Tài liệu Semiconductor MaterialsS. K. Tewksbury Microelectronic Systems Research Center Dept. of Electrical ppt

electron) valence band. The bonding is largely covalent, though the shift of valence charge
from the Group V atoms to the Group III atoms induces a component of ionic bonding to
the crystal (in contrast to the elemental semiconductors which have purely covalent bonds).
Representative III-V compound semiconductors are GaP, GaAs, GaSb, InP, InAs, and InSb.
GaAs is probably the most familiar example of III-V compound semiconductors, used
for both high speed electronics and for optoelectronic devices. Optoelectronics has taken
advantage of ternary and quaternary III-V semiconductors to establish optical wavelengths
and to achieve a variety of novel device structures. The ternary semiconductors have the
general form (A
x
,A

1−x
)B (with two group III atoms used to fill the group III atom positions
in the lattice) or A(B
x
,B

1−x
) (using two group V atoms in the Group V atomic positions
in the lattice). The quaternary semiconductors use two Group III atomic elements and
two Group V atomic elements, yielding the general form (A
x
,A

1−x
)(B
y
,B

1−y
). In such
constructions, 0 ≤ x ≤ 1. Such ternary and quaternary versions are important since the
mixing factors (x and y) allow the bandgap to be adjusted to lie between the bandgaps
of the simple compound crystals with only one type of Group III and one type of Group
V atomic element. The adjustment of wavelength allows the material to be tailored for
particular optical wavelengths, since the wavelength λ of light is related to energy (in this
case the gap energy E
g
)byλ=hc/E
g
, where h is Plank’s constant and c is the speed
of light. Table 2 provides examples of semiconductor laser materials and a representative
optical wavelength for each, providing a hint of the vast range of combinations which are
available for optoelectronic applications. Table 3, on the other hand, illustrates the change
in wavelength (here corresponding to color in the visible spectrum) by adjusting the mixture
of a ternary semiconductor.
Table 2: Semiconductor optical sources and representative wavelengths.
Material layers used wavelength
ZnS 454 nm
AlGaInP/GaAs 580 nm
AlGaAs/GaAs 680 nm
GaInAsP/InP 1580 nm
InGaAsSb/GaSb 2200 nm
AlGaSb/InAsSb/GaSb 3900 nm
PbSnTe/PbTe 6000 nm
Table 3: Variation of x to adjust wavelength in GaAs
x
P
1−x
semiconductors.
Ternary Compound Color
GaAs
0.14
P
0.86
Yellow
GaAs
0.35
P
0.65
Orange
GaAs
0.6
P
0.4
Red
5
FCC Lattice
A
FCC Lattice
B
(a) (b) (c)
In contrast to single element elemental semiconductors (for which the positioning of
each atom on a lattice site is not relevant), III-V semiconductors require very good control
of stoichiometry (i.e., the ratio of the two atomic species) during crystal growth. For example,
each Ga atom must reside on a Ga (and not an As) site and vice versa. For these and other
reasons, large III-V crystals of high quality are generally more difficult to grow than a large
crystal of an elemental semiconductor such as Si.
2.1.3 Compound II-VI Semiconductors
These semiconductors are based on one atomic element from Group II and one atomic el-
ement from Group VI, each type being bonded to four nearest neighbors of the other type
as shown in Figure 1c. The increased amount of charge from Group VI to Group II atoms
tends to cause the bonding to be more ionic than in the case of III-V semiconductors. II-VI
semiconductors can be created in ternary and quaternary forms, much like the III-V semi-
conductors. Although less common than the III-V semiconductors, the II-VI semiconductors
have served the needs of several important applications. Representative II-VI semiconductors
are ZnS, ZnSe,and ZnTe (which form in the zinc blende lattice structure discussed below);
CdS and CdSe, (which can form in either the zinc blende or the wurtzite lattice structure)
and CdTe which forms in the wurtzite lattice structure.
2.2 Three-Dimensional Crystal Lattice
The two-dimensional views illustrated in the previous section provide a simple view of the
sharing of valence band electrons and the bonds between atoms. However, the full 3-D lattice
structure is considerably more complex than this simple 2-D illustration. Fortunately, most
semiconductor crystals share a common basic structure, developed below.
Figure 2: Three-dimensional crystal lattice structure. (a) Basic cubic lattice. (c) Face-
centered cubic (fcc) lattice. (c) Two interpenetrating fcc lattices. In this figure, the dashed
lines between atoms are not atomic bonds but instead are used merely to show the basic
outline of the cube.
The crystal structure begins with a cubic arrangement of 8 atoms as shown in Figure
2a. This cubic lattice is extended to a face-centered cubic (fcc) lattice, shown in 2b, by
6

adding an atom to the center of each face of the cube (leading to a lattice with 14 atoms).
The lattice constant is the side dimension of this cube.
The full lattice structure combines two of these fcc lattices, one lattice interpenetrating
the other (i.e., the corner of one cube is positioned within the interior of the other cube,
with the faces remaining parallel), as illustrated in Figure 2c. For the III-V and II-VI
semiconductors with this fcc lattice foundation, one fcc lattice is constructed from one type
of element (e.g., type III) and the second fcc lattice is constructed from the other type of
element (e.g., group V). In the case of ternary and quaternary semiconductors, elements
from the same atomic group are placed on the same fcc lattice. All bonds between atoms
occur between atoms in different fcc lattices. For example, all Ga atoms in the GaAs crystal
are located on one of the fcc lattices and are bonded to As atoms, all of which appear on
the second fcc lattice. The interatomic distances between neighboring atoms is therefore less
than the lattice constant. For example, the interatomic spacing of Si atoms is 2.35
˚
A but
the lattice constant of Si is 5.43
˚
A.
If the two fcc lattices contain elements from different groups of the periodic chart,
the overall crystal structure is called the zinc blende lattice. In the case of an elemental
semiconductor such as silicon, silicon atoms appear in both fcc lattices and the overall
crystal structure is called the diamond lattice (carbon crystallizes into a diamond lattice
creating true diamonds, and carbon is a group IV element). As in the discussion regarding
III-V semiconductors above, the bonds between silicon atoms in the silicon crystal extend
between fcc sublattices.
Although the common semiconductor materials share this basic diamond/zinc blende
lattice structure, some semiconductor crystals are based on a hexagonal close-packed (hcp)
lattice. Examples are CdS and CdSe. In this example, all the Cd atoms are located on
one hcp lattice while the other atom (S or Se) is located on a second hcp lattice. In the
spirit of the diamond and zinc blende lattices above, the complete lattice is constructed by
interpenetrating these two hcp lattices. The overall crystal structure is called a wurtzite
lattice. Type IV-VI semiconductors (PbS, PbSe, PbTe, and SnTe) exhibit a narrow band
gap and have been used for infrared detectors. The lattice structure of these example IV-VI
semiconductors is the simple cubic lattice (also called an NaCl lattice).
2.3 Crystal Directions and Planes
Crystallographic directions and planes are important in both the characteristics and the ap-
plications of semiconductor materials since different crystallographic planes can exhibit sig-
nificantly different physical properties. For example, the surface density of atoms (atoms/cm
2
)
can differ substantially on different crystal planes. A standardized notation (the so-called
Miller indices) is used to define the crystallographic planes and directions normal to those
planes.
The general crystal lattice defines a set of unit vectors (a,b, and c) such that an entire
crystal can be developed by copying the unit cell of the crystal and duplicating it at integer
offsets along the unit vectors, i.e., replicating the basis cell at positions n
a
a + n
b
b + n
c
c,
where n
a
, n
b
, and n
c
are integers. The unit vectors need not be orthogonal in general. For
7
(a) (b) (c)
x
z
y
the cubic foundation of the diamond and zinc blende structures, however, the unit vectors
are in the orthogonal x, y, and z directions.
Figure 3 shows a cubic crystal, with basis vectors in the x,y, and z directions. Su-
perimposed on this lattice are three planes (Figures 3a, b and c). The planes are defined
relative to the crystal axes by a set of three integers (h, k, l) where h corresponds to the
plane’s intercept with the x-axis, k corresponds to the plane’s intercept with the y-axis and
l corresponds to the plane’s intercept with the z-axis. Since parallel planes are equivalent
planes, the intercept integers are reduced to the set of the three smallest integers having the
same ratios as the above intercepts. The (100), (010) and (001) planes correspond to the
faces of the cube. The (111) plane is tilted with respect to the cube faces, intercepting the
x, y, and z axes at 1, 1, and 1, respectively. In the case of a negative axis intercept, the
corresponding Miller index is given as an integer and a bar over the integer,e.g., (
¯
100), i.e.,
similar to (100) plane but intersecting x-axis at -1.
Figure 3: Examples of crystallographic planes within a cubic lattice organized semiconductor
crystal. (a) (010) plane. (b) (110) plane. (c) (111) plane.
Additional notation is used to represent sets of planes with equivalent symmetry and to
represent directions. For example, {100} represents the set of equivalent planes (100), (
¯
100).
(010), (0
¯
10), (001), and (00
¯
1). The direction normal to the (hkl) plane is designated [hkl].
The different planes exhibit different behavior during device fabrication and impact electrical
device performance differently. One difference is due to the different reconstructions of the
crystal lattice near a surface to minimize energy. Another is the different surface density of
atoms on different crystallographic planes. For example, in Si the (100), (110), and (111)
planes have surface atom densities (atoms per cm
2
)of6.78×10
14
,9.59×10
14
, and 7.83×10
14
,
respectively.
3 Energy Bands and Related Semiconductor Parameters
A semiconductor crystal establishes a periodic arrangement of atoms, leading to a periodic
spatial variation of the potential energy throughout the crystal. Since that potential energy
8

varies significantly over interatomic distances, quantum mechanics must be used as the basis
for allowed energy levels and other properties related to the semiconductor. Different semi-
conductor crystals (with their different atomic elements and different inter-atomic spacings)
lead to different characteristics. However, the periodicity of the potential variations leads
to several powerful general results applicable to all semiconductor crystals. Given these
general characteristics, the different semiconductor materials exhibit properties related to
the variables associated with these general results. A coherent discussion of these quantum
mechanical results is beyond the scope of this chapter and we therefore take those general
results as given.
In the case of materials which are semiconductors, a central result is the energy-
momentum functions defining the state of the electronic charge carriers. In addition to
the familiar electrons, semiconductors also provide holes (i.e. positively charged particles)
which behave similarly to the electrons. Two energy levels are important: one is the energy
level (conduction band) corresponding to electrons which are not bound to crystal atoms and
which can move through the crystal and the other energy level (valence band) corresponds to
holes which can move through the crystal. Between these two energy levels, there is a region
of “forbidden” energies (i.e., energies for which a free carrier can not exist). The separation
between the conduction and valence band minima is called the energy gap or band gap. The
energy bands and the energy gap are fundamentally important features of the semiconductor
material and are reviewed below.
3.1 Conduction and Valence Band
In quantum mechanics, a “particle” is represented by a collection of plane waves (e
j(ωt−

k·x)
)
where the frequency ω is related to the energy E according to E =¯hω and the momentum
p is related to the wave vector by p =¯h

k. In the case of a classical particle with mass m
moving in free space, the energy and momentum are related by E = p
2
/(2m) which, using
the relationship between momentum and wave vector, can be expressed as E =(¯hk)
2
/(2m).
In the case of the semiconductor, we are interested in the energy/momentum relationship for
a free electron (or hole) moving in the semiconductor, rather than moving in free space. In
general, this E-k relationship will be quite complex and there will be a multiplicity of E-k
“states” resulting from the quantum mechanical effects. One consequence of the periodicity
of the crystal’s atom sites is a periodicity in the wave vector k, requiring that we consider
only values of k over a limited range (with the E-k relationship periodic in k).
Figure 4 illustrates a simple example (not a real case) of a conduction band and a valence
band in the energy-momentum plane (i.e., the E vs k plane). The E vs k relationship of the
conduction band will exhibit a minimum energy value and, under equilibrium conditions,
the electrons will favor being in that minimum energy state. Electron energy levels above
this minimum (E
c
) exist, with a corresponding value of momentum. The E vs k relationship
for the valence band corresponds to the energy-momentum relationship for holes. In this
case, the energy values increase in the direction toward the bottom of the page and the
“minimum” valence band energy level E
v
is the maximum value in Figure 4. When an
electron bound to an atom is provided with sufficient energy to become a free electron, a
9
ΓΚL [111] direction [100] direction
kk
Energy E
E
c
E
v
Conduction band
minimum
(free electrons)
Valence band
minimum
(free holes)
Energy gap
E
g
Figure 4: General structure of conduction and valence bands.
hole is left behind. Therefore, the energy gap E
g
= E
c
−E
v
represents the minimum energy
necessary to generate an electron-hole pair (higher energies will initially produce electrons
with energy greater than E
c
, but such electrons will generally lose energy and fall into the
potential minimum).
The details of the energy bands and the bandgap depend on the detailed quantum
mechanical solutions for the semiconductor crystal structure. Changes in that structure
(even for a given semiconductor crystal such as Si) can therefore lead to changes in the
energy band results. Since the thermal coefficient of expansion of semiconductors is non-
zero, the band gap depends on temperature due to changes in atomic spacing with changing
temperature. Changes in pressure also lead to changes in atomic spacing. Though these
changes are small, the are observable in the value of the energy gap. Table 4 gives the room
temperature value of the energy gap E
g
for several common semiconductors, along with the
rate of change of E
g
with temperature (T ) and pressure (P ) at room temperature.
The temperature dependence, though small, can have a significant impact on carrier
densities. A heuristic model of the temperature dependence of E
g
is E
g
(T )=E
g
(0
o
K) −
αT
2
/(T + β). Values for the parameters in this equation are provided in Table 5. Between
0K and 1000K, the values predicted by this equation for the energy gap of GaAs are accurate
to about 2 ×10
−3
eV (electron volts).
10

Table 4: Variation of energy gap with temperature and pressure. Adapted from [5].
Semiconductor E
g
(300K) dE
g
/dT (meV/
o
K) dE
g
/dP (meV/kbar)
Si 1.110 -0.28 -1.41
Ge 0.664 -0.37 5.1
GaP 2.272 -0.37 10.5
GaAs 1.411 -0.39 11.3
GaSb 0.70 -0.37 14.5
InP 1.34 -0.29 9.1
InAs 0.356 -0.34 10.0
InSb 0.180 -0.28 15.7
ZnSe 2.713 -0.45 0.7
ZnTe 2.26 -0.52 8.3
CdS 2.485 -0.41 4.5
CdSe 1.751 0.36 5.
CdTe 1.43 -0.54 8
Table 5: Temperature dependence parameters for common semiconductors. Adapted from
[4].
E
g
(0
o
K) α(×10
−4
) β E
g
(300K)
GaAs 1.519 eV 5.405 204 1.42 eV
Si 1.170 eV 4.73 636 1.12 eV
Ge 0.7437 eV 4.774 235 0.66 eV
11
Energy gap
Conduction
band
minimum
Valence band
minimum
0
1
2
3
4
5
6
-1
-2
-3
[111]
[100]
Γ
LK
[111]
[100]
Γ
LK
[111]
[100]
Γ
LK
Germanium Silicon GaAs
Wave Vector
Energy (eV)
(a) (b) (c)
3.2 Direct Gap and Indirect Gap Semiconductors
Figure 5 illustrates the energy bands for Ge, Si and GaAs crystals. In Figure 5b, for silicon,
the valence band has a minimum at a value of

k different than that for the conduction band
minimum. This is an indirect gap, with generation of an electron-hole pair requiring an
energy E
g
and a change in momentum (i.e., k). For direct recombination of an electron-hole
pair, a change in momentum is also required. This requirement for a momentum change (in
combination with energy and momentum conservation laws) leads to a requirement that a
phonon participate with the carrier pair during a direct recombination process generating
a photon. This is a highly unlikely event, rendering silicon ineffective as an optoelectronic
source of light. The direct generation process is more readily allowed (with the simultaneous
generation of an electron, a hole, and a phonon), allowing silicon and other direct gap
semiconductors to serve as optical detectors.
Figure 5: Conduction and valence bands for (a) germanium, (b) silicon, and (c) GaAs.
Adapted from [4].
12

In Figure 5c, for GaAs, the conduction band minimum and the valence band minimum
occur at the same value of momentum, corresponding to a direct gap. Since no momentum
change is necessary during direct recombination, such recombination proceeds readily, pro-
ducing a photon with the energy of the initial electron and hole (i.e., a photon energy equal
to the bandgap energy). For this reason, direct gap semiconductors are efficient sources of
light (and use of different direct gap semiconductors with different E
g
provides a means of
tailoring the wavelength of the source). The wavelength λ corresponding to the gap energy
is λ = hc/E
g
.
Figure 5c also illustrates a second conduction band minimum with an indirect gap, but
at a higher energy than the minimum associated with the direct gap. The higher conduction
band minimum can be populated by electrons (which are in an equilibrium state of higher
energy) but the population will decrease as the electrons gain energy sufficient to overcome
that upper barrier.
3.3 Effective Masses of Carriers
For an electron with energy close to the minimum of the conduction band, the energy vs
momentum relationship is approximately given by E(k)=E
0
+a
2
(k−k
∗
)
2
+a
4
(k−k
∗
)
4
+
Here, E
0
= E
c
is the “ground state energy” corresponding to a free electron at rest and k
∗
is
the wave vector at which the conduction band minimum occurs. Only even powers of k −k
∗
appear in the expansion of E(k) around k
∗
due to the symmetry of the E-k relationship
around k = k
∗
. The above approximation holds for sufficiently small increases in E above
E
c
. For sufficiently small movements away from the minimum (i.e., sufficiently small k −k
∗
),
the terms in k−k
∗
higher than quadratic can be ignored and E(k) ≈ E
0
+a
2
k
2
, where we have
taken k
∗
= 0. If, instead of a free electron moving in the semiconductor crystal, we had a free
electron moving in free space with potential energy E
0
, the energy-momentum relationship
would be E(k)=E
0
+(¯hk)
2
/(2m
0
), where m
0
is the mass of an electron. By comparison of
these results, it is clear that we can relate the curvature coefficient a
2
associated with the
parabolic minimum of the conduction band to an effective mass m
∗
e
, i.e., a
2
=(¯h
2
)/(2m
∗
e
)or
1
m
∗
e
=
2
¯h
2
·
∂
2
E
c
(k)
∂k
2
.
Similarly for holes, an effective mass m
∗
h
of the holes can be defined by the curvature of
the valence band minimum, i.e.,
1
m
∗
h
=
2
¯h
2
·
∂
2
E
v
(k)
∂k
2
.
Since the energy bands depend on temperature and pressure, the effective masses can also
be expected to have such dependencies, though the room temperature and normal pressure
value is normally used in device calculations.
The above discussion assumes the simplified case of a scalar variable k. In fact, the wave
vector

k has three components (k
1
,k
2
,k
3
), with directions defined by the unit vectors of the
underlying crystal. Therefore, there are separate masses for each of these vector components
13

of

k, i.e., masses m
1
,m
2
,m
3
. A scalar mass m
∗
can be defined using these directional masses,
the relationship depending on the details of the directional masses. For cubic crystals (as in
the diamond and zinc blende structures), the directions are the usual orthonormal directions
and m
∗
=(m
1
·m
2
·m
3
)
1/3
. The three directional masses effectively reduce to two components
if two values are equal (e.g., m
1
= m
2
), as in the case of longitudinal and transverse effective
masses (m
l
and m
t
, respectively) seen in silicon and several other semiconductors. In this
case, m
∗
=[(m
t
)
2
·m
l
]
1/3
.If all three values of m
1
,m
2
,m
3
are equal, then a single value m
∗
can be used.
An additional complication is seen in the valence band structures in Figure 5. Here,
two different E-k valence bands have the same minima. Since their curvatures are different,
the two bands correspond to different masses, one corresponding to heavy holes with mass
m
h
and the other to light holes with mass m
l
. The effective scalar mass in this case is
m
∗
=(m
3/2
h
+m
3/2
l
)
2/3
.Such light and heavy holes occur in several semiconductors, including
Si.
Values of effective mass are given in Tables 8 and 13.
3.4 Intrinsic Carrier Densities
The density of free electrons in the conduction band depends on two functions. One is the
density of states D(E) in which electrons can exist and the other is the energy distribution
function F (E,T) of free electrons.
The energy distribution function (under thermal equilibrium conditions) is given by the
Fermi-Dirac distribution function
F (E)=

1 + exp

E − E
f
k
B
T

−1
which, in most practical cases, can be approximated by the classical Maxwell-Boltzmann
distribution. These distribution functions are general functions, not depending on the specific
semiconductor material.
The density of states D(E), on the other hand, depends on the semiconductor material.
A common approximation is
D
n
(E)=M
c
√
2
π
2
(E−E
c
)
1
/2
¯h
3
(m
∗
e
)
3
/2
for electrons and
D
p
(E)=M
v
√
2
π
2
(E
v
−E)
1
/2
¯h
3
(m
∗
h
)
3
/2
for holes. Here, M
c
and M
v
are the number of equivalent minima in the conduction band
and valence band, respectively. Note that necessarily E ≥ E
c
for free electrons and E ≤ E
v
for free holes due to the forbidden region between E
c
and E
v
.
The density n of electrons in the conduction band can be calculated as
n =

∞
E=E
c
F (E,T)D(E)dE.
14

Xem chi tiết: Tài liệu Semiconductor MaterialsS. K. Tewksbury Microelectronic Systems Research Center Dept. of Electrical ppt