)
= E1jJ(z)
{) 2

2m*

Z

When two such materials are placed adjacent to each other to form a

( 1.25)

heterojunction,

then this equation is valid within each, remembering of course that the effective mass
could be a function of position. However the bandgaps of the materials can also be
different (see Fig.


1.7).

The discontinuity in either the conduction or the valence band can be represented
by a constant potential term. Thus the SchrOdinger equation for any one of the bands,
taking the effective mass to be the same in each material, would be generalised to

( 1.26)

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HETEROSTRUCTURES

v

9

conduction
band

valence band

v

Figure 1.8

The one-dimensional potentials V (z ) in the conduction and valence band as might
occur at a heterojunction (marked with a dashed line) between two dissimilar materials

In the above example, the one-dimensional potentials V (z) representing the band

discontinuities at the heterojunction would have the form shown in Fig.

1.8, noting

that increasing hole energy in the valence band is measured downwards.

1 .6

HETEROSTRUCTURES
.,;

�

B

l�
Egap

B

conduction band

�_1����

__

Figure 1.9

B


A

V_al_e_nCe _b_an_d

____

�

B

���

____

_

______

The one-dimensional potentials V(z) in the conduction and valence bands for a
typical single quantum well (left) and a stepped quantum well (right)

Heterostructures are formed

from multiple heterojunctions, and thus a myriad of

If a thin layer of a narrower-bandgap material 'A' say, is sandwiched
between two layers of a wider-bandgap material 'B', as illustrated in Fig. 1.9 (left)
possibilities exist.

then they form a double heterojunction. If layer 'A' is sufficiently thin for quantum

properties to be exhibited, then such a band alignment is called a single quantum well.
If any charge carriers exist in the system, whether thermally produced intrinsic or
extrinsic as the result of doping, they will attempt to lower their energies. Hence in
this example, any electrons (solid circles) or holes (open circles) will collect in the
quantum well (see Fig.

1.9). Additional semiconductor layers can be included in the

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SEMICONDUCTORS AND HETEROSTRUCTURES

10

B

A

A

B

•

B

•

•


•

The one-dimensional potentials V(z) in the conduction and valence band for
typical symmetric (left) and asymmetric (right) double quantum wells

Figure 1.10

heterostructure, for example a

stepped

or

asymmetric quantum well

can be formed

by the inclusion of an alloy between materials A and B, as shown in Fig.

A

B

•

B

A
•


B

A
•

B

A

1.9 (right).

B

•

Figure 1.11 The one-dimensional potentials V(z) in the conduction and valence band for a
typical multiple quantum well or superlattice

Still more complex structures can be formed, such as symmetric or asymmetric

double quantum wells, (see Fig. 1.10) and multiple quantum wells or superlattices (see
Fig.

1. 11). The difference between the latter is the extent of the interaction between

the quantum wells; in particular, a multiple quantum well exhibits the properties of
a collection of isolated single quantum wells, whereas in a superlattice the quantum
wells do interact. The motivation behind introducing increasingly complicated struc­
tures is an attempt to tailor the electronic and optical properties of these materials for

exploitation in devices. Perhaps the most complicated layer structure to date is the

chirped superlattice active region of a mid-infrared laser [16].
All of the structures illustrated so far have been examples of Type-I systems. In this
type, the bandgap of one material is nestled entirely within that of the wider-bandgap
material. The consequence of this is that any electrons or holes fall into quantum wells
which are

within the

same layer of material. Thus both types of charge carrier are

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THE ENVELOPE FUNCTION APPROXIMATION

B

A

B

C

A

11

C


VV
Type I

Figure 1.12

The one-dimensional potentials V (z) in the conduction and valence bands for
a typical Type-I single quantum well (left) compared to a Type-II system (right)

localised in the same region of space, which makes for efficient (fast) recombination.
However other possibilities can exist, as illustrated in Fig.

B

A

B

•

A

B

C

A

•


1.12.

C

A

C

•

Type I

Figure 1.13

The one-dimensional potentials V (z) in the conduction and valence bands for
a typical Type-I superlattice (left) compared to a Type-II system (right)

In Type-II systems the bandgaps of the materials, say 'A' and 'C', are aligned such
that the quantum wells formed in the conduction and valence bands are in different
materials, as illustrated in Fig.

1.12 (right). This leads to the electrons and holes being

confined in different layers of the semiconductor. The consequence of this is that the
recombination times of electrons and holes are long.

1 .7 THE ENVELOPE FUNCTION APPROXIMATION
Two important points have been argued:

1. The effective mass approximation is a valid description of bulk materials.


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SEMICONDUCTORS AND HETEROSTRUCTURES

12

2. Heterojunctions between dissimiliar materials, both of which can be well rep­
resented by the effective mass approximation, can be described by a material
potential which derives from the difference in the bandgaps.
The logical extension to point 2 is that the crystal potential of multiple heterojunctions
can also be described in this manner, as illustrated extensively in the previous section.
Once this is accepted, then the electronic structure can be represented by the simple
one-dimensional Schrodinger equation that has been aspired to:

/i2
8
--2m* z8

(1.27)

The envelope function approximation is the name given to the mathematical justifi­
cation for this series of arguments (see for example works by Bastard [17,18] and
Burt [ 19,20)). The name derives from the deduction that physical properties can be
derived from the slowly varying envelope function, identified here as 1/J(z), rather
than the total wave function 1/J(z)u(z) where the latter is rapidly varying on the scale
of the crystal lattice. The validity of the envelope function approximation is still an
active area of research [20]. With the line of reasoning used here, it is clear that
the envelope function approximation can be thought of as an approximation on the

material and not the quantum mechanics.
Some thought is enough to appreciate that the envelope function approximation
will have limitations, and that these will occur for very thin layers of material. The
materials are made of a collection of a large number of atomic potentials, so when
a layer becomes thin, these individual potentials will become significant and the
global average of representing the crystal potential by a constant will breakdown, for
example [21]. However, for the majority of examples this approach works well; this
will be demonstrated in later chapters, and, in particular, a detailed comparison with
an alternative approach which does account fully for the microscopic crystal potential
will be made in Chapter 12.

1 .8

THE RECIPROCAL LATTICE

For later discussions the concept of the reciprocal lattice needs to be developed. It has
already been shown that considering electron wave functions as plane waves (eik•r),
as found in a vacuum, but with a correction factor called the effective mass, is a useful
method of approximating the electronic bandstructure. In general, such a wave will
not have the periodicity of the crystal lattice; however, for certain wave vectors it will.
Such a set of wave vectors G are known as the reciprocal lattice vectors with the set
of points mapped out by these primitives known as the reciprocal lattice.
If the set of vectors G did have the periodicity of the lattice then this would imply
that:

( 1.28)

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THE RECIPROCAL LATIICE

13

i.e. an electron with this wave vector G would have a wave function equal at all points
in real space separated by a Bravais lattice vector R. Therefore:

(1.29)
which implies that

G.R = 211"n,

nEZ

(1.30)

Now learning from the form for the Bravais lattice vectors R given earlier in equa­
tion (1.22), it might be expected that the reciprocal lattice vectors G could be con­
structed in a similar manner from a set of three primitive reciprocal lattice vectors,
I.e.
(1.31)
With these choices then, the primitive reciprocal lattice vectors can be written as
follows:

8 2 X 83
81.(82 X 83)
83 X 81
b2 = 211"
81.(82 X 83)
81 X 82

b3 = 211"
81.(82 X 83)
b1

=

211"

(1.32)
(1.33)
(1.34)

It is possible to verify that these forms do satisfy equation (1.30):

(1.35)
Now b1 is perpendicular to both 82 and 83, and so only the product of b1 with 81 is
non-zero, and similarly for b2 and b3; hence:

(1.36)
and in fact, the products bj.8j

=

211"; therefore:
(1.37)

£l1 + +

Clearly 131
f32£l2 f33£l3 is an integer, and hence equation (1.30) is satisfied.

Using the face-centred cubic lattice vectors defined in equation (1.21), then:

81.(82

X

alx
83) = a2x
a 3x

which gives:

81.(82 x 83)

=

0x

a1y
a2y
a3y

alz
a2z
a3z

&2 &l.2
&2 2
-k
&

0

0

0

I� &o 1 -2Ao l &l. &o 1 + 2Ao l &l. �I
2

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(1.38)

2

(1.39)


14

SEMICONDUCTORS AND HETEROSTRUCTURES

. 31.(32 33)
X

=0

-

�o [


0-

( �o) 2] �o [(�o r ]
+

-

. . 31.(32 33) (AO)3
X

(1.40)

0

= 2 """2

(1.41)

Therefore, the first of the primitive reciprocal lattice vectors follows as:

j

o
&

k

&


(1.42)

2 2

o

(1.43)
(1.44)
A similar calculation of the remaining primitive reciprocal lattice vectors
gives the complete set as follows:

b2 b3
and

which are of course equivalent to the body-centred cubic Bravais lattice vectors (see
reference

[ 1], p. 68). Thus the reciprocal lattice constructed from the linear combi­

nations:

(1.46)
is a body-centred cubic lattice with lattice constant

47r/Ao.

cubic primitve reciprocal lattice vectors in equation

(1.45), then:


Taking the face-centred

(1.47)
(1.48)
The specific reciprocal lattice vectors are therefore generated by taking different
combinations of the integers

fh,

/32, /33.
and

This is illustrated

in Table 1.1.

It was shown by von Laue that when waves in a periodic structure satisfied the
following:

1

k.G = 21G1
A

[1], p. 99). Thus the 'free' electron
1.5), will be perturbed when the electron wave

then diffraction would occur (see reference
dispersion curves of earlier (Fig.


(1.49)

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THE RECIPROCAL LATIICE

15

Table 1.1 Generation of the reciprocal lattice vectors for the face-centred cubic crystal
by the systematic selection of the integer coefficients (31, f32, and (33

(31

(32

(33

0

0
0

0
0
0
1
0
0


1

0
0
-

1

0
0

-1
1
1

0

-1

1

0
0

-1

0

-1


1
-1
1

-1

0

1

G (211"/Ao)

(0,0,0)

(1,1, 1)
(1, 1, 1)
(1, 1,T)
(1, I, I)
(1, 1,I)
(I, 1, 1)
(1, 1 , 1)
(1,1,1)

1

(0,0,2)
( 0,2,0)
(2,0,0)
(2,2,2)


0
1
0

1

1

(etc.)

vector satisfies equation (1.49). Along the [001] direction, the smallest reciprocal
lattice vector G is (0,0,2) (in units of 211" /A o). Substituting into equation ( 1.49) gives:

,

,

,

k.(Oi+Oj+k)

= 2" x 2 x

1

211"
Ao

( 1 .50 )


This then implies the electron will be diffracted when:

k

=

211"
k
Ao

(1.51)

Fig. 1.14 illustrates the effect that such diffraction would have on the 'free-electron'
curves. At wavevectors which satisfy von Laue's condition, the energy bands are
disturbed and an energy gap opens. Such an improvement on the parabolic dispersion
curves of earlier, is known as the nearly free electron model.
The space between the lowest wavevector solutions to von Laue's condition is
called the first Brillouin zone. Note that the reciprocal lattice vectors in any particular
direction span the Brillouin zone. As mentioned above a face-centred cubic lattice
has a body-centred cubic reciprocal lattice, and thus the Brillouin zone is therefore
a three-dimensional solid, which happens to be a 'truncated octahedron' (see, for
example reference [1], p. 89). High-symmetry points around the Brillouin zone are
often labelled for ease of reference, with the most important of these, for this work,
being the k = 0 point, referred to as T', and the < 001 > zone edges, which are
called the 'X' points.

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16


SEMICONDUCTORS AND HETEROSTRUCTURES

Brillouin zone

o

21t/.4o

k

Figure 1.14 Comparison of the free and nearly free electron models

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CHAPTER 2

SOLUTIONS TO SCHROOINGER'S
EQUATION

2.1

THE INFINITE WELL

The infinitely deep one-dimensional potential well is the simplest confinement poten­
tial to treat in quantum mechanics. Virtually every introductory level text on quantum
mechanics considers this system, but nonetheless it is worth visiting again as some
of the standard assumptions often glossed over, do have important consequences for
one-dimensional confinement potentials in general.

The time-independent Schrodinger equation summarises the wave mechanics anal­
ogy to Hamilton's formulation of classical mechanics [22], for time-independent po­
tentials. In essence this ·states that the kinetic and potential energy components sum
to the total energy; in wave mechanics, these quantities are the eigenvalues of linear
operators, i.e.
(2.1)
where the eigenfunction 1/J describes the state of the system. Again in analogy with
classical mechanics the kinetic energy operator for a particle of constant mass is given
Quantum Wells, Wires and Dots, Second Edition. P.
©2005 John Wiley & Sons, Ltd.

Harrison

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17


18

SOLUTIONS TO SCHRODINGER'S EQUATION

by the following:

T=2m
p2

(2.2)

where P is the usual quantum mechanical linear momentum operator:

p = -inV' = -in

(�i
ox �j
+

8

+

�k)
z
8

(2.3)

By using this form for the kinetic energy operator T, the SchrOdinger equation then
becomes:
(2.4)

where the function Vex, y, z) represents the potential energy of the system as a func­
tion of the spatial coordinates. Restricting this to the one-dimensional potential of
interest here, then the SchrOdinger equation for a particle of mass m in a potential
well aligned along the z-axis (as in Fig. 2.1) would be:
n2

2
8

- 2m z8


V(z)=0

V(z)=injinity

Figure 2.1

(2.5)

I�

V(z)=infinity

The one-dimensional infinite well confining potential

Outside of the well, V (z) = 00, and hence the only possible solution is 1/J(z) = 0,
which in tum implies that all values of the energy E are allowed. Within the potential
well, the Schrodinger equation simplifies to:

-

8
z
8
2m
n2

(2.6)

which implies that the solution for 1/1 is a linear combination of the functions f(z)

which when differentiated twice give -f(z). Hence try the solution:
1/J(z)

= A sin kz + B cos kz

(2.7)

Substituting into equation (2.6) then gives:
n2k2
-(Asinkz + B cos kz) = E (Asin kz + B cos kz)
2m

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(2.8)


THE INFINITE WELL

fi?k2

.. 2m

=

E

19
(2.9)


Consideration of the boundary conditions will yield the, as yet unknown, constant k.
With this aim, consider again the kinetic energy term for this system, i.e.

fi? fP
T = ---1jJ(z)
2m {)z2

which can be rewritten as

T

= --- (

1i,2 {) {)
-1jJ(z)
2m {)z {)z

(2.10)

)

(2.11)

The mathematical form of this implies that, as a minimum, 1jJ( z) must be continuous.
If it is not, then the first derivative will contain poles which must be avoided if the
system is to have finite values for the kinetic energy. Given that 1jJ(z) has already
been deduced as zero outside of the well, then 1jJ(z) within the well must be zero at
both edges too.

n=3


sin(31tzll)

n=2

n=l

o

Figure 2.2

20

60
40
z-axis (A)

80

Solutions to the one-dimensional infinite well confining potential

If the origin is taken as the left hand edge of the well as in Fig. 2.2, then 1jJ (z)
as defined in equation (2.8) can contain no cosine terms, i.e. B = 0, and hence
1jJ(z) = A sin kz. In addition, for 1jJ(0) = 1jJ(lw) = 0:
(2.12)
where n is an integer, representing a series of solutions. Substituting into equa­
tion (2.9), then the energy of the confined states is given by:
(2.13)

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20

SOLUTIONS TO SCHROOINGER'S EQUATION

The only remaining unknown is the constant factor A, which is deduced by considering
the normalisation of the wave function; as 'l/J*(z)'l/J(z) represents the probability of
finding the particle at a point z, then as the particle must exist somewhere:

which gives A

=

11W'l/J*(Z)'l/J(Z) dz = 1

(2.14)

J(2/lw), and therefore

(2.15)

Under the effective mass and envelope function approximations, the energy of an
electron or hole in a hypothetical infinitely deep semiconductor quantum well can be
calculated by using the effective mass m * for the particle mass m of equation (2. 13).

100000

.....-. n=3
---- n=2


,-.,

10000

--n=l

�

;>

S

>.

'-'

i

�

1000
100
10

Figure 2.3

0

Well width lw (A)


50

100

150

First three energy levels versus well width for

200

an electron in a GaAs infinite

potential well

Figure 2.3 displays the results of calculations of the lowest three energy states of
an electron in a GaAs well of width lw surrounded by hypothetical infinite barriers
(for these and all material parameters see Appendix A). All three states show the
same monotonic behaviour, with the energy decreasing as the well width increases.
The sine function solutions derived for this system are completely standard and
found extensively in the literature. Although it should be noted that the arguments
developed for setting the boundary conditions, i.e. 'l/J( z) continuous, also implied that
the first derivative should be continuous too, although use is never made of this second
boundary condition, The limitations of solution imposed by this are avoided by saying

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IN-PLANE DISPERSION


21

that not only is the potential infinite outside the well, but in addition the Schrodinger
equation is not defined in these regions-a slight contradiction with the deduction of
the first boundary condition_ This point, Le_ that there is still ambiguity in the choice
of boundary conditions for commonly accepted solutions, will be revisited later in
this chapter_

2.2

IN-PLANE DISPERSION

Figure 2.4

A GaAs/Gal - ",AI",As layered structure and the in-plane motion of a charge carrier

V(z)

If the one-dimensional potential
is constructed from alternating thin layers
of dissimilar semiconductors, then the particle, whether it be an electron or a hole,
can move in the plane of the layers (see Fig 2-4)_
In this case, all of the tenns of the kinetic energy operator are required, and hence
the SchrOdinger equation would be as follows:

As the potential can be written as a sum of independent functions, Le.
=
+
= 0, the eigen­
+

where it just happens in this case that
function of the system be written as a product:

Vex) = V(y) V Vex)

V(y) V(z),

(2.17)
Using this in the above Schrodinger equation then:
-

/i2 (EP'l/J
8x2x 'l/Jy'l/Jz [P'l/J
8y2y 'l/Jx'l/Jz + [P8z'l/J2z 'l/Jx'l/Jy) + V(z)'l/Jx'l/Jy'l/Jz = E'l/Jx'l/Jy'l/Jz
E,
x-, y-,
E Ex Ey + Ez.

2m

+

(2.18)
It is then possible to identify three distinct contributions to the total energy
one
from each of the perpendicular
and z-axes, i.e.
=
It is said
+


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22

SOLUTIONS TO SCHR60INGER'S EQUATION

that the motions 'de-couple' giving an equation of motion for each of the axes:

fJ2,¢x 'l/Jy'l/Jz = Ex'¢x'¢y'l/Jz
2
h,2 02ox'l/Jy
-2
'¢x'¢z Ey'l/Jx'¢y'¢z
m oy2
02,¢z
- n}
m
2 OZ2 'l/Jx'l/Jy + V(z)'l/Jx'¢y'¢z = Ez'¢x'¢y'¢z
- 2n?
m

=

(2.19)
(2.20)
(2.21)

Dividing throughout, then:


(2.22)
(2.23)
(2.24)

17=0
Figure 2.5

E

=

EJI +

r/k.�.
2m*

Schematic showing the in-plane (k""y) dispersion curves and the subband structure

The last component is identical to the one-dimensional SchrOdinger equation for
a confining potential

V(z)

as discussed, for the particular case of an infinite well, in

the last section. Consider the first and second components. Again, an eigenfunction
f is sought which when differentiated twice returns - f; however, in this case

it


must be remembered that the solution will represent a moving particle. Thus the
eigenfunction must reflect a current flow and have complex components, so try the

exp

(ikxx).
fj,2 02
exp (ikxx) = Ex exp (ikxx)
2m ox2

standard travelling wave,

Then:

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(2.25)


DENSITY OF STATES

23
(2.26)

which is clearly just the kinetic energy of a wave travelling along the x-axis. A
similar equation follows for the y-axis, and hence the in-plane motion of a particle in
a one-dimensional confining potential, but of infinite extent in the x-y plane can be
summarised as:
and


(2.27)

Therefore, while solutions of Schrooinger's equation along the axis of the one­
dimensional potential produce discrete states of energy Ez = En, in the plane of
a semiconductor quantum well there is a continuous range of allowed energies, as
illustrated in Fig. 2.5. In bulk materials, such domains are called 'energy bands',
while in quantum well systems these energy domains associated with confined levels
are referred to as 'subbands'. Therefore the effect of the one-dimensional confining
potential is to remove a degree of freedom, thus restricting the momentum of the
charge carrier from three-dimensions to two. It is for this reason that the states within
quantum well systems are generally referred to as two-dimensional.
Later in this text, quantum wires and dots will be considered which further restrict
the motion of carriers in two and three dimensions respectively, thus giving rise to
the terms one- and zero-dimensional states.
Summarising then, within a semiconductor quantum well system the total energy of
an electron or hole, of mass m*, with in-plane momentum kx,y , is equal to Ez + Ex,y ,
which is given by:
(2.28)

2.3

DENSITY OF STATES

Therefore the original confined states within the one-dimensional potential which
could each hold two charge carriers of opposite spin, from the Pauli exclusion princi­
ple, broaden into subbands, thus allowing a continuous range of carrier momenta. In
order to answer the question 'Given a particular number of electrons (or holes) within
a subband, what is the distribution of their energy and momenta?', the first point that
is required is a knowledge of the density of states, i.e. how many electrons can exist

within a range of energies. In order to answer this point for the case of subbands in
quantum wells, it is necessary first to understand this property in bulk crystals.
Following the idea behind Bloch's theorem (see reference [l] p. 133) that an eigen­
state within a bulk semiconductor, which can be written as 'l/J = (l/n) exp (ik.r),
must display periodicity within the lattice, then if the unit cell is of side L:

:. 'l/J (x, y, z) =

'l/J(x, y, z) = 'l/J(x +

�

exp {i [k x( x +

L, y + L, Z + L)

L) + ky(Y + L) + kz(z + L)]}

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(2.29)
(2.30)


24

SOLUTIONS TO SCHRODINGER'S EQUATION

:. 'IjJ(x, y , z) = n exp [i(kxx + kyY + kzz) ] exp [i(kxL + kyL + kzL)]


1

(2.31)

Thus for the periodicity condition to be fulfilled, the second exponential tenn must
be identical to I, which implies that:
(2.32)

where nx, ny and nz are integers. Each set of values of these three integers defines
a distinct state, and hence the volume of k-space occupied by one state is (27r /
These states fill up with successively larger values of nx, ny and nz, i.e. the lowest
energy state has values (000), then pennutations of (100), (110), etc., which gradually
fill a sphere. At low temperatures, the sphere has a definite boundary between states
that are all occupied followed by states that are unoccupied; the momentum of these
states is called the Fermi wave vector and the equivalent energy is the Fermi energy.
At higher temperatures, carriers near the edge of the sphere are often scattered to
higher energy states, thus 'blurring' the boundary between occupied and unoccupied
states. For a more detailed description see, for example, Ashcroft and Mermin [1].
Many of the interesting phenomena associated with semiconductors derive from
the properties of electrons near the Fermi energy, as it is these electrons that are able
to scatter into nearby states thus changing both their energy and momenta, In order
to be able to progress with descriptions of, transport for example (later in this book),
it is necessary to be able to describe the density of available states.
The density of states is defined as the number of states per energy per unit volume
of real space:
dN
(2.33)
p(E) = dE

L) 3.


In k-space, the total number of states N is equal to the volume of the sphere of radius
k, divided by the volume occupied by one state and divided again by the volume of
real space, i.e.
(2.34)

(2.35)

where the factor 2 has been introduced to allow for double occupancy of each state
by the different carrier spins, Returning to the density of states, then:
( )
dN
dN dk
p E = dE = dkdE

(2.36)

Now equation (2.35) gives
(2.37)

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DENSITY OF STATES

( )

25

In addition, the parabolic bands of effective mass theory give:

.
.. k =

. dk =
..
dE

( )

2m*E !
fi2

(2.38)

2m ! E -!
*
fi2
2

(2.39)

Which finally gives the density of states in bulk as:

p(E)

=

( )

_1_ 2m

27r2

fi2

3

2"

E!

(2.40)

Thus the density of states within a band, and around a minimum where the energy can
be represented as a parabolic function of momentum, is continual and proportional
to the square root of the energy.

radiusk

./

...-1-

e--

"

�

�


I

�

\
\

II

.......

Figure 2.6

--

--

v

II

area

Illustration of filling the two-dimensional momenta states in a quantum well

The density of states in quantum well systems follows analogously; however this
time, as there are only two degrees of freedom, successive states represented by values
of nx and ny fill a circle in k-space, as illustrated in Fig. 2.6. Such a situation has

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