1
J. Humlíček FKL II
7. Impurity states
Semiconductors: a small concentration of (suitable) impurities has a strong effect on the transport of electric current. The modification of pristine
("pure","intrinsic") material is usually denoted as doping.
Point (vacancies, substitutional or interstitial) and line defects (dislocations), complexes (e.g., Frenkel's pair of vacancy-interstitial), precipitates.
Donors and acceptors, in Si mainly P, As, Sb and B, Ga, In.
The energy levels within gap are usually denoted as shallow (hydrogen-like) or deep.
Shallow states, effective mass approximation
Screened Coulomb potential of a donor:
0
ˆ ( ) ,
4 
d
r
e
V r
r
(7.1)
where r is the relative permittivity of the host crystal. Electron belonging to the donor atom obeys the Schroedinger equation
0ˆ ˆ( ) ( ) ( ) ,dH e V r E r   (7.2)
where H0
je is the one-electron Hamiltonian of the unperturbed crystal.
2
Wannier functions are defined via the discrete Fourier transform of the Bloch functions (eigenfunctions of the one-electron Hamiltonian); for n-th
band, this and inverse Fourier transform reads
1
( ) exp( ) ( ) ,
1
( ) exp( ) ( ) .
l
j
n j l j nk
k
j n jnk
R
a r R ik R r
N
r ikR a r R
N


  
 


(7.3)
The sum for Wannier functions an extends over the quasicontinuum of the wavefunctions from the first Brillouin zone, in the inverse trnasform
the sum extend over the lattice points of the basic Born-Kármán volume of the crystal, se the definition of Eq. (1.13); N=N1N2N3 is the number
of primitive cells in this volume.
3
Bloch (left part) and Wannier (right part) functions. The circles denote lattice points, the
green lines are plane waves eikx
.
Wannier functions are localized around the lattice points, as shown schematically in the above figure. The localization can be fairly strong
(exponential in one dimension: W. Kohn, Phys. Rev. 115, 809 (1959)), see also Marzari et al., REVIEWS OF MODERN PHYSICS, VOLUME 84,
OCTOBER–DECEMBER 2012.
Wannier functions ( )n ja r R are eigenfunctions of the vector operator
ˆ
R , belonging to the eigenvalue Rj (i.e., to the j-th lattice vector):
ˆ
( ) ( ) .  n j j n jRa r R R a r R (7.4)
4
Let us assume a general wavefunction in the form of the linear combination of the stationary states of one-electron Hamiltonian without
impurities, i.e., the Bloch functions,
, ,
1
( ) ( ) ( ) ( ) exp( ) ( ) ;
jl l
n l n l l j n jnk
Rn k n k
r A k r A k ik R a r R
N
      (7.5)
the action of the operator
ˆ
R leads to
 
,
,
1ˆ
( ) ( ) exp( ) ( )
1
= ( ) exp( ) ( ) .
jl
jl
n l l j j n j
Rn k
n l l j n jk
Rn k
R r A k ik R R a r R
N
A k i ik R a r R
N
  
  
 
 
(7.6)
We will approximate the possible values of the vector k l by a continuous set; consequently, (in this limit of infinite crystal) the action of the
gradient operator is
exp( ) exp( )l j j l jk
ik R iR ik R  (7.7)
for any vector kl. Additionally, Eq. (7.6) can be brought to the following form:
 
, ,
,
ˆ
( ) ( ) ( ) ( ) ( )
( ) ( ) .
l l
l l
l
l
n l n lk nk k nk
n k n k
n lk nk
n k
R r i A k r i A k r
i A k r
  

          
   
 

(7.8)
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The first sum in the upper row of Eq. (7.8) vanishes; in fact, in the approximation of infinite crystal the sum over k is replaced by the volume
integral over the Brillouin zone and the resulting value is the difference of the products A in the two equivalent points at the opposite
boundaries of the BZ.
The wavefunction of Eq. (7.5) can also be expressed as the following linear combination of Wannier functions,
,
( ) ( ) ( ) ;  
j
n j n j
n R
r C R a r R (7.9)
the coefficients Cn are usually denoted as envelope functions. Rearranging Eq. (7.5), we obtain
,
1
( ) ( )exp( ) ( ) ;
j l
n l l j n j
n R k
r A k ik R a r R
N

 
  
  
  (7.10)
the envelope functions Cn are therefore Fourier transforms of the coefficients An. In analogy with (7.7), we will further approximate the changes
of wavefunctions at the distances much larger than those of the nearest-neighbors,
exp( ) exp( ) .l j l l jR
ik R ik ik R  (7.11)
In searching for the electronic structure of the crystal including defects (solving the wave equation (7.2)), we make use the electronic structure of
the unperturbed crystal. Let us select the n-th band, with the Bloch eigenfunctions and eigenenergies 0
( )nE k :
0 0
, ,
ˆ ( ) ( ) ( ) .nn k n k
H r E k r  (7.12)
Using Eq. (7.11), the operator acting on the envelopes of the Wannier functions belonging to the n-th band can be expressed as
6
 0 0ˆ .w n R
H E i   (7.13)
The next step consists in adding the perturbation due to the impurity atom; a simple description is based on the assumption of smooth changes of
the potential (7.1) at the distances comparable with the lattice constant. In this case, we substitute the position dependence of ˆ
dV in Eq. (7.2) by
the dependence on the vector R and the equation for the envelope functions and the eigenenergies E becomes
 0
( ) ( ) ( ) .n d n nR
E i V R C R EC R      (7.14)
In order to use the last equation, we have to specify the functional dependence 0
( )nE k of the dispersion of n-th band, do replacing its argument by
the gradient operator. Here, we choose the simple parabolic dependence of the isotropic non-degenerate band (which is typical of the conduction
band minima in the center of the Brillouin zone, e.g., in GaAs):
2
2
0
*
( ) (0) .
2
n c
k
E k E
m
  (7.15)
Equation (7.14) for the envelopes of Wannier functions reads
 
2
*
( ) ( ) (0) ( ) .
2
d cR
V R C R E E C R
m
 
     
 
(7.16)
It has the form of Schroedinger equation for a particle of the mass m*, moving in the potential dV , and its energy starting at the conduction band
minimum. The correspondingf wavefunction is the product of the envelope and Wannier functions according to Eq. (7.9). This approach is
usually called effective mass approximation.
7
Equation (7.16) with centrosymmetric potential of Eq. (7.1) is equivalent to the quantum description of the electron in isolated hydrogen atom.
The differences consist in using the effective mass of the electron, and the Coulombic attraction to the positively charged impurity is reduced due
to the screening by the electrons of the host crystal. The latter is introduced by the permittivitty in the denominator of Eq. (7.1). Stationary states
are both discrete (bound) and continuum (delocalized); the latter lie above the bottom of the conduction band of the host crystal. The analogy
with the hydrogen atom allows us to find the energies En of the bound states as
 
4*
0
22 2 22
0 0
1 1
(0) = , 1,2,...,
2 4
n c
r
e mm R
E E n
m n n 
     (7.17)
where n is the principal quantum number, e is the elementary charge, and m0 is the mass of free electron; R denotes the Rydberg constant of the
donor electron. For m*=r=1 (isdolated hydrogen atom), the value of R is 13.6 eV, pro m*≈0.1 a r≈10, this value is reduced to about 14 meV.
Bohr radius of the donor state is
2
* 0 0
* 2
0
4
,r m
a
m e m
 
 (7.18)
and the envelope function of the lowest state (1s) is approximated by the exponential,
*
1 * 3
1
( ) .
( )
R
a
sC R e
a

 (7.19)
Bohr radius (7.18) is much larger than that of the hydrogen atom (0.0529 nm), due to the large values of permittivity and small effective masses.
The envelopes of Wannier functions (i.e., the coefficients C) are non-negligible for a large number of the lattice points. On the contrary, the
coefficients A from Eq. (7.5) multiplying the Bloch functions are non-negligible for a small number of the (discrete) values of the wave vector.
This can be explained by the Fourier transform relating the two systems of coefficients, see (7.9) and (7.10). In order to estimate the exponential
behavior of the envelope C, we can utilize the cosine Fourier transform
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 
   
*
* *
2 2* 2 *
0
1/
( ) cos .
1 1/
R
a
a a
A k e kR dR
a k k a


  
 
 (7.20)
At *
1/k a , the value of A is a half of its maximum (at k = 0); this function is "narrower" for "wider" envelope C. Equation (7.20) represents a
special case of the more general uncertainty principle for a function f and its Fourier transform F:
1 1
( ) ( ) , ( ) ( ) ;
2 2
i t i t
F e f t dt f t e F d 
  
 
 

 
   (7.21)
for two real numbers, T and W, defined as
2 2 2 2 2 21 1
| ( ) | , | ( ) | ,T t f t dt W F d
E E
  
 
 
   (7.22)
where
2 2
| ( ) | | ( ) | ,E f t dt F d 
 
 
   (7.23)
the following inequality holds:
1
.
2
TW  (7.24)
This result means that both functions, f and F, cannot be simultaneously "small", and, at the same time, "narrowing" one of them leads to
"widening" of the other. This is important also in our case of the donor states: with increasing quantum number n, the binding energy drops as
9
1/n2
, the wavefunction lies in the volume proportional to n2
. Higher excited states are therefore described better by our model than the ground
state.
Donor states for anisotropic conduction bands of several host crystals are of considerable practical importance (e.g., Si, Ge, and GaP). Due to the
crystal symmetry, the minima of conduction bands are degenerated and the position dependence of the interaction in the neighborhood of the
donor atom leads to the so called "valley-orbit coupling“. The above approximation requires modifications.
Let us mention silicon, having six equivalent minima of the conduction band, located close to the edge of the Brillouin zone X (in the [100]
direction, labeled usually ). The first step consists in te modification of Eq. (7.16) for the envelopes; for any of the minima j = 1,...,6 we have
the equation
2
0*
2
( ) ( ) ( ) ( ) ,
2
t l
d j c j
t l
V R R E E k R
m m m
   
           
  
(7.25)
where mt and ml are transverse and longitudinal effective masses, respectively (0.19m0 and 0.92m0 for Si). Approximate solutions can be found
variationally, using a suitable test function reflecting the lowering of symmetry (spherical to cylindrical). A popular approach is due to Kohn and
Luttinger (1955), using the trial function with two parameters,
 2 2 2
( ) ,
a y z bx
x R e
  
  (7.25)
where the x direction lies alog the longitudinal axis of the corresponding band minimum.
10
Schematics of the band structure of a direct semiconductor, donor states of n=1,2 a 3.
Binding energy (the ground state, n=1) of shallow donors in zincblende semiconductors, in
meV. The values in the middle column are results from Eq. (6.17), those in the right column
are measured data for the most common dopants.
11
Calculated and measured energies of shallow donor states in Si.
12
Real part of conductivity resulting from transmission spectra of Si:P (5.59E16 cm-3
) at
different temperatures.
0 200 400
0.0
0.5
1.0
1.5
1s(A1
)->cb
->2p+/-
1s(A1
)->2p0
->2p+/-
1s(T2
)
1s(E)
->2p0
1s(T2
)
15K
25
45
75
TransN7-Sig1TD
200
65
300K
100
WAVENUMBER (cm
-1
)
Si:P 7.87 
-1
cm
-1
5.59x10
16
cm
-3
1
(
-1
cm
-1
)
1s(E)
13
Energy of the lowest bound acceptor state (in meV), compared with the calcvulation usin
spherical symmetry.
14
Calculated and measured shallow acceptor levels in Si.