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November 9, 2010
Class 18
Equation of Motion for Electrons
By definition v g 
Thus
vg 
d
and =ε/ħ
dk
1 d
1
or in more than 1-D v g   k  k 
 dk

The effect of the crystal on the electrons is contained in the dispersion relation (k)= ħ-1 ε(k)
Let’s consider now an electron being acted upon by a force F, the work done on the electron by
the electric field E is
d  Fvg dt On the other hand d 
d
dk  vg dk
dk
Combining the two equation
dk

 F Which also applies in more than one dimension where k and F are vectors. This
dt
relationship can be readily obtained by using the linear momentum is p= ħk
dk
  eE
For an electric field: 
dt
dk
e
 e v g  B    k   B
For a magnetic field 
dt

Effective mass
Assume we insist in considering these electrons as classical free
electrons. The effective velocity of electrons in a box is the group
d 1 d
F

velocity vg 
. Classically m 
dk  dk
a
a
dvg
dt

dvg dk 1 d 2 dk 

dk dt  dk 2 dt  F
2
   m*  d 2
a
dk 2

dk
F 

dt

m*=m near the bottom of the band where ε(k) is approximately a parabola and thus its second
derivative is a constant, but the value of the second derivative decreases and become negative
near the top of the band.
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Physically, negative mass only means that when an impulse is applied to the particle, this particle
responds decreasing its momentum rather than increasing it, this can be understood considering
that the closer the electron wave length is to the characteristic distance in the lattice, the larger
the lattice-electron interaction is. When an impulse is given to an electron, the results is an
increasing k that is now closer to that of the lattice, thus more momentum than before is
transferred to phonons and the only way to explain such a behavior from a classical particle is to
assume their mass is negative (not too classical after all!)
Holes
If we consider an almost full band, then we have a few electrons missing at the top, those
missing electrons would have a negative effective mass. Missing particles with negative mass
and negative charge are equivalent to particles with positive mass and positive charge, thus
instead of solving the problem for (2N-x) electrons we solve the problem for x holes.
When an electron is removed from an otherwise full band, 5 things happen:
1. kh=–ke
A full band (due to symmetry), has a total momentum of zero (there are as many electrons
with +k as there are with –k), so by removing 1 electron with (let’s say) momentum –ke, the
entire band is left with a total momentum +ke, thus within the model where a missing
electron is associated with a hole kh=–ke,
2. Considering the top of the band as the origin for energy, εh(kh)= -εe(ke).
An electron in a band is bound and energy needs to be added to remove the electron from the
band into a zero energy state, the deeper in the band, the more energy is needed to remove it.
That energy will be recovered if the electron is allowed back in the band. Thinking in terms
of holes, more work is needed to produce a hole deeper in the band, thus the energy must be
positive. Also, allowing an electron back in the band, is equivalent to remove the hole from
the band to an state of zero energy, in this process, energy is released, thus the hole must
have had positive energy. (See figure 8, chapter 8, page 196)
In other words, if the band is symmetric εe(ke)= εe(-ke)=- εh(-ke)= - εh(kh) or εh(kh)= -εe(ke)
3. v(kh)= v(ke)
Assuming the band is symmetric, we have
εe(ke)=-εe(-ke)= (-εe(-ke))= (-εe(kh))= εh(kh) Thus v(kh)= v(ke). Where we have
used that v=ħ-1kε(k)
4. mh=-me
An electron near the top of the band has a negative effective mass, thus the effective mass of
the band increases by removing that electron (which simply means that there is one less
electron given momentum away to the lattice. Another way to see this is to notice that the
effective mass of an electron is inversely proportional to the (2ε/k2), since the energy is
inverted, thus (2εe/k2)= (2(-εh)/k2)= -(2εh/k2),
2
dp h
 eE
dt
Thus the equation of motion for a hole is that of a positive charge particle. This comes from
dp e
dk
d  k h 
dk
  e  eE  
 eE   h  eE
dt
dt
dt
dt
This is true also if a magnetic field is present
5.
dp h
dk
  h  eE  v h  B  since vh=ve
dt
dt
From all of the above, a missing electron can be regarded as a particle (hole) which is positive, it
has a positive effective mass equal in magnitude to that of the missing electron (which was
negative) and moving with the same speed but in opposite direction as the missing electron.
Metals and Insulators
The distance between points in the k space is 2/L. The length of the entire band is 2N/L=2/a,
thus we have (2/a)/(2/L)=L/a=N states in the first Brillouin zone (one per atom). The same is
observed at k=2/a, 3/a, etc. Each primitive cell contribute exactly one independent value of k
to each energy band, so each band can accommodate up to 2N electrons, this is the same we
found for FE within the first Brillouin zone, the difference is now that electrons in the highest
energy levels at k=/a cannot increase their energy continuously.
If the band is full (case observed when each electrons contribute with an even number of
electrons) and the gap is large (~5eV for instance) for reasonable electric field values, electrons
cannot increase their energy and thus there is no effect on the electrons. The material is an
insulator.
If instead for a bivalent materials, two bands overlap, the result may be two partly filled bands
(one almost full and the other almost empty) what gives a metal for large overlap and a semimetal for small overlap.
For atom with an odd number of electrons, like alkali metals (1 electrons) they have to be metals
since the band will always be partly full (half full if only 1 band is involved)
The actual band structure is quite complicated due different reasons, for instance the different
directions an electron can move in a crystal (determining the Bragg condition at different values
of k) and the coexistence of several valence bands, for instance, tetravalent material have 4
valence bands (show figure 14, chapter 8, page 203). Since the equivalent mass is related to the
slope of the band at ka=, the difference curvatures of each of the bands determine electrons of
different effective masses leading to holes of different masses. It is common to talk about heavy
holes, or light holes.
Semiconductors
At zero Kelvin, a perfect crystal of most semiconductors can be regarded as insulators, arbitrarily
defined as those with resistivity above 1014 -cm.
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To fix ideas, the band gap is defined as the energy gap between two bands taken between the
lowest point of the upper band and the highest point of the lower band, particularly important
(and what band gap most often refers to), is the gap between the valence band and the conduction
band. This gap is what defines the conductive properties of any material. The lowest point in
the conduction band is called the conduction band edge, while the highest point in the valence
band is known as the valence band edge. Semiconductors are characterized by intermediate band
gaps (roughly less than 3eV) between those of a metal (no band gap) and insulators. At room
temperature, semiconductors have resistivity in the range of 10-2 and 109 -cm.
Conductivity in semiconductors arises in two ways, by thermal or optical pumping of electrons
into the conduction band or by the addition of impurities, the former is known as intrinsic
conduction while the later is known as extrinsic conduction. Intrinsic conduction arises when
electrons are promoted from the valence to the conduction band, leading to an almost empty
conduction band, where electrons can move as free electrons and an almost full valence band
where holes are available to move. Extrinsic conduction arises when impurities are added that
act as donor (acceptors) near the conduction (valence) band.
Semiconductors are often classified by the number of electrons they have in their valence band
as group II, III, IV, V, and IV. Each group corresponds to a column in the periodic table. A
good number of most common semiconductors are made as a combination of two different atoms,
considering a semiconductors of chemical formula AB, compounds with A trivalent and B
pentavalent are known as III-V semiconductors, the II-VI are compounds made of a the divalent
and a hexavalent atom, and IV-IV are made only of tetravalent elements.
Semiconductor Type
III-V (trivalent-pentavalent)
Al, Ga, In
N, As, P
II-VI (divalent-hexavalent)
Zn, Cd, Hg
S, Se, Te, O
IV-IV (Diamond Type)
C, Si, Ge, Sn
C, Si, Ge, Sn
Intrinsic conduction
In semiconductors, a large change in conductivity with temperature is observed, however unlike
metals for which conductivity decreases with increasing temperature due to an increase
interaction between electrons and the lattice (ions vibrate with higher amplitudes),
semiconductors increase their conductivity when temperature increases, the reason is because a
larger number of electrons is promoted from the valence band to the conduction band, thus more
negative carriers (electrons) in the conduction band and more positive carriers (holes) in the
valence band are available. Notice in figure 3, chapter 8, page 188, conductivity increases
between 3 and 4 orders of magnitude when the temperature increases ~100K. Also notice that at
a given temperature, Ge has larger conductivity than Si due to the smaller band gap (0.66 vs 1.11
at room temperature).
Direct gap
When the valence and conduction band edges correspond to states with the same k, the gap is
known as direct gap, excitation produced by photons consist of a simple excitation that takes the
electron from the valence band and place it in the conduction band with the same k this electron
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had in the valence band. Notice that any incoming photon will have a wavevector a lots smaller
than the electron at the top of the band (the energy is related to k and the energy of the photon is
equal to that of the gap which is a lot smaller than the band width. So energy is conserved since
the energy of the electron in the upper band differs from that on the lower band by an energy
equal to that of the photon. Momentum (wavevector) remains unchanged for the electron while
the absorbed photon has a very small k.
Indirect gap
When the electrons at the top of the valence band have a wavenumber that is different from that
they would have at the bottom of the conduction band, direct transition is not possible since the
photon cannot account for the large change in momentum. This transition is only possible if
mediated by a phonon. The reaction includes the absorption of a photon and a creation of three
particles, an electron-hole pair and a phonon. In a case like this, the energy of the incoming
photon will have to be equal to the energy of the gap plus the energy of the created phonon
(which is usually small compared to the gap 0.01eV-0.03eV).
  Eg   Where  si the phonon frequency.
The phonon wavevector K is such that kc+K~0 where kc is the difference in wavevectors
between the bottom of the conduction and the top of the valence bands. See figure 4 and 5,
chapter 8, page 189.
Intrinsic Carrier Concentration
An intrinsic semiconductor is a pure semiconductor where conduction depends on electrons from
the valence band promoted to the conduction band thus leaving holes in the valence band and
making electrons available in the conduction band. These holes in the valence band and
electrons in the conduction band are known as carrier because electricity can be carried by
making them move in an electric field. The holes are positive carriers and the electrons are
negative carriers.
The idea of carrier concentration is fundamental to semiconductor physics. At zero Kelvin, most
semiconductors behave as insulators, with full valence band (s) and empty conduction bands. As
temperature increases, some electrons are pumped to the conduction band leaving holes behind
and thus creating carriers.
In semiconductor Physics, the chemical potential  is called the Fermi level (not the Fermi
energy) and we will assume that for electron in the conduction band, (ε-)>>kBT what simply
means that the distance between the Fermi level and the conduction band edges is large
compared to the thermal energy. This is for the most part an appropriate approximation since the
band gap is from tenth of eV to a couple of eV when thermal energy at room temperature is 26
meV.
Within this approximation,
fe  e
at T.

   
k BT
which is the probability that a given energy level in the conduction band is occupied
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The energy of electrons in the conduction band has the form
2k 2
Where Ec is the energy of the conduction band edge and me is the electron
 k  Ec 
2me
effective mass.
The density of states is given by
3
1
1  2m  2
De    2  2 e    Ec  2 Where the energy has being referred to the band edge since
2   
when we defined the density of states we assumed the bottom of the band had an energy of zero.
The concentration of electrons in the conduction band is then
3




1
1  2m  2
n   De   f e  d  2  2 e  e k B T    Ec  2 e k B T d
Ec
Ec
2   


 
By integrating  using that  xe  x dx 
0
2 


3
m k T  2
n  2 e B2  e
 2 
  Ec
k BT
We can do the same for holes. The concentration of holes is given by
   
1
f h  1  fe  1 

   
e
1
k BT
e
k BT
e
   
e
k BT
   
k BT
1
The density of holes is given by
1  2m 
Dh    2  2 h 
2   
3
2
Ev   12
Where Ev is the energy of the valence band edge
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Ev  
m k T  2
p   Dh   f h  d  2 h B2  e kBT
Ec
 2 

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