Transcript Vibrational and Thermal properties of Semiconductors

Modern Computational condensed Matter
Physics:
Basic theory and applications
Prof. Abdallah Qteish
Department of Physics, Yarmouk University,
21163-Irbid, Jordan
Chemistry Dept, YU, 14 May 2007
Starting from first-principles, can we efficiently and accurately
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Calculate the various properties (structural, electronic structure,
vibrational, thermal, elastic, magnetic, …, etc) of bulk solids;
Investigate the surface and interface properties of solids;
Study defects;
Construct the phase diagrams of alloys;
Study the properties of liquids and amorphous materials;
Investigate the material properties under extreme condition (very high
temperature and pressure);
Deal with biological systems;
Others ??
Answer:
YES
Direct application of Standard QM !!
In Standard QM, Ψ which is the solution of the many-body Schrödinger Eq.
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N
 1 N 2 N
1
    i   Vext (r i )   1
 2 i 1
2 i j | r i  r j
i 1

is the basic variable
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
 (r1 , r 2 ,...,r N )  E (r1 , r 2 ,...,r N ) ...... (1)


X  X [ ];
X is any observablephysicalquantity
Main problem: Ψ is a function of 3N variables, and N is of order of 1024 for a
realistic condensed matter sample.
Thus, direct application of Standard QM is simply impossible.
Remark: In Eq. (1) the nuclei are assumed to be at fixed positions
adiabatic or Born-Oppenhiemer approximation.
Density Functional Theory (DFT)
Hohenberg and Kohn, PRB 136, 864 (1964)
{about 500 citations per year}
Nobel Prize in Chemistry in
1998, for his development
of DFT.

DFT is based on two theorems:
– The charge density, n(r) is a basic variable
E=E[n].
– Variational principle: E[n] has a minimum at the ground
state n(r), nGS(r), or
E GS [n GS ]   Vextn GS (r )dr  F [n GS ]  E[n]
n(r) as a basic variable
Standard QM
Ψ(r1, … rN)
solve
M.B. Schr.
Eq.
n(r1 )   |  (r1 ,...,rN ) |2 dr2 ...drN
n(r)
V(r)
DFT: one-to-one
correspondence
• Since n determines V (to an additive constant), Ψ and hence the K.E. (T)
and the e-e interaction energy (U) are functionals of n.
• One can then define a universal energy function ≡ F[n] = <Ψ| T + U| Ψ>. So,
E[n]   Vextn(r )dr  F [n]
{unkown functional
of n}
Kohn-Sham formalism of DFT
Kohn and Sham, PRA 140, 1133 (1965)
 KS have introduced the following separation of F[n]
F[n]  To [n]  EH [n]  EXC
where,
 1 
To    i* (r )   2  i (r )dr ,
 2 
i 1
N
E H [ n] 
1 n(r )n(r ' )
drdr '
2  | r  r ' |
K.E. of non-interacting e-system.
Classical e-e interaction energy.
and
EXC is called exchange correlation energy
EXC=EX+EC+(T-To)
{the only unknown
or difficult to calculate terms ==
to be approximated}
Exact self-consistent single-particle equations
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Varying E[n] with respect to n(r) under the constraint of
constant number of electrons



To   Vextn(r )dr  EH [n]  E XC [n]    n(r )dr  0
n(r )
E[n] To

 Vext  VH  VXC   (2)
n(r ) n(r )
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Now, suppose that we have a non-interacting electronic system with the
same density n(r), sustained by an effective potential Veff. Then,



To   Veff n(r )dr    n(r )dr  0
n(r )
To
E[n]

 Veff   (3)
n(r ) n(r )

Eqs. (2) and (3) are mathematically equivalent, and
Veff  V KS  Vext  VH  VXC
...
(4)
This leads to exact (no approximation is used so far for EXC) transform of
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to
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Therefore, EGS and nGS(r) can be obtained by solving a set of N singleparticle Schrödinger like equations (known as KS equations):
 1 2

KS
    V [n]  i (r )   i i (r )
 2


Note that
n( r )   f i |  i ( r ) | 2 ,
......... (5)
where i runs overall
i
occupiedor partiallyoccupiedstates,and f i is
theoccupationno.of statei.
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Thus, equations 3 to 5 have to solved self-consistently.
Periodic Boundary Conditions and Bloch’s Theorem
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Periodic Boundary Conditions: Finite systems are assumed to be
periodically repeated to fill the whole space
An efficient recipe to
study atoms, molecules,
surfaces,
Interfaces, … etc
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Bloch’s Theorem: The wave-functions of the electrons moving in a
periodic potential are given as
 nk (r )  unk (r )eik.r
unk(r) have the same periodicity as the potential.
n is the band index
k is a wave-vector inside the 1st BZ.
This transforms the problem into calculating few wavefunctions for,
in principle, infinite number of k points.
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The great simplification comes from the fact that Ψnk are weakly
varying functions with respect to k … only few carefully chosen kpoints (known as special k-points) are required.
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Convergence test: Si in the diamond structure
Mesh
2x2x2
4x4x4
8x8x8
No. special
K-points
2
10
60
Example: 2x2 mesh
For 2D square lattice
E (H)
(a=10.4 Bohr)
-7.930764
-7.936765
-7.936879
Expt.
Lattice
constant (Å)
5.392
5.384
5.384
5.431
Bulk
modulus (Mbar)
0.959
0.956
0.954
0.99
Approximations to EXC
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Local density approximation (LDA)
– Assumption: EXC depends locally on ρ( r )
E
LDA
XC
   (r )
hom
XC
(r )dr
– Recommended LDA functional: Perdew-Wang (PRB
45, 13244, 1992)
– LDA is currently being used to study fundamental
problems in physics, chemistry, geology, material
science and pharmacy.

Generalized gradient approximation
(GGA)
– Assumption:
GGA
E XC
   (r ) XC (  (r ),  (r )) dr
– Recommended GGA functional: PerdewBurke-Ernzerhof (PBE) [PRL 77, 3865
(1996)].
– GGA is found to improve the binding
energies, but not the band gaps.
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Meta-GGA (MGGA)
– Assumption:
MGGA
E XC
   (r ) XC (  (r ),  (r ),  (r )) dr
– Here, τ is the kinetic energy density
occ
 (r )   | i (r ) |2
i
– Recommended MGGA functional: ToaPerdew-Staroverov-Scuseria (TPSS) [PRL
91, 146401 (2003)]
– Self-interaction free correlation. Not well
tested yet.
Main problem with LDA, GGA and MGGA
– They allow for spurious self-interaction
(SI).
– Exact DFT is SI free:
1
E x    
2 vv'kk '
 v*k (r ) vk (r ' ) v k (r ' ) v* k (r )
' '
' '
| r  r' |
For vk  v 'k ' ,
'

(
r
)

(
r
)
1
self
vk
vk
E x   
drdr '   EHSelf
2
|rr|
 Exact cancelation between self VH and VX
 T hisis not thecase for LDA and GGA.
drdr ' .
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Theory of Exact-exchange (EXX)
[Stadele et al. PRB 59, 10 031 (1999)]
– Total energy
1  (r )  (r ' )
'
Etot [  ]  To  Ee i  
d
r
d
r

'
2
|r r |
 v*k (r )vk (r ' )v 'k ' (r ' ) v*'k ' (r )
1
'
d
r
d
r
 EC [  ]

'

2 vv'kk '
|r r |
– Single-particle equations
 1 2




V

V
[

]

V
[

]

V
[

]

 i   i i , with
ion
H
X
C
 2

E X
VX (r ) 

 (r )
 E X [  ]  vk (r ' )
 VKS (r '' )
 c.c.


'
''

vk
  vk (r ) VKS (r )
  (r )

Hybrid DFT/HF functionals
– Adiabatic connection formula
1
E XC   d E XC ,
, with
0
E XC ,    | Vee |    
1 3
 (r )  (r ' )
3
d
r
d
r
'

2
| r  r '|
– Three empirical parameters hybrid fucntionals
ACM 3
LSD
EXC
 EXC
 a1 ( E HF  EXLSD )  a2 EXGGA  a3 ECGGA
Example: B3LYB (Becke exchange and Lee-Yang-Parr Corr.)
– One empirical parameter hybrid fucntionals
ACM 1
GGA
EXC
 EXC
 a1 ( E HF  EXGGA )
Example: B1LYB
– Parameter free hybrid fucntionals
1
ACM 1
GGA
E XC
 E XC
 ( E HF  E XGGA )
4
Examples: B0LYB
PBE0
Single-particle energies
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Whence a certain approximation for EXC is adopted, one has to solve selfconsistently the Schrödinger like single-particle equations
 1 2

KS
    V [n]  nk (r )   nk nk (r )
 2

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What is the physical meaning of εnk ?
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Answer: two points of view
-
According to the KS derivation of the single-particle equations:
εnk are mathematical construct {Lagrange multipliers} -- no physical
meaning.
- According to the optimized effective potential (OEP) approach: VKS is
the best local approximation to the non-local energy dependent
electron self-energy operator (in many-body quasi-particle theory) -εnk are approximate quasi-particle energies --- can be used to
interpret band structure data.
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Si band structure
Computational approaches
All-electron:
Pseudopotential:
- all the electron are explicitly included
- the space is separated in core are
interstitial regions.
- electrons = valence+
core.
- only Valence electrons
are explicitly included.
interstitial
- Two main approaches
core
I- LAPW {partial waves (core) and PW
(interstitial)}
II- LMTO {partial waves (core) and Hankel
functions (interstitial)}
- effective potential
(pseudopotential)
due to the nucleus
are the core electrons
- PW basis sets to
expand Ψnk
Some results
I.
Phase stability and structural properties
{example ZnS}
[Qteish and Parrinello, PRB 61, 6521 (2000)]
E vs V curves of ZnS
The ZB structure is the most
stable phase of ZnS, in
agreement with experiment
Zincblende (cubic –
2 atom unit cell)
Rocksalt (cubic –
2 atom unit cell)
SC16 (cubic –
16 atom unit cell)
Cinnabar (hexagonal –
6 atom unit cell)
Structural Properties: ZnS
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Zinc-blende structure (equilibrium phase)
Structural Parameter
Lattice constant (Å)
Bulk modulus (GPa)
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Theory
5.352
83.4
Expt.
5.401
76.9
Error (%)
0.9
8.5
Expt.
5.060
103.6
Error (%)
0.8
0.7
Rocksalt structure (high pressure phase)
Structural Parameter
Lattice constant (Å)
Bulk modulus (GPa)
Theory
5.017
104.4
The theoretical values are obtained by fitting the calculated E to
Murnaghan’s EOS.
BoV  (Vo / V ) Bo 
E (V )  ' 
 1  Cons.
Bo  Bo  1

'
II. Structural phase transformation under high pressure
•
Enthalpy (H) vs Pressure for ZnS
H ( p)  EV ( p)  pV ( p)
•
Transition pressure (GPa)
Transition
ZB to RS
ZB to SC16
ZB to cinnabar
SC16 to RS
Theory
14.5
12.5
16.4
16.2
Expt.
15
-------
III. Phonons: inter-planer force constant approach
• IPFC’s are calculated
by displacing the
atoms of one layer
by small amount
Fi = -kiu
• IPFC’s are then used
to calculate the
phonon spectra along
some high-symmetry
direction.
Ben Amar, Qteish and Meskini, PRB 53, 5372 (1996)
IV. Elastic constants
• Direct method: applying proper strain and calculate the
corresponding stress [Nielson and Martin
PRB 32, 3792 (1985)]
• Using density functional perturbation theory (Lec. 3)
DFPT
b Direct method
a
Elastic constant
Of ZnSe
Hamdi, Aouissi, Qteish and Meskini, PRB 73, 174114 (2006)
V. Thermal Properties (details are in Lecture III)
• Linear thermal
expansion coefficient
of ZnSe
• Constant pressure heat
capacity at of ZnSe
Hamdi, Aouissi, Qteish and Meskini, PRB 73, 174114 (2006)
Conclusions
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DFT is a very powerful tool in theoretical/computational
condensed matter physics.
It has wide applications in physics, chemistry, material science,
geophysics, … etc.
Exciting and continuous progress on the level of theory,
algorithms and applications.
Highly suitable for scientists working in developing countries –
workstations are enough.
End