Transcript Vibrational Properties of the Lattice

VI. Electrons in a Periodic Potential
A. Energy Bands and Energy Gaps in a Periodic Potential
B. Metals, Insulators, and Semiconductors
C. Energy Bands and Fermi Surfaces in 2-D and 3-D Systems
D. Bloch’s Theorem
E. Using Bloch’s Theorem: The Kronig-Penney Model
F. Empty Lattice Bands and Simple Metals
G. Density of States for a Periodic Potential
H. E(k) and N(E) for d-electron Metals
I.
Dynamics of Bloch Electrons in a Periodic Potential
J.
Effective Mass of Electrons
K. Electrons and Holes
A. Energy Bands and Energy Gaps in a Periodic Potential
Recall the electrostatic potential energy in a crystalline solid along a line passing
through a line of atoms:
bare ions
solid
Along a line parallel to this but running between atoms, the divergences of the periodic
potential energy are softened:
U
x
A simple 1–D mathematical model that
 2x 
U
(
x
)

U

U
cos


0
1
captures the periodicity of such a potential is:
 a 
U 0  U1  0
Electron Wavefunctions in a Periodic Potential
Consider the following cases:
U1  0 Wavefunctions are plane waves
and energy bands are parabolic:
U1  0
k  a
2k 2
E
2m
i ( kxt )
  Ae
Electrons wavelengths much larger than a, so wavefunctions and
energy bands are nearly the same as above
U1  0 Electrons wavelengths approach a, so waves begin to be strongly
back-scattered :
k  a
B A
   Aei ( kxt )  Bei ( kxt )
U1  0 Electrons waves are strongly back-scattered (Bragg scattering) so
standing waves are formed:
k  a
   C ei ( kxt )  ei ( kxt )  12 A eikx  eikx eit




Electron Energies in a Periodic Potential
There are two such standing waves possible:
 
1
2
 
1
2

Ae

e
A eikx  eikx eit 
ikx
 eikx
it

1
2
1
2
2 A cos(kx)eit
 *   2 A2 cos2 ( ax )
2iA sin(kx)eit
 *   2 A2 sin 2 ( ax )

These two approximate solutions to the S. E. at k  a have very different
potential energies.  *  has its peaks at x = a, 2a, 3a, …at the positions of
the atoms, where U is at its minimum (low energy wavefunction). The other
solution,  *  has its peaks at x = a/2, 3a/2, 5a/2,… at positions in between
atoms, where U is at its maximum (high energy wavefunction).
We can do an approximate calculation of the energy difference between these
two states as follows. Letting U0 = 0 for simplicity, and remembering U1 < 0:
a


a


E  E   U ( x)  *   *  dx  2 A2  U1 cos(2ax ) cos2 ( ax )  sin 2 ( ax ) dx
x 0
x 0
A
1
a
cos 2ax 
Origin of the Energy Gap
a
E  E   a2  U1 cos2 ( 2ax )dx
x 0
Now use the
identity:
cos2 x  12 1  cos2x
a
E  E  
U1
a
U
4x
a
4x a




1

cos(
)
dx


x

sin(
a
a
4
a ) 0

1
 U1  Eg
“energy gap”
x 0
E
In between the two energies
there are no allowed energies;
i.e., an energy gap exists. We
can sketch these 1-D results
schematically:
The periodic potential U(x)
splits the free-electron E(k) into
“energy bands” separated by
gaps at each BZ boundary.
EEg
E+
 a
kx

a
Different Representations of E(k)
If we apply periodic boundary
conditions to the 1-D crystal, the
energy bands are invariant under a
reciprocal lattice translation vector:
 

E(k  G)  E(k )
 2n
G  a iˆ
The bands can be graphically
displayed in either the (i) extended
zone scheme; (ii) periodic zone
scheme; or (iii) reduced zone
scheme.
(i)

G1
extended zone scheme: plot E(k) from k = 0 through all possible BZs (bold curve)
(ii) periodic zone scheme: redraw E(k) in each zone and superimpose
(iii) reduced zone scheme: all states with |k| > /a are translated back into 1st BZ
B. Metals, Insulators, and Semiconductors
It is easy to show that the number of k
values in each BZ is just N, the
number of primitive unit cells in the
sample. Thus, each band can be
occupied by 2N electrons due to their
spin degeneracy.
E
A monovalent element with one atom
per primitive cell has only 1 valence
electron per primitive cell and thus N
electrons in the lowest energy band.
This band will only be half-filled.
The Fermi energy is the energy
dividing the occupied and unoccupied
states, as shown for a monovalent
element.
EEg
E+
kx
EF
 a

a
Metals, Insulators, and Semiconductors
For reasons that will be explained more fully later:
• Metals are solids with incompletely filled energy bands
• Semiconductors and insulators have a completely filled or empty bands and an energy
gap separating the highest filled and lowest unfilled band. Semiconductors have a small
energy gap (Eg < 2.0 eV).
Quick quiz: Does this mean a divalent
element will always be an insulator?
Answer: In 1-D, yes, but not
necessarily in 2-D or 3-D! Bands
along different directions in k-space
can overlap, so that electrons can
partially occupy both of the
overlapping bands and thus form a
metal.
But it is true that only crystals with an
even number of valence electrons in a
primitive cell can be insulators.
C. Energy Bands and Fermi Surfaces in 2-D and 3-D Lattices
Let’s analyze a simple square lattice
of atoms with interatomic distance a.
Its reciprocal lattice will also be
square, with reciprocal lattice base
vector of length 2/a.
The Brillouin zones of the reciprocal
lattice can be identified with a simple
construction:
Now consider what happens if we
have a weak periodic potential and
have atoms with 1, 2, and 3 valence
electrons per atom. A truly free
electron system would have a Fermi
circle to define the locus of states at
the Fermi energy.
Let’s see how the shape of the Fermi
circle is distorted by the periodic
petential.
3
3
2
3
3
2
2
1
3
3
2
3
3
2 /a
2 /a
2-D Energy Bands and Fermi Circles
This figure shows the
Fermi circles
corresponding to 2-D
crystals with one, two
and 3 valence electrons
per atom.
Note how the divalent
crystal has a Fermi
circle that cuts across
the 1st BZ boundary
(in red).
The different Fermi
circles correspond to
systems with
successively higher EF
values, as shown in this
2D band diagram.
EF3
EF2
EF1
Shape of 2-D Energy Bands and Fermi “Circles”
This 2D band diagram shows how a
weak periodic potential introduces
gaps in the free-electron bands at
the BZ boundary. Here, bands are
shown along the [10] and [11]
directions in k-space.
EF2
Thus, there is a similar
discontinuity introduced into a
free-electron Fermi circle any time
it crosses a BZ boundary. This is
illustrated for a 2-D divalent
crystal:
Note that the Fermi circle does not completely fill the
1st BZ, so some electrons are in the 2nd BZ.
Schematic Shape of a 3-D Fermi Surface
In 3D crystals the periodic potential
distorts the shape of a Fermi sphere
in the vicinity of the BZ boundary.
A schematic example for a simple
cubic lattice and a crude model E(k)
function is shown here:
Note that the Fermi circle does not
completely fill the 1st BZ but makes
contact with the 1st BZ boundary
along the [100] directions.
Shape of 3-D Energy Bands in a Real Metal
In 3D the energy bands are plotted
along the major symmetry
directions in the 1st BZ. Many of
the high symmetry points on the 1st
BZ boundary are labeled by letters.
Free-electron
bands in an fcc
crystal
The gamma point (  ) is always the
zone center, where k = 0.
Electron
bands in Al
Shape of 3-D Fermi Surface in a Real Metal
Even if the free-electron Fermi sphere does not intersect a BZ boundary, its shape can
still be affected at points close to the boundary where the energy bands begin to
deviate from the free-electron parabolic shape. This is the case with Cu.
Just a slightly perturbed free-electron sphere!
D. Bloch’s Theorem and Bloch Wavefunctions
This theorem is one of the most important formal results in all of solid state physics
because it tells us the mathematical form of an electron wavefunction in the presence of
a periodic potential energy. We will prove the 1-D version, which is known as Floquet’s
theorem.
What exactly did Felix Bloch prove in 1928? In the
independent-electron approximation, the timeindependent Schrodinger equation for an electron in
a periodic potential is:
where the potential energy
is invariant under a lattice

translation vector T :
Bloch showed that the
solutions to the SE are the
product of a plane wave
and a function with the
periodicity of the lattice:
 

U (r  T )  U (r )



ik r
 k (r )  uk (r )e
“Bloch functions”
 2 2




U
(
r
)  E

 2m

and
where

 

T  ua  vb  wc



uk (r  T )  uk (r )
Heisenberg and Bloch
Who says physicists don’t have any fun?
Proof of Bloch’s Theorem in 1-D
1. First notice that Bloch’s theorem implies:
  
 

 ikr ikT




ik r ik T
ik T
 k (r  T )  uk (r  T )e e
 uk (r )e e
  k (r )e
 



It is easy to show that this equation formally
ik T
Or just:  k (r  T )   k (r )e
implies Bloch’s theorem, so if we can prove it
we will have proven Bloch’s theorem.
2. Prove the statement shown above in 1-D:
Consider N identical lattice points around a circular ring,
each separated by a distance a. Our task is to prove:
 ( x  a)   ( x)eika
1
2
N
Built into the ring model is the
periodic boundary condition:
 ( x  Na)   ( x)
The symmetry of the ring
implies that we can find a
solution to the wave equation:
 ( x  a)  C ( x)
3
Proof of Bloch’s Theorem in 1-D: Conclusion
If we apply this translation N times we will
return to the initial atom position:
And has the most
This requires C N  1
general solution:
Or:
C e
2ni / N
e
ika
 ( x  Na)  C N ( x)   ( x)
C N  e2ni n  0,  1,  2,...
Where we define the Bloch wavevector: k 
Now that we know C we can rewrite
It is not hard to generalize this to 3-D:
 ( x  a)  C ( x)  eika ( x) Q.E.D.
 



ik T
 k (r  T )   k (r )e



ik r
 k (r )  uk (r )e
But what do these functions look like?
2n
Na
Bloch Wavefunctions
This result gives evidence to support the nearly-free electron approximation, in which the
periodic potential is assumed to have a very small effect on the plane-wave character of a
free electron wavefunction. It also explains why the free-electron gas model is so
successful for the simple metals!
A Remarkable Consequence of Bloch’s Theorem:
In the case of lattice vibrations in a crystal, we can calculate the propagation speed
(group velocity) of a wave pulse from a knowledge of the dispersion relation:
group velocity: (1-D)
d
vg 
dk
(3-D)
   
vg (k )  k (k )
Similarly, it can be shown using Bloch’s theorem that the propagation speed of an
electron wavepacket in a periodic crystal can be calculated from a knowledge of the
energy band along that direction in reciprocal space:
d
1 dE
electron velocity: (1-D) v g 

dk
 dk
  1  
(3-D) v g (k )   k E (k )

This means that an electron (with a specified wavevector) moves through a perfect
periodic lattice with a constant velocity; i.e., it moves without being scattered or in any
way having its velocity affected!
But wait…does Bloch’s theorem prove too much? If this result is true, then what is the
origin of electrical resistivity in a crystal?
E. Using Bloch’s Theorem: The Krönig-Penney Model
Bloch’s theorem, along with the use of periodic boundary conditions, allows us to
calculate (in principle) the energy bands of electrons in a crystal if we know the
potential energy function experienced by the electron. This was first done for a simple
finite square well potential model by Krönig and Penney in 1931:
U
U0
x
-b 0
a a+b
2a+b 2(a+b)
We can solve the SE in each region of space:
ix
ix
0<x<a
 I ( x)  Ae  Be
-b < x < 0
 II ( x)  Ce  De
Qx
Qx
Each atom is represented by a finite
square well of width a and depth
U0. The atomic spacing is a+b.
  2 d 2



U
(
x
)

  E
2
2
m
dx


 2 2
E
2m
 2Q 2
U0  E 
2m
Now do you
remember how
to proceed?
Boundary Conditions and Bloch’s Theorem
The solutions of the SE require that the wavefunction
and its derivative be continuous across the potential
boundaries. Thus, at the two boundaries (which are
infinitely repeated):
x=0
x=a
 I ( x)  Aeix  Beix
 II ( x)  CeQx  DeQx
i ( A  B)  Q(C  D) (2)
A  B  C  D (1)
Aeia  Beia   II (a)
Now using Bloch’s theorem for a
periodic potential with period a+b:
 II (a)   II (b)eik ( ab)
k = Bloch wavevector
Now we can write the boundary conditions at x = a:
Aeia  Beia  (CeQb  DeQb )eik ( ab)
(3)
i ( Aeia  Beia )  Q(CeQb  DeQb )eik ( ab)
(4)
The four simultaneous
equations (1-4) can be written
compactly in matrix form
Results of the Krönig-Penney Model
 1
 i

 eia
 ia
ie
1
 i
e ia
 ie ia
1
Q
 e Qb eik ( a b )
 QeQb eik ( a b )
1


Q

 eQb eik ( a b ) 

QeQb eik ( a b ) 
 A
B
 0
C 
 
 D
Since the values of a and b are inputs to the model, and Q depends on U0 and
the energy E, we can solve this system of equations to find the energy E at any
specified value of the Bloch wavevector k. What is the easiest way to do this?
Taking the determinant and setting it equal to zero gives:
 Q2   2 

 sin a sinhQb  cosa  coshQb  cosk (a  b)
 2Q 
Numerical Solution of the Krönig-Penney Model
Using the values of a, b, and U0 we can construct the energy bands as follows:
1. Choose a value of k
2. Solve for the allowed energies E1(k), E2(k), E3(k) … numerically
3. Choose a new value of k
4. Repeat step 2
2
1
2m
Using “atomic units” where
, length is measured in Bohr radii (a0 = 0.529Å)
and energy has units of Rydbergs (1 Ry = 13.6 eV), I chose arbitrary values of:
a = 5.0 a0
b = 0.5 a0
U0 = 5.0 Ry
and obtained the following set of energy bands:
Numerical Results of the Krönig-Penney Model
The 1st BZ has a
maximum k value
given by:
kmax 
You can see the same qualitative 1-D band structure we
deduced from the approximate band gap calculation earlier!

ab


5.5a0
 0.57
rad
a0
F. Empty Lattice Bands and Simple Metals
We expect that for a very weak periodic potential the energy bands of a solid should
look like those of a free-electron gas system. The limit of a vanishing potential is called
the “empty lattice”, and the empty-lattice bands are often plotted for comparison with
the energy bands of real solids. But how are the empty lattice bands calculated? It is
desirable to plot them in the reduced zone scheme, so we need to represent all of the
translations of the fully periodic E(k) back into the 1st BZ.
The symmetry of the reciprocal lattice
requires:
So to represent all of the periodic
bands in the reduced zone scheme:

 
E(k )  E(k1  G)
Where k1 is a wavevector
lying in the 1st BZ.

 
2   2
E (k )  E (k1  G) 
k1  G
2m
The  sign is redundant, and Myers chooses to use only the minus sign. In principle
all reciprocal lattice vectors G will be represented, but electrons can only propagate
when the G vector satisfies the Bragg condition (i.e.,only those G that give nonzero
structure factors).
Empty Lattice Bands for bcc Lattice
Myers illustrates this procedure for the bcc lattice by plotting the empty lattice bands
along the [100] direction in reciprocal space. (For HW you will do this for the fcc lattice
and the [111] direction)
The general reciprocal lattice translation vector:
In our analysis of the structure factor, we
used a simple cubic lattice, for which the
reciprocal lattice is also simple cubic:


 
Ghkl  hA  kB  lC
 2
 2
 2
ˆ
ˆ
A
x B
y C
zˆ
a
a
a
And thus the general reciprocal
lattice translation vector is:

2
2
2
Ghkl 
h xˆ 
k yˆ 
l zˆ
a
a
a
We write the reciprocal lattice
vectors that lie in the 1st BZ as:
 2
2
2
k1 
x xˆ 
y yˆ 
z zˆ
a
a
a
The maximum value(s) of x, y, and z depend on the reciprocal lattice type and the
direction within the 1st BZ, as we can see…..
k Values in 1st BZ for bcc Lattice
 2
2
2
k1 
x xˆ 
y yˆ 
z zˆ
a
a
a
We write the reciprocal lattice
vectors that lie in the 1st BZ as:
The maximum value(s) of x, y, and z depend on
the reciprocal lattice type and the direction
within the 1st BZ. For example:

Remember that the reciprocal
lattice for a bcc direct lattice
is fcc! Here is a top view,
from the + kz direction:
2
a
H
kx
[100]
0<x<1
[110]
0<x<½
0<y<½
2
a
H
N
ky
Empty Lattice Bands for bcc Lattice
Thus the empty lattice energy bands are given by:

2

2   2 2  2 
2
2
2
E (k ) 
k1  Ghkl 
  x  h    y  k   z  l 
2m
2m  a 

We can enumerate the lowest few bands for the y = z = 0 case, using
only G vectors that have nonzero structure factors (h + k + l = even):
{G} = {000}
 2  2  2
2
E
  x  E0 x
2m  a 
{G} = {110}
(110) (1 1 0) (101) (101)
2
 
 
(1 1 0) (1 10) (1 0 1) (1 01)
(011) (011) (0 1 1) (0 1 1)
{G} = {200}
(200)
( 200)
(020) (020) (002) (002)

 

E  E x 1 1   E x 1 1
E  E x 1 1   E x  2
E  E x  2 
E  E x  2 
E  E x  2   E x  4
E  E0 x 1 12  E0 x 1 1
2
2
2
2
2
0
0
2
2
2
2
0
0
2
0
2
0
2
0
2
2
0
Empty Lattice Bands for bcc Lattice: Results


2
E(k )  E0 x  h  k 2  l 2
Thus the lowest energy empty lattice
energy bands for the bcc lattice are:

6
5
4
Series1
E/E0
Series2
Series3
3
Series4
Series6
Series7
Series8
2
1
0
0
0.2
0.4
0.6
0.8
1
x
These are identical to the bands on p. 196 of Myers, except one higher
lying band is missing from my plot.
G. Density of States for a Periodic Potential
A weak periodic potential only perturbs the freeelectron energy bands in the vicinity of the Brillouin
zone boundaries, so the density of states N(E) closely
resembles the ideal free-electron form:
2k 2
E (k ) 
2m
N ( E)  CE1/ 2
However, it is clear that the perturbation
of free-electron bands near the BZ
boundaries will introduce sharp features
in the N(E) where the gradient of E(k)
goes to zero. Thus we can
schematically plot N(E) for the simple
metals:
N ( E) 
for a 3-D system
2V
dS

8 3  k E
H. Energy Bands and N(E) for Transition Metals
One new feature arises in the transition
metals—the appearance of a series of
partially filled d-electron bands. The sbands overlap the d-bands in the first row
transition metals, leading to band
hybridization, as we can see in the energy
bands of the transition metals:
For elements with the same type of
crystal lattice, an increasing Z is
accompanied by an increase in the
Fermi energy, populating more and
more of the d-electron states in the
lattice.
(Note the series V  Cr  Fe and also
the series Co  Ni  Cu)
N(E) for Transition Metals
The d-bands are much narrower than s- and p-bands due to the greater localization of the
d-electrons. As a result, the transition metals with partially filled d-bands have a much
higher N(EF) than the simple metals. This means that these transition metals:
• are often very chemically reactive (especially with O)
• make good catalysts (supply electrons to molecules that stick to metal surface)
• have very large heat capacities and magnetic susceptibilities
• often display superconductivity
I. Dynamics of Bloch Electrons in an E Field
Earlier we noted that metals have partially-filled upper bands, while semiconductors and
insulators have completely filled upper bands. What is the qualitative difference between
filled and partially-filled bands?


For a single electron: j  nev
For a collection of electrons:

 
J  ne
v (k )

k
 1  
For each electron: v   k E ( k )

But the symmetry of the
energy bands requires:


E ( k )  E ( k )

 
Thus we conclude: v (k )  v(k )
So for a filled band, which has 
 
J  ne v (k )  0
an equal number of electrons
1stBZ
with k positive and negative,

Filled energy bands carry no
current! We will see that this is
true even when an electric field
is applied.
Note: the electrons in filled bands are not stationary…there are just the same number
moving in each direction, so the net current is zero.
Dynamics of Bloch Electrons in an E Field
The same argument that we made for filled bands also applies to a partially filled band in
the absence of an electric field. However, when an external electric field is applied to a
periodic solid, the electrons all experience a force F that causes a change in their k values:

 
The rate at which each electron absorbs
dW F  dr  
dE(k )

 F  vg 
energy from the field is the absorbed power:
dt
dt
dt


 dE(k ) 
 dk
 1
Substituting for the group velocity and


F   k E (k ) 
  k E (k ) 
using the chain rule, we have:

dt
dt


dk
By comparing both sides, we have:
The acceleration theorem
F 
dt
 dp c


F
We could go on to write:
where pc  k “crystal momentum”
dt
But remember…pc is not the electron’s momentum, because F is only the external force and
does not include the force of the lattice on the moving electron.
The Bottom Line
Now we see that the external electric field causes a change in the k vectors of all electrons:



dk
F 
 eE
dt
E
 a
v


dk  eE

dt

If the electrons are in a partially filled band, this
will break the symmetry of electron states in the
1st BZ and produce a net current. But if they are in
a filled band, even though all electrons change k
vectors, the symmetry remains, so J = 0.

a
kx
kx
When an electron reaches the 1st BZ edge (at k =
/a) it immediately reappears at the opposite edge
(k = -/a) and continues to increase its k value.
As an electron’s k value increases, its velocity
increases, then decreases to zero and then becomes
negative when it re-emerges at k = -/a!!
Thus, an AC current is predicted to result from a
DC field! (Bloch oscillations)
Do We Ever Observe This?
Not until fairly
recently, due to the
effect of collisions
on electrons in a
periodic but
vibrating lattice.
However…
In both experiments the periodic potential was fabricated in an
artificial way to minimize the effect of collisions and make it
possible to observe the Bloch oscillations of electrons (or
atoms!).
But in Reality…
Bloch oscillations are not routinely observed because the electrons in a periodic system
undergo collisions with ions in the lattice much too frequently. In practice this occurs on
the time scale of the collision time  (10-14 s). Let’s analyze the effect of an external
electric field in this more realistic case:



dk  eE k


dt


 eE x
k x 


k
y
kx
The steady-state situation can be
represented (for a nearly free electron
metal) by a very small shift in the Fermi
sphere:
This leads to a net current, since there
is no longer perfect cancellation of
terms corresponding to ±k values, as
when E = 0.

J x  ne  vx (k )
1stBZ
We can also write this as:
kx
Ex
Jx =
density of
e
uncompensated e-
J x  N ( EF )E evFx 
velocity of
uncompensated eE  energy shift of Fermi sphere
Nearly-free Electron Conductivity
We can simplify this expression
through the chain rule:
Now using the approximation of a
free-electron band E(k):
 dE 
 k x
J x  N ( EF )E evFx   evFx N ( EF )
 dkx  EF
  eEx 
2
J x  evFx N ( EF )vFx k x  evFx
N ( EF )
F 
 

2
J x  e2vFx
 F N (EF )Ex
For a spherical Fermi surface, we can write
Which then yields:
1
2
vFx
 vF2
3
2
2
2
vF2  vFx
 vFy
 vFz2  3vFx
1
J x  e 2 vF2 F N ( EF ) E x  E x
3
1
3
  e 2 vF2 F N ( EF )
This result reduces to the free-electron gas expression (see right) when the
FEG value of N(EF) is substituted, but it is more general and highlights the
importance of the N(EF) and F in determining the conductivity of a metal.
 ne 2
  
 m



J. Band Effective Mass of an Electron
We can use our previous results to write the equation of motion of a Bloch electron in 1-D:
dv
dv dk
ax  x  x x
dt dkx dt
This gives: ax 
And we can write:
dvx dvg
d  1 dE  1 d 2 E

 


dkx dkx dkx   dkx   dkx2
Also, from the
acceleration theorem:
dk x Fx

dt

1 d E  Fx 
 
 dkx2   
2
Or, in the form of
Newton’s 2nd law:
This allows us to define a “band effective mass”
2
Fx  2 a x
d E
dkx2
2
m*  2
d E
dkx2
or
1
1 d 2E
 2
m *  dkx2
By contrast to the free-electron mass, the “band effective mass” varies depending on
the electron’s energy and thus its location in the band!
Physical Meaning of the Band Effective Mass
 2 k x2
Of course, for a free electron, E 
2m
In a 3-D solid we would find that m* is a
second-order tensor with 9 components:
2
and m* 
m
2
 
 
m
1
1 2E
 2
m *  ki k j
i, j  x, y, z
The effective mass concept if useful because it allows us to retain the notion of a freeelectron even when we have a periodic potential, as long as we use m* to account for
the effect of the lattice on the acceleration of the electron.
But what does it mean to have a varying “effective mass” for different materials?
Physical Meaning of the Band Effective Mass
The effective mass is
inversely proportional to the
curvature of the energy band.
Near the bottom of a nearly-free electron band m* is approximately
constant, but it increases dramatically near the inflection point and
even becomes negative (!) near the zone edge.
K. Electrons and Holes
Remember that the Hall Effect gives us a way to measure the sign and concentration of the
charge carriers in a metal. If electrons are the charge carriers, we expect the Hall
coefficient RH = 1/ne to be negative. However, for some elements (Zn, Cd, Bi) RH is
positive!
To understand this behavior, first consider the
excitation of an electron from a filled band into
an unfilled band:
By the
band
symmetry,
k-j = -kj
This is what happens in a semiconductor. The
electron in the upper band can now conduct
electric current under a field. But how does the
lower band now behave?
When the lower band was filled, we had a net
electron wavevector of zero:
N /2

i  1

ki  0
Electrons and Holes
N /2
This can also be written:

i j
 
ki  k j  0
N /2

i j

 
ki   k j  k  j
So when the state +j is empty in a band, the (incomplete) band has effective wavevector k-j.
Now we analyze the current flow in the
incomplete band under the influence of a field E:
N /2

i  1
N /2

 evi     evi   ev j  0
i j

ev j
This shows that an incomplete band (with state +j empty) behaves just like a positive
charge moving with the same velocity an electron would have in that state.
Thus the properties of all of the remaining electrons in the incomplete band are equivalent
to those of the vacant state j if the vacant state has:
a.
A k-vector k-j
b.
A velocity v+j
c.
A positive charge +e
We call this vacant state a positive “hole” (h+)
Dynamics of Electrons and Holes
If this hole is accelerated in an applied electric field:
The corresponding equation for the electron is:
But earlier we deduced that the hole velocity is the
same as that of the corresponding “missing” electron:
So by equating the derivatives we find:


eE
eE

mh
me


dv h
mh
  eE
dt


dve
me
 eE
dt
 
vh  ve
mh  me
However, note that near the top of a band the band curvature is negative, so the effective
electron mass is also negative. The corresponding hole mass is then positive!
So the equation of motion of a “hole”
in an electromagnetic field is:


   This explains why
dk h
F 
 eE  vh  B  some metals have
dt
positive R !
H
Dynamics of Electrons and Holes in K-Space
Remember that in a metal with the Fermi energy near the top of the band, we can
EITHER describe the properties of the electrons OR the properties of the hole(s). The
hole has no tangible existence apart from the electrons in the band.
The motion of electrons and holes in k-space is tricky to
understand, because a hole has the opposite wavevector as
the missing electron. So we need to use the following
notation to describe hole motion in the presence of an
electric field:
position of electron in k-space
position of hole at ke (site of hole)
position of hole at kh
So we see that electrons and holes in the same band move rigidly in the same direction in
k-space without altering their relative position--“like beads on a string”.