Summary of Bloch’s Theorem
 
ik  R
e ,
The eigenvalue corresponding to translation symmetry is

not k The eigenvalue and eiegenstates are periodic
not . The eigenvalue
and eiegenstates
are periodic functions functions
 
of , unchanged when
K
k  k  K.

st index describes Energy states are indexed by and n. The 1
k
nd
symmetry properties during translation, while the 2
y
yp p
g
distinguishes energy states with the same symmetry.
Essential result: 
 
 ( q  K )  0
(E  E ) (q )  
U
K

0

q
K
1
Our road map
Computations
Schrodinger Eq.
+
Lattice symmetry
approximations: i
i
nearly free and tightly‐bound
tightly
bound electrons
Electronic structures
Properties p
of solids
Electron‐electron interactions
2
Two approximations
Nearly free electrons. Electrons are non‐interacting in a p
periodic crystal potential which is relatively weak and can be y
p
y
treated as a perturbation. As in the free‐electron‐gas model, they are still subject to the Pauli exclusion principle.
Free electron gas : The interactions between electrons and between electrons and nuclei are turned off
between electrons and nuclei are turned off, subject only to the Pauli exclusion principle. Tightly bound electrons.
Tightly‐bound
electrons.
Electrons are tightly bound to particular atoms, overlapping only weakly with neighbors.
3
Nearly Free and Tight‐Bound Electrons l
d h
d l
(Ch 8 of CMP)
(Ch 8 of CMP)
• Nearly Free Electrons
Nearly Free Electrons
– Degenerate Perturbation Theory
– Fermi Surface
Fermi Surface
• Tight‐Bound Electrons
– Wannier
W
i Functions
F ti
– Tight‐Binding Model
– Berry Phases
B
Ph
4
Nearly Free Electrons
Consider an electron traveling in a weak periodic potential. 
 
0
 ( q  K )  0.
The Schrodinger equation is (E q  E ) (q )  
U
K

K
Because the crystal potential is small, we assume Because
the crystal potential is small we assume U K   wK ,
where  is a small dimensionless parameter.

 (q )
Expend and E
in powers of  as follows.

(0) 
(1) 
 (q )   (q )  (q )  ..... and E  E (0)  E (1)   .....
Substituting these expansions into the Schrodinger Eq. gives 
0
(0)
(1)
(0) 
(1) 
(0 ) 
 (q  K )  .. wK   0.
(E q  E  E   ..)  (q )  (q )  ..  

 
K

 0

0
(0)
(0) 
(0)
(1) 
(1) (0) 
(0) 
   0.
(E q  E ) (q )  (E q  E ) (q )  E  (q )  

(
q

K
)
w
K



K
5
(E  E )
0

q
(0)
(0)

  0

(0)
(1) 
(1) (0) 
(0) 


(q )  (E q  E ) (q )  E  (q )  
 (q  K ) wK    0.


K


Zero‐order approximation. (E q0  E (0) ) (0) (q )  0.
In the extended zone scheme, 

(0) 
(0)
ik
 k (q )   k ,q ,
 k (r )  e r
 1
 iqi r
  (r )   (q )e
V q
2 2

k
0
(0)
0



E

.
Ek  Ek
k
2m

st BZ, In the reduced zone scheme, must remain in the 1
k
and  acquires an additional index n, i.e.,

 nk(0) (q )   K
 
n  k ,q
,

(0)

nk
  

i ( K n  k ) r
(r )  e
(0)
0
  E 
E nk
k K
n
6
First‐order approximation.

0
0
(1) 
(1) (0 ) 
(0) 
(E q  E k ) (q )  E  (q )  
 (q  K ) wK  0.

K
(0) 
)
0
  k (q )   k ,q ,
  E
E k((0)
k
0
0
(1) 
   w   0.

(E q  E k ) (q )  E (1) k ,q  

k
,q  K K

K

(0)

k


ik r
(r )  e
2 2

k
.
E k0 
2m
2m
 
qk
E (1)  w0 .
For , only K=0 survives the sum, and   
For , and K≠0, qk K
wK  k ,q  K

 (q )  
.
0
0


 
K 0 E k  E k  K
U K  k ,q  K

 k (q )   k ,q  
.
0
0


 
K 0 E k  E k  K
(1)

(0) 
(1) 
 (q )   ( q )  ( q ) 
The breakdown condition is when the denominator vanishes, i.e., E k0  E k0  K .
7
Degenerate perturbation.
For a set of degenerate eigenfunctions  j , j  1, 2, ....., l
l
  Cj  j
Hˆ   E 
 i  j   ij
j 1
i
 i Hˆ   E  i  ,
Applying on both sides of the Eq. gives
(H 11  E )C1 
H 12C2 
H 13C3 
...  0
H 21C1 
(H 22  E )C2 
H 23C3 
..  0
H 31C1 
H 32C2 
...
...
(H 33  E )C3  ..  0
Nontrivial solutions exist only when the determinant of the l × l matrix i  i (Hˆ  E )  j
vanishes, i.e, the secular Eq.,
..
;
Hij   i Hˆ  j
...  0
H 11  E
H 12
H 13
...
H 21
H 22  E
H 23
...
0
H 31
H 32
H 33  E ...
...
...
...
...
8

 
Consider two states and 1  k
2  k  K
The matrix elements are

 


k (Hˆ  E ) k , k (Hˆ  E ) k  K ,
 
 
k  K (Hˆ  E ) k  K ,...
To calculate the matrix elements we recall that
To calculate the matrix elements, we recall that
Hˆ  

q
 0 

 
q E q q   q U K  q  K  .

qK 
The matrix elements are
E k  E k  K  E appears 



ˆ
ˆ
only in the diagonal k (H  E ) k  k H k  E k  E k0  U 0  E
elements.
l
t

 

 
ˆ
ˆ
In the summation, K’=-K
k (H  E ) k  K  k H k  K  U  K
 


  *
 iK r 3 
1
*
ˆ
ˆ
k  K H  E k  k H k  K  U  K  U K U K  unit U (r )e d r
 cell
 
 
k  K (Hˆ  E ) k  K  E k0  K  U 0  E
The secular equation is
E k0  U 0  E
U K
U  K
E
0
 
k K
 U0  E
 0.
0
9
The eigenenergies are E  U 0 
If
E k0  E k0  K , E  E k0  U 0  U K .
E E
0

k
2
0
 
k K
E  E

4
0

k
0
 
k K
2

2

 UK .
The energy gap is E g  2 U K .
At the point where the degeneracy occurs, 

K

k
  2



k
 (k  K )
1
ˆ

,  k  K  K ( K    K ).
2m
2m
2
2
2
2
Electrons scatter El
t
tt
strongly off the lattice when k is at the BZ when k
is at the BZ
boundary, and a band gap forms.
k
reciprocal
lattice point
10
11
12
How does the Fermi surface cross the BZ boundary of a square 2D lattice has two electrons per lattice site? What is kF?
Assuming N = No. of total sites
= k points in the 1st BZ.
(Each k point takes two electrons)
The 1st BZ contains all 2N electrons. l t
kF
0
1st BZ
2/a
Fermi surface
2/a
 2 
2
 k F Dk    Dk
 a 
2
4 2
k  2
a
2
F
 kF 
reciprocal lattice points of a 2D square lattice
2 

 1.128
a
 a
13
A free‐electron Fermi surface completely encloses the 1st BZ.
The portion of the Fermi surface in the 2nd BZ is mapped b k i t th 1st BZ so that the back into the 1
BZ th t th
energy surface is continuous.
14
Near‐free‐electron Fermi surface for 3 electrons per site in an fcc lattice
an fcc
Electrons residing at the Fermi surfaces determine transport phenomena of solids.
15
Near‐free‐electron Fermi surface for fcc latices
16
Tight‐Binding Approximation
Electrons are tightly bound to particular atoms, overlapping
Electrons are tightly bound to particular atoms, overlapping only weakly with neighbors.
To establish the tight
To
establish the tight‐binding
binding approximation, we first construct approximation, we first construct
a set of orthornormal wave functions from Bloch functions and which are localized on atomic sites. Let’s first express Bloch functions as a Fourier series

 


1
 ik R R denotes the position of an atomic site. denotes the position of an atomic site


 n ,k ( r ) 
w
R
,
r
e


n
r is the electron position; n is the band 
N R
index. N = No. of lattice sites
Using the inverse Fourier transform ,
 
 
1

 ik R
N = No. of k points in the 1st BZ, over  ( r )e
wn  R, r  


N k n ,k
which the summation runs.  

wn R, r is a wave function localized to for the n‐th
R
energy band. 

17
Wannier functions centered at lattice site R is defined to be
 

 
 

1
 ik  R i ( k )

r R  wn ( R, r ) 
e
 nk (r ),


N k

 (k )
where is a completely arbitrary real function. By manipulating , one can optimize the Wannier
one can optimize the Wannier functions and make them drop off functions and make them drop off

as fast as possible when moves away from
R.
r
The Wannier
Th
W i functions are real‐space representations of the f
i
l
i
f h
bands, and they form an orthornormal set because
  *   3
 wn ( R, r )wm ( R, r )d r   R ,R m,n .
See the textbook for proof.
HW #4‐1: Find the Wannier
HW
#4 1 Fi d th W i function of a 1D lattice composed f ti
f 1D l tti
d
of N atoms with a lattice constant a, assuming that the Bloch wave function is  k ( x )  eikx uk ( x ) and the phase factor (k)
wave function is and the phase factor is neglected.
18
Tight‐Binding Model


 
A projection operator : selects the portion of a ket
function parallel to  .

That is, if  
 c

 ,
    c  .

If the Wannier function centered at does decay exponentially R


once r leaves site R , one can write the Hamiltonian restricted
once leaves site , one can write the Hamiltonian restricted to the nth band as Hˆ ( n )  

RR
 
 
R R Hˆ R R ,

where R denotes the
where denotes the Wannier
function
 Wannier function corresponding to wn ( R, r ).
Note that     1,

if  form a complete
if form a complete orthornormal set.
19
 
 
(n)
ˆ
ˆ
R R H R R ,
H 

The matrix element is
RR
2 2



  







n
n
(
)
(
)
3
*
ˆ R  d r w ( R, r )[
   R H
 U (r )]wn ( R, r ) 
Hˆ RR

  n
2m

  
 2
1
 ik ( R  R )
 

 
1

   (r ) ,
 ik R i ( k )
 
e
E
wn  R, r  
 n ,k ( r )e
nk
nk


N k
N k


R
R
which is negligible if and are not nearest neighbor.


if R and R are nearest neighbors
 t if and are nearest neighbors,
(n)
ˆ
Assuming that
 


H RR  
U if R  R,
one therefore has the tight‐binding Hamiltonian Hˆ TB  

R

 


R t R   
R U R.

R
20

 


ˆ
H TB  
R t R   
R U R.


R
R
The 1st term is a hopping term that gives rise to an interaction energy between electrons of nearest neighbors.
The 2nd term is an on‐site term that describes the energy of placing an electron at a lattice site.
The tight‐binding Hamiltonian has a simple exact solution,

1
k 
N
e

R
 
ik  R

R ,

f k i th 1st BZ.
for in the 1
BZ

1
R

The inversion is N
Hˆ TB
e

k
 
 ik  R

k ,
   
 ikR ik( R  ) 

1
1
 i ( k  k  ) R
    k Ue
.

k
te
k
k





R N kk 
R N kk 
21
Hˆ TB  

kk 
     
   

 1
1
 i ( k  k  ) R ik 
 i ( k  k  ) R
k t
e
e
k  
U k
e
k .






N R
 N R
kk 
 
 k
E

k,
k

k
e
with E k  U  t 

 
ik 
Note that
e

R
 
 iq  R

 N (q )
.


Hence is an eigenfunction of
k
Hˆ TB ,

  

ˆ
 k k k  E  k
 H TB k  
E
k
k

k
Assuming z is the number of nearest neighbors, the maxima and minima of Ek are and the maxima and minima of E
are U  z t , and the
bandwdith is 2 z t .
22
For example, using the tight binding model, the energy of 1D For
example using the tight binding model the energy of 1D
lattice with a lattice constant a is
E k   U  t
e


 
ik 
 U  t  eika  eika 
 U  2t cos(ka )
HW # 4‐2: Using the tight binding method, considering only the s

orbital at each atomic
orbital at each atomic site, what is the band energy site
what
is
the
band
energy
E (k )

k
as a function of for a fcc
structure with a lattice constant a?
23
Geometric (Berry) Phases
Both vectors stay tangential to the curved surface at all times, and remain as parallel as possible to the direction they were pointing at before each infinitesimal displacement.
After completing the closed path, or loop, the vectors go back to the original point, but they find i i l i t b t th fi d
themselves rotated with respect to the directions they were pointing
the directions they were pointing at when the journey started.
Such rotation angles of geometrical Such
rotation angles of geometrical
origin are known as Berry phases. States (wave functions) in quantum mechanics are represented by vectors in a linear space; they also acquire phases after parallel transport along loops. parallel transport
along loops
24
Berry Phases in Q. M.
For a parametric Hamiltonian with For
a parametric Hamiltonian with  being a parameter being a parameter
defined in a suitable domain, Ĥ       E     
   is the ground state non degenerate for any  . Assume is the ground state, non‐degenerate for any Assume
The phase difference  is defined as follows.
The phase difference 
is defined as follows
 2   ei  1 
 is a real number. is a real number.
12
e
 i12
  1   2 

 12   Im ln  1   2 

Note that any QM state vector i
is arbitrary by a constant bit
b
t t
phase factor; one can have a choice of the gauge.
choice of the gauge.
25
 4 
  3 
In the continuum limit with an infinitesimal change of parameter 
ei         
1  i  1        
N
    s ,s1
 1 
 2 
s 1
   i
       
C
The total phase difference along a closed path
The Berry phase, a gauge invariant,   12  23  34  41
is potentially a physical observable
is potentially a physical observable.
i
is gauge invariant.
i
i
26
Berry Phases in One Dimension
Recall that
Recall that and u
and uk(x+a) = uk(x).
(x)
Assuming k to be continuous and periodic in k, and
 k  ei ( k ) k , uk ( x)  ei ( k )uk ( x)
in which  is some phase function to be determined,
in which is some phase function to be determined
there is a uniquely determined phase , known as the Berry phase, that satisfies .
The phase 
The
phase  is related to the allowed eigenvalues for the is related to the allowed eigenvalues for the
a
Wannier functions,
R
 la,
2
where l is an integer.
27
Define the projection operator on the nth Bloch band
on the nth Bloch band, Pˆn    nk  nk .
k
Define also the Wannier function to be eigenstates of the ˆˆˆ R
position operator projected onto this band, i.e.,
iti
t
j t d t thi b d i
Rˆ R  PRP
and define w( R, k )   k R .
Note that R is the eigenvalue of R ,
corresponding to location of the di t l ti
f th
Wannier function. One can show that Rw( R, k )    k Rˆ  k  w( R, k ')
k
ˆˆˆ R
 Rw( R, k )  R  k R   k PRP
   k  k   k  R
R̂  k   k  R
k k 
   kk   k  Rˆ  k  w( R, k )
k k 
28
H.W.
Because of the phase arbitrariness of the Bloch function, there is some , with which
Rw( R, k )    k Rˆ  k  w( R, k ')
k


Na
dk 

2
a dx
Na



k
  dk  2 i   (k  k )  
uk* ( x)uk ( x) w( R, k )
2
 k
 0 a
integration by parts
integration
by parts
1

 Rw( R, k )  i w( R, k )
 w( R, k )  eikR
k
w( R, x)  x R   x  k  k R   eikR k ( x)
k
k
just the same Wannier functions defined previously.
29
Since by assumption,  k  2 / a ( x)   k ( x),
one has  k  2 / a ( x)  e
 i ( k  2a )
e
 k  2 / a ( x)
 k ( x )e
 i ( k )
  k ( x)e
In addition
i[ ( k )  ( k  2a )]
i[ ( k )  ( k  2a )]
 = (k )   (k  2a ).
 w( R, k )   k R ,
w( R, k  2 / a)   k  2 / a R  ei  k R .
w( R, k )  eikR  w( R, k  2 / a)  ei 2 R / a w( R, k )
2 R
 
 l 2 .
a
a
R
 la
2
w( R, x) decays exponentially if decays exponentially if x >> R.
R
30