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Chap 3. Introduction to Quantum Theory of
Solids
 Allowed and Forbidden Energy Bands
 k-space Diagrams
 Electrical Conduction in Solids
 Density of State Functions
 Statistical Mechanics
 Homework
Solid-State Electronics
Chap. 3
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Preview
 Recall from the previous analysis that the energy of a bound electron is
quantized. And for the one-electron atom, the probability of finding the
electron at a particular distance from the nucleus is not localized at a
given radius.
 Consider two atoms that are in close proximity to each other. The wave
functions of the two atom electrons overlap, which means that the two
electrons will interact. This interaction results in the discrete quantized
energy level splitting into two discrete energy levels.
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Formation of Energy Bands
 Consider a regular periodic arrangement of atoms in which each atoms
contains more than one electron. If the atoms are initially far apart, the
electrons in adjacent atoms will not interact and will occupy the
discrete energy levels.
 If the atoms are brought closer enough, the outmost electrons will
interact and the energy levels will split into a band of allowed energies.
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Formation of Energy Bands
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Chap. 3
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Kronig-Penny Model
 The concept of allowed and forbidden energy levels can be developed
by considering Schrodinger’s equation.
Kronig-Penny Model
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Kronig-Penny Model
 The Kronig-Penny model is an idealized periodic potential
representing a 1-D single crystal.
 We need to solve Schrodinger’s equation in each region.
 To obtain the solution to the Schrodinger’s equation, we make use of
Bloch theorem. Bloch states that all one-electron wave functions,
involving periodically varying potential energy functions, must be of
the form, (x) = u(x)ejkx, u(x) is a periodic function with period (a+b)
and k is called a constant of the motion.
 The total wave function (x,t) may be written as (x,t) = u(x)ej(kx-(E/ħ)t).
 In region I (0 < x < a), V(x) = 0, then Schrodinger’s equation becomes
d 2u1 ( x)
du ( x)
2m E
 2 jk 1
 (k 2   2 )u1 ( x)  0 ,  2  2
dx
dx

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Kronig-Penny Model
 The solution in region I is of the form,
u1 ( x)  Ae j ( k ) x  Be j ( k ) x for 0  x  a
 In region II (-b < x < 0), V(x) = Vo, and apply Schrodinger’s eq.
2m V
d 2u 2 ( x )
du ( x)
 2 jk 2
 (k 2   2 )u2 ( x)  0 ,  2   2  o2 o
dx
dx

The solution for region II is of the form,
u2 ( x)  Ce j (  k ) x  De j (  k ) x for -b  x  0
 Boundary conditions:
u1 (0)  u2 (0)  A  B  C  D  0
du1
dx

x 0
du2
dx
   k A    k B    k C    k D  0
x 0
u1 (a)  u2 (b)  Ae j ( k ) a  Be j ( k ) a  Ce j (  k )b  De j (  k )b  0
du1
dx

xa
du2
dx
   k Ae j ( k ) a    k Be j (  k ) a    k Ce  j (  k )b    k De j (   k )b  0
x b
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Dept. of E. E. NCU
Kronig-Penny Model
 There is a nontrivial solution if, and only if, the determinant of the
coefficients is zero. This result is


 2   2
 1  f (  E / Vo ) 
(sin a)(sin b)  (cosa)(cos b)  cos k (a  b)  1
2
 The above equation relates k to the total energy E (through ) and the
potential function Vo (through ). The allowed values of E can be
determined by graphical or numerical methods.
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Kronig-Penny Model
 Recall -1cosk(a+b)1, so E-values which cause f() to lie in the range
-1 f() 1 are the allowed system energies.—
 The ranges of allowed energies are called energy bands; the excluded
energy ranges (|f()|1) are called the forbidden gaps or bandgaps .
 The energy bands in a crystal can be visualized by

Energy
4
3
2
1
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Dept. of E. E. NCU
E-k Diagram
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Chap. 3
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Dept. of E. E. NCU
k-space Diagram
 Consider the special case for which Vo = 0, (free particle case)
 cos(a+b) = cosk(a+b), i.e.,  = k,
 
2mE
2

1
2m( mv 2 )
p
2
 k

2
,where p is the particle momentum
and k is referred as a wave number.
 We can also relate the energy and momentum as E = k2ħ2/2m
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Chap. 3
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E-k diagram
 More interesting solution occur for E < Vo ( = j), which applies to
the electron bound within the crystal. The result could be written as
 2  2
(sin a)(sinh b)  (cosa)(cosh b)  cos k (a  b)
2
 Consider a special case, b0, Vo , but bVo is finite, the above eq.
becomes
m V ba
sin a
P'

 cosa  cos ka, P'  
a

o
2




 The solution of the above equation results in a band of allowed
energies.
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Chap. 3
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E-k diagram
 Consider the function of f (a)  P'
Solid-State Electronics
Chap. 3
13
sin a
 cos a graphically,
a
Instructor: Pei-Wen Li
Dept. of E. E. NCU
E-k diagram
 E-k diagram could be generated from the above figure.
 This shows the concept of the allowed energy bands for the particle
propagating in the crystal.
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Chap. 3
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Dept. of E. E. NCU
Reduced k-space
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Chap. 3
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Dept. of E. E. NCU
Electrical Conduction in Solids
 the Bond Model
 Energy Band
E-K diagram of a semiconductor
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Chap. 3
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Dept. of E. E. NCU
Drift Current
 If an external force is applied to the electrons in the conduction band
and there are empty energy states into which the electrons can move,
electrons can gain energy and a net momentum.
n
 The drift current due to the motion of electrons is J  e vi
i 1
where n is the number of electrons per volume and vi is the electron
velocity in the crystal.
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Chap. 3
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Dept. of E. E. NCU
Electron Effective Mass
 The movement of an electron in a lattice will be different than that of
an electron in free space. There are internal forces in the crystal due to
the positively charged ions or protons and electrons, which will
influence the motion of electrons in the crystal. We can write
Ftotal  Fext  Fint  ma
 Since it is difficult to take into account of all of the internal forces, we
can write
F  m*a
ext
 m* is called the effective mass which takes into account the particle
mass and the effect of the internal forces.
Solid-State Electronics
Chap. 3
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective mass, E-k diagram
 Recall for a free electron, the energy and momentum are related by
p 2  2k 2
dE  2 k p
1 dE p
E





 v
2 m 2m
dk
m
m
 dk m
– So the first derivative of E w.r.t. k is related to the velocity of the particle.
 In addition,
d 2E 2
1 d 2E 1

 2

dk 2
m
 dk 2 m
– So the second derivative of E w.r.t. k is inversely proportional to the mass
of the particle.
 In general, the effective
mass could be related to
1
1 d 2E
 2
*
m
 dk 2
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Chap. 3
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective mass, E-k diagram
 m* >0 near the bottoms of all band; m* <0 near the tops of all bands
 m* <0 means that, in response to an applied force, the electron will accelerate
in a direction opposite to that expected from purely classical consideration.
 In general, carriers are populated near the top or bottom band edge in a
semiconductor—the E-k relationship is typically parabolic and, therefore,
d 2E
 constant ...E near Eedge
dk 2
thus carriers with energies near the top or bottom of an energy band typically
exhibit a CONSTANT effective mass
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Chap. 3
20
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Dept. of E. E. NCU
Concept of Hole
Solid-State Electronics
Chap. 3
21
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Extrapolation of Concepts to 3-D
 Brilliouin Zones: is defined as a Wigner-Seitz cell in the reciprocal lattice.
  point: Zone center (k = 0)  (0 0 0 )
 X point: Zone-boundary along a <1 0 0 >
2
direction  a (1,0,0) 6 symmetric points
(1 0 0) (-1 0 0) (0 1 0) (0 -1 0) (0 0 1) (0 0 -1)
 L point: Zone-boundary along a <1 1 1>
direction  2 ( 1 , 1 , 1 ) 8 symmetric points
a 2 2 2
 , X, and L points are highly symmetric  energy stable states  carriers
accumulate near these points in the k-space.
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Chap. 3
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Dept. of E. E. NCU
E-k diagram of Si, Ge, GaAs
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Chap. 3
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Energy Band
 Valence Band:
– In all cases the valence-band maximum occurs at the zone center, at k = 0
– is actually composed of three subbands. Two are degenerate at k = 0,
while the third band maximizes at a slightly reduced energy.
The k = 0 degenerate band with the smaller curvature about k = 0 is called
“heavy-hole” band, and the k = 0 degenerate band with the larger
curvature is called “light-hole” band. The subband maximizing at a
slightly reduced energy is the “split-off” band.
– Near k = 0 the shape and the curvature of the subbands is essentially
orientation independent.
Solid-State Electronics
Chap. 3
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Energy Band
 Conduction band:
– is composed of a number of subbands. The various subbands exhibit
localized and abssolute minima at the zone center or along one of the
high-symmetry diirections.
– In Ge the conduction-band minimum occurs right at the zone boundary
along <111> direction. ( there are 8 equivalent conduction-band minima.)
– The Si conduction-band minimum occurs at k~0.9(2/a) from the zone
center along <100> direction. (6 equivalent conduction-band minima)
– GaAs has the conduction-band minimum at the zone center directly over
the valence-band maximum. Morever, the L-valley at the zone boundary
<111> direction lies only 0.29 eV above the conduction-band minimum.
Even under equilibrium, the L-valley contains a non-negligible electron
population at elevated temp. The intervalley transition should be taken
into account.
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Chap. 3
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Dept. of E. E. NCU
Metal, Semiconductor, and Insulator
Insulator
Solid-State Electronics
Chap. 3
Semiconductor
26
Metal
Instructor: Pei-Wen Li
Dept. of E. E. NCU
The k-space of Si and GaAs
 Direct bandgap: the valence band maximum and the conduction band
minimum both occur at k = 0. Therefore, the transition between the
two allowed bands can take place without change in crystal momentum.
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Chap. 3
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Dept. of E. E. NCU
Constant-Energy Surfaces
 A 3-D k-space plot of all the allowed k-values associated with a given
energy E. The geometrical shapes, being associated with a given
energy, are called constant-energy surfaces (CES).
 Consider the CES’s characterizing the conduction-band structures near
Ec in Ge, Si, and GaAs.
(a) Constant-energy surfaces
Solid-State Electronics
Chap. 3
(b) Ge surface at the
Brillouin-zone boundaries.
28
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Constant-Energy Surfaces of Ec
 For Ge, Ec occurs along each of the 8 equivalent <111> directions; a Si
conduction band minimum, along each of 6 equivalent <100>
directions. For GaAs, Ec is positioned at the zone center, giving rise to
a single constant-energy surface.
 For energy slightly removed from Ec:
E-Ec  Ak12+Bk22+Ck32,
where k1, k2, k3 are k-space coordinates measured from the center of a band
minimum along principle axes.
For example: Ge, the k1, k2, k3 coordinate system would be centered at the
[111] L-point and one of the coordinate axes, say k1-axis, would be directed
along the kx-ky-kz [111] direction.
 For GaAs, A = B = C, exhibits spherical CES;
 For Ge and Si, B=C, the CES’s are ellipsoids of revolution.
Solid-State Electronics
Chap. 3
29
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective Mass
 In 3-D crystals the electron acceleration arising from an applied force
is analogously by
dv
1
 * F
dt m
where
1
1
 mxx
mxy
mxz1 
 1

1
1
1
  m yx m yy m yz 
m *  1
1
1 
 mzx mzy mzz 
1 2 E
m  2
 kik j
1
ij
..i, j  x, y, z
 For GaAs, E  Ec  A(kx2  k y2  kz2 ) , so mij = 0 if ij, and mxx1  m yy1  mzz1 
2A
2
therefore, we can define mii=me*, that is the the effective mass tensor reduces
to a scalar, giving rise to an orientation-indep. equation of motion like that of a
classical particle.
2
2
2
2
 E  Ec 
Solid-State Electronics
Chap. 3
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2
e
2m
(k x  k y  k z )
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective Mass
 For Si and Ge:
so mij = 0 if ij, and
E-Ec = Ak12+B(k22+k32)
1
m xx

2A
2B
1
1
,
m

m

yy
zz
2
2
 Because m11 is associated with the k-space direction lying along the
axis of revolution, it is called the longitudinal effective mass ml*.
Similarly, m22 = m33, being associated with a direction perpendicular to
the axis of revolution, is called the transverse effective mass mt*.
2 2 2
 E  Ec 
k 
(k22  k32 )
2 1
2
2ml
2mt
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Chap. 3
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Effective Mass
 The relative sizes of ml* and mt* can be deduced by inspection of the Si
and Ge constant-energy plots.


lengthof theelliosoid


m 
along theaxis of revolution



m
max. widt h of theellipsoid


perpendicu
lar
t
o
the
axis
of
revolution


*
l
*
t
 For both Ge and Si, ml* > mt*. Further, ml*/mt* of Ge > ml*/mt* of Si.
 The valence-band structure of Si, Ge, and GaAs are approximately
spherical and composed of three subbands. Thus, the holes in a given
subband can be characterized by a single effective mass parameter, but
three effective mass (mhh*, mlh*, and mso*) are required to characterize
the entire hole population. The split-off band, being depressed in
energy, is only sparsely populated and is often ignored.
Solid-State Electronics
Chap. 3
32
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Dept. of E. E. NCU
Effective Mass measurement
 The near-extrema point band structure, multiplicity and orientation of
band minima, etc. were all originally confirmed by cyclotron
resonance measurement.
 Resonance experiment is performed in a microwave resonance
cavity at temperature 4K. A static B field and an rf E-field
oriented normal to B are applied across the sample. The carriers
in the sample will move in an orbit-like path about the direction
of B and the cyclotron frequency c = qB/mc. When the B-field
strength is adjusted such that c = the  of the rf E-field, the carriers
absorb energy from the E-field (in resonance). m= qB/c
 Repeating the different B-field orientations allows one to separate out the
effective mass factors (ml* and mt*)
Solid-State Electronics
Chap. 3
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Dept. of E. E. NCU
Effective Mass of Si, Ge, and GaAs
Solid-State Electronics
Chap. 3
34
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of State Function
 To calculate the electron and hole concentrations in a material, we
must determine the density of these allowed energy states as a function
of energy.
 Electrons are allowed to move relatively freely in the conduction band
of a semiconductor but are confined to the crystal.
 To simulate the density of allowed states, consider an appropriate
model: A free electron confined to a 3-D infinite potential well, where
the potential well represents the crystal.
 The potential of the well is defined as
V(x,y,z) = 0 for 0<x<a, 0<y<a, 0<z<a, and V(x,y,z) =  elsewhere
 Solving the Schrodinger’s equation, we can obtain
2
2m E
2
2
2
2
2
2
2  


2
Solid-State Electronics
Chap. 3
 k  k x  k y  k z  (nx  n y  nz ) 2 
a 
35
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of State Function

 The volume of a single quantum state is Vk =(/a)3, and the differential
volume in k-space is 4k2dk
 Therefore, we can determine the density of quantum states in k-space
2
2
as
 1  4k dk k dk 3
gT (k )dk  2 
 2 a
3
8

   
 
a
– The factor, 2, takes into account the two spin states allowed for each
quantum state; the next factor, 1/8, takes into account that we are
considering only the quantum states for positive values of kx, ky, and kz.
Solid-State Electronics
Chap. 3
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Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of State Function
 Recall that
k2 
2mE
2
 We can determine the density of states as a function of energy E by
gT ( E )dE 
3
4
2

(
2
m
)
 E  dE  a 3
3
h
 Therefore, the density of states per unit volume is given by
gT ( E )dE 
3
4
2

(
2
m
)
 E  dE
3
h
 Extension to semiconductors, the density of states in conduction band
3
is modified as
4
* 2
gc (E) 
h
3
 (2mn )  E  Ec
and the density of states in valence band is modified as
gv (E) 
Solid-State Electronics
Chap. 3
3
4
*
2

(
2
m
)
 E  Ev
p
3
h
37
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of State Function
 mn* and mp* are the electron and hole density of states effective masses.
In general, the effective mass used in the density of states expression
must be an average of the band-structure effective masses.
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Chap. 3
38
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of States Effective Mass
 Conduction Band--GaAs: the GaAs conduction band structure is
approximately spherical and the electronss within the band are
characterized by a single isotropic effective mass, me*,  mn*  me*...GaAs
 Conduction Band--Si, Ge: the conduction band structure in Si and Ge
is characterized by ellipsoidal energy surfaces centered, respectively, at
points along the <100> and <111> directions in k-space.
mn*  6 3 ml*mt*2  3 ...Si
1
2
m 4
*
n
2
3
m m 
*
l
1
*2
t
3
...Ge
 Valence Band--Si, Ge, GaAs: the valence band structures are al
characterized by approximately spherical constant-energy surfaces
(degenerate).
*
*
*

m p  mhh   mlh 
Solid-State Electronics
Chap. 3
3
39
2
3

2
2
3
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Density of States Effective Mass
Solid-State Electronics
Chap. 3
40
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Statistics Mechanics
 In dealing with large numbers of particles, we are interested only in the
statistical behavior of the whole group rather than in the behavior of
each individual particle.
 There are three distribution laws determining the distribution of
particles among available energy states.
 Maxwell-Boltzmann probability function:
– Particles are considered to be distinguishable by being numbered for 1 to
N with no limit to the number of particles allowed in each energy state.
 Bose-Einstein probability function:
– Particles are considered to be indistinguishable and there is no limit to the
number of particles permitted in each quantum state. (e.g., photons)
 Fermi-Dirac probability function:
– Particles are indistinguishable but only one particle is permitted in each
quantum state. (e.g., electrons in a crystal)
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Chap. 3
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Dept. of E. E. NCU
Fermi-Dirac Distribution
 Fermi-Dirac distribution function gives the probability that a quantum
state at the energy E will be occupied by an electron.
f (E) 
1
E E F
1  exp(
)
kT
 the Fermi energy (EF) determine the statistical distribution of electrons
and does not have to correspond to an allowed energy level.
 At T = 0K, f(E < EF) = 1 and f(E >EF ) = 0, electrons are in the lowest
possible energy states so that all states below EF are filled and all states
above EF are empty.
Solid-State Electronics
Chap. 3
42
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Fermi-Dirac Distribution, at T=0K
Solid-State Electronics
Chap. 3
43
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Fermi-Dirac Distribution
 For T > 0K, electrons gain a certain amount of thermal energy so that
some electrons can jump to higher energy levels, which means that the
distribution of electrons among the available energy states will change.
 For T > 0K, f(E = EF) = ½
Solid-State Electronics
Chap. 3
44
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Boltamann Approximation
 Consider T >> 0K, the Fermi-Dirac function could be approximated by
f (E) 
1
  ( E  EF ) 
 exp 

E E F
kT

1  exp(
)
kT
which is known as the Maxwell-Boltzmann approximation.
Solid-State Electronics
Chap. 3
45
Instructor: Pei-Wen Li
Dept. of E. E. NCU
Homework
 3.5
 3.8
 3.16
Solid-State Electronics
Chap. 3
46
Instructor: Pei-Wen Li
Dept. of E. E. NCU