Theory of electronic states in quantum systems

1
Theory of electronic states in quantum systems

• Aldo Di Carlo
• Diparimento di Ing. Elettronica,
• Università di Roma Tor Vergata, Roma (Italy)
• aldo.dicarlo_at_uniroma2.it

The atomistic way
2
Can we describe nanostructures starting from the
atomic structure ?
Si(111)77 Surface
TEM image from Skolnick lecture of a InAs/GaAs
dot
GaN
InGaN
HRTEM image Indium segregation inGaN/InGaN
Quantum Well
GaN
3
What about realistic nanostructures ?
Inorganics
3D (quantum wells) 1-10 atoms in the unit cell
2D (quantum wells) 10-100 atoms in the unit cell
1D (quantum wires) 1000-10000 atoms in the
unit cell
0D (quantum dots) 100000-1000000 atoms in
the unit cell
Organics
Nanotubes, DNA 100-1000 atoms (or more)
4
Many Body Hamiltonian

• Many body Hamiltonian in the adiabatic
  approximation and fix ion charge

The total energy of the many body system is given
by
Where the many body wave function is
Too difficult, we need approximations
5
One electron Hamiltonian

• Typical approximations
• Hartree, Hartree-Fock, Local Density
  approximation (LDA) .
• These approximations reduce the many body system
  to
• the problem of one electron moving in an
  effective field

V has the crystal symmetry
V(rR)V(r)
To find the one electron energy En, one has to
solve the Schroedinger equation
6
k.p method and Envelope Function approaches
d) Other k.p-EFA are Kane Hamiltonian, Luttinger
Hamiltonian etc.
7
How good is effective mass aprox. ?
GaAs
AlGaAs
EC
E1
E1
d
Effective mass
Exact
d nm
1
2
3
4
5
6
7
8
9
10
11
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The atomic orbital basis

• We attempt to solve the one electron Hamiltonian
  in terms of
• a Linear Combination of Atomic Orbitals (LCAO)

Cia coefficients, fia atomic orbitals (s,p,d)
Lowdin theorem
One can perform the orthogonalization procedure
in such a manner as to preserve the symmetry
properties of the original atomic basis
with fia Lowdin orbitals (s,p,d,...)
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The Schroedinger equation in LCAO basis
10
The empirical tight binding
The calculation of these integrals requires the
knowledge of both basis function and potentials
even though they do not appear separately in the
final matrix element
Empirical tight binding develops approximations
only for the Hamiltonian matrix elements Hia,jb
themselves without attempting to model the
potential and the explicit form of the basis
functions
11
Scalability of TB approaches
Empirical Tight-Binding
Hamiltonian matrix elements are obtained by
comparison of calculated quantities with
experiments or ab-initio results. Very efficient,
Poor transferability.
Semi-Empirical Hartree-Fock
Methods used in the chemistry context (INDO, PM3
etc.). Medium transferability.
Density Functional based Tight-Binding (DFTB,
FIREBAL, SIESTA)
DFT local basis approaches provide transferable
and accurate interaction potentials. The
numerical efficiency of the method allows for
molecular dynamics simulations in large super
cells, containing several hundreds of atoms.
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Why Tight-Binding ?

• Allows us to describe the band structure over
  the entire Brillouin zone
• Relaxes all the approximations of Envelope
  Function approaches
• Allows us to describe thin layer perturbation
  (few Å)
• Describes correctly band mixing
• Gives atomic details
• The computational cost is low
• It is a real space approach
• Molecular dynamics
• Scalability (from empirical to ab-initio)

13
The sp3s Hamiltonian Vogl et al. J. Phys. Chem
Sol. 44, 365 (1983)
In order to reproduce both valence and conduction
band of covalently bounded semiconductors a s
orbital is introduced to account for high energy
orbitals (d, f etc.)
14
The sp3d5s HamiltonianJancu et al. PRB 57
(1998)
Many parameters, but works quite well !
15
Tight-Binding sp3d5s model for nitrides
Ab-Inito Plane Wave DFT-LDA Band Structure for
GaN Wurtzite
TB Wurtzite GaN Band Structure
Nearest-neighbours sp3d5s tight-binding
parametrization for wurtzite GaN, AlN and InN
compare well with Ab-Initio results.
16
Tight-binding for quantum wells (I)
For a quantum well and similar 2D
heterojunctions, the symmetry is broken in one
direction, thus the Bloch theorem cannot apply in
this direction. We define the Bloch sums in the
m-th atomic plane
LCAO wavefunction
In this case the number of Cs is related to the
number of planes
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Tight-binding for quantum wells (II)
Schroedinger equation
Matrix form
where M is the number of interacting planes
For M1 we have
but boundary conditions ?
18
Tight-binding for quantum wells (III)Boundary
conditions
Periodic
After P planes the structure repeats itself.
Suitable for superlattices


Finite chain
After P planes the structure end. Suitable for
quantum wells
Open boundary conditions
After P planes there is a semiinfinite
crystal Suitable for current calculations
BULK
BULK
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Where do we put the atoms ?
To describe the electronic and optical properties
of a nanostructure we need to knowwhere the
atoms are.
1) We know a priori where the atoms are (for
example X-ray informations)
2) We need to calculate the atomic positions
Simple analytic espressions
Full calculation
Continum theory
Classical calculations
Atomistic (Valence Foce Field etc.)
Quantum calculation
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Example Strain and Pseudomorphic growth
An epitaxial layer is grown, on a substrate with
different lattice constant. The epilayer deforms
(strain)
as
Strain tensor
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Calculation of strain tensor
SiN
GaN
Strain Map
AlGaN
22
Strain in a AlGaN/GaN nanocolumn
20nm
Al0.28Ga0.72 N
GaN
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Problem free standing heterojunction
AlGaN
GaN
1mm
1mm
Strain is present !! The system tends to deform
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Strain calculation and determination of
equilibrium geometrical shape.
define shape
start
yes
stop
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The final shape
We consider a GaN/Al0.3Ga0.7N heterostructure.
The GaN layer thickness is 100 nm, the
Al0.3Ga0.7N layer thickness is 23 nm. The lateral
dimension of the structure is 1?m ? 1?m. The
heterostructure is free-standing.
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AlGaN/GaN nanocolumns
Potential
The Poisson equation
piezo-electric polarization
pyro-electric polarization
piezo-electric moduli tensor
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How do we describe alloys ?
Usually, tight-binding parameterizations are made
for single elments and binary compounds (Si, Ge,
GaAs, InAs etc.). However, nanostructure are
usually build by using also ternary (AlGaAs
etc.) and quatrnary (InGaAsP etc.) alloys.
1) Supercell calculations
A0.5B0.5C
Average over an ensamble of configurations
3) Other methods (Modified VCA, CPA, T-matrix
etc.)
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Self-Consistent Tight-Binding
Charge transfer is important in semiconductor
nanostructures.
Self-consistent solution of Schreodinger and
Poisson equations are common in envelope
function approaches
Tight-binding allows for a full (with all the
electrons) self-consistent solution of the
nanostructure problem
Full self consistent approach only suitable for
small systems like molecules
Here we develop a self-consistent approach for
only the free charge
With the aim of Self-Consistent treatment of
external electrostatic potential, Tight-Binding
can be applied to semiconductor device
simulations.
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Self-Consistent Tight-Binding
A. Di Carlo et. al., Solid State Comm. 98, 803
(1996) APL 74, 2002 (1999)
The electron and hole densities in each 2D layer
are given by
The influence of free carrier charge
redistribution and macroscopic polarization
fields are included by solving the Poisson
equation
boundary conditions

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Bidimensional k// Integration
Wurtzite Brillouin Zone
Grid in the 2D irreducible wedge each point has
a different weight according to symmetry
operations.
M

• SCF Carrier Density

Only a small portion of the 2D irreducible wedge
is considered typically we use from 5 points
(for the first SCF cycles) up to 15 points (for
the final SCF cycle.
K
G

• Optical Properties

We use a larger portion and an interpolated fine
k// grid typically we use 1000 real k// and
1001000 k// points for interpolated quantities.
Background colours eigenvalue of the GaN top
valence band.
0.0 -0.5 -1.0 -1.5 -2.0 eV
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AlGaAs/InGaAs/GaAs quantum well
It is easy to account for G and X related
electron density
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AlGaAs/InGaAs/GaAs quantum well
Splitting of LH and HH bands due to the strain
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Optical Properties
It is possible to generalize Graf-Vogl theory
PRB 51, 4940 (1995), even when Bloch symmetry
is lost in one direction.
Absorption/gain coefficient
Distance between atoms
TB matrix elements
No additional empirical parameters are required !
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Application GaN based nanostructure
Spontaneous Pol. C/m2
Piezoelectric Pol. C/m2
-0.0368
-0.029
-0.0368
Al0.15Ga0.85N relaxed
Al0.15Ga0.85N relaxed
GaN Strained
- - - -

Surface charge density
0001
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Self-consistent results for AlGaN/GaN
P
APL 74, 2002 (1999)
Al0.2Ga0.8N
Al0.2Ga0.8N
GaN
Increasing the free charge density in the well
(sheet density) the polarization field is
screened.
4.5
4.0
n2D 1013 cm-2
For high sheet density a flat quantum well
shape is recovered.
3.5
Conduction Band Edge eV
Pumping power
3.0
BLU SHIFT vs Excitation Power Free Charge
Screening
2.5
n2D0
2.0
50
100
150
200
250
300
350
Depth Å
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Al0.15Ga0.85N/GaN Single Quantum Well
a
Oscillator strength
Fundamental transition energy
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Doping screening of polarization fields
Which is the effect of the doping with respect to
the screening of the polarization fields ?
Ionized doping density and electron density in a
high n doped AlGaN/GaN MQW
Di Carlo et al APL 76, 3950 (2000)
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Doping screening AlGaN/GaN MQW
Transition energy 1V-1C
Oscillator Strength 1V-1C
Al0.15Ga0.85N(10nm)/GaN(4nm)
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Doping screening vs. experiments
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Radiative and non-radiative recombinations
time
n increases
Field decreases
Electric Field is high
Electric Field is low
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Which is the value of the electric field?
Bernardini-Fiorentini field value
Low field value
Calculated mean field in the well of a MQW
structure with 15 nm Al0.15Ga0.85N barriers under
CW pumping (as before)
The evaluation of electric fields from optical
data is tnr dependent
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Comparison with experiments
tnr0.1 ns unscreened QWs. tnr?
pure radiative recombination The 1c-1v
transition shows non monotonic behavior. This is
a consequence of a competition between the role
of charge accumulation (mesoscopic capacitor
effect) and radiative and non-radiative
recombinations
Di Carlo et al. PRB 63, 235305 (01)
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Time Resolved PL
Decay times of photoluminescence after short
pulse excitation present varying time constant in
the decay rate, depending on the well width.
Simulation data suggest that the faster decay in
the smaller QW is due to strong spontaneous
recombination overwhelming non radiative
recombination rate.
A. Reale et al. JAP 91, 400 (03)
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Excitons in GaN MQWsTraetta, et al. APL 76, 1042
(2000)
We use the Bethe-Salpeter equation for the two
particle Green function Traetta et al. PRB59,
13196 (99)
Wave functions are calculated with both
self-consistent Envelope Function Approach and
sp3d5s tight-binding

• The exciton binding energy is reduced by
  spontaneous
• and piezoelectric polarization fields.
• Envelope Function Approximation fails for narrow
  QWs
• Crossing of the exciton binding energy for
  different
• alloy concentration induced by the
  polarization fields.

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Indium fluctuations in InGaN/Gan QWs
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Indium fluctuations
We treat the system at atomic level
(Tight-Binding) In fluctuations are obtained from
experimental results
10.000 100.000 atoms !!!!!
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Diagonalization of the Hamiltonian Matrix
If we consider many atoms the tight-binding
hamiltonian matrix become very large. For
100.000 atoms one gets an hamiltonian dimension
of 1 million
Diagonalization of such large hamiltonian matrix
can be obtained via the Lanczos iteration method
Diagonalization is achieved for states near a
given energy by using the Folded Spectrum Method
(A. Zunger)
We diagonalize the matrix
V
C
C
To diagonalize A one can use the Lanczos method
(iterative)
l
l
V
H
A
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Conduction Band Edge Profile
100 meV
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GaN/InGaN well TB results
Gleize et al. phys. stat. solidi C, 0, 298 (2003)
InGaN quantum dot in a 15 InGaN matrix 54 000
atoms Dot size 1.5 x 1.5 x 1.5 nm3
In composition of InGaN
Typical photoluminescence results show the
presence of these two peaks
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Quantum wires grown on (N11) surfaces.
Strain map
Piezoelectric charge
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Atomistic description (method)
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TB wavefunctions of the 311 wire
d gt d0
d0
EC-V(without piezo)- EC-V(with piezo) 13 meV
(15 meV experimental)
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Applications Charge Transport
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Boundary conditions for transport
active region where symmetry is lost contact
regions (semi-infinite bulk)
The transport problem is
activeregion
contact
contact
contact
Open-boundary conditions can be treated within
several schemes

• Transfer matrix
• Green Functions

These schemes are well suited for localized
orbital approach like TB
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Transfer Matrix
The idea is to define proper boundary condition
to the Schoedinger Equation in order to describe
current carrying states
Let us consider a simple 1D chain of one-orbital
atoms in the tight-binding context
where e is the orbital energy and t the
interaction
L
R
t
e
The Schroedinger eq. looking at the Ci atorm
reads
Ci-2
Ci-1
Ci
Ci1
Ci ei t Ci1 t Ci-1 E Ci
(1)
from Eq. (1)
Boundary conditions
In the left contact (L)
(2)
In the right contact (R)
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Tunneling Current
The transmitted and reflected amplitudes, T and
R, are obtained by solving the linear system in
Eq. (2).
From the transmission amplitude T we can easily
obtain the transmission coefficient T
The tunneling current is
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Transfer Matrix and TB
We can express the Schroedinger equation in a
form suited for open boundary conditions
p
where a is the orbital label (a s, px, py, pz,
)
i-th atom
s
R
L
a
transfer matrix G
Boundary conditions are defined in the
semi-infinite L and R bulk.
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Si/SiO2 tunneling Transfer Mat. sp3d5s TB
b-critobalite
b-quartz
tridymite
Staedele, et al. J. Appl. Phys. 89 348 (2001 )
Empirical parameterizations are necessary due to
the band gap problem of ab-initio approaches
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Tunneling Current Comparison with experimental
data
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Device description Green Function Approach
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Carbon Nanotubes Greens Function Molecular
dynamics
We have performed Molecular Dynamics simulations
of a reactive collision of a biased nanotube
(V100mV) and benzyne and we have calculated the
current flowing in the nanotube at each MD step
Following the NASA MD simulations (J. Han, A.
Globus, R. Jaffe, G. Deardorff)
v 0.6 Å/ps
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Time Dependent Current
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Conclusions

• Microscopic approaches such as tight-binding can
  be applied to calculate electronic and optical
  properties of realistic nanostructures.
• Tight-binding approaches can range from
  empirical to ab-initio
• For inorganic nanostructures TB avoid all the
  Envelope function approximations
• For organic nanostructures TB is the simplest
  and light quantum method
• Being a localized basis approach Green-Function
  or Transfer matrix techniques can be easily
  implemented to calculate electron transport
• Molecular dynamics and current calculations can
  be coupled together
• Non-adiabatic processes can be easily accounted
  within TB

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Where do we find these programs ?

• http//icode.eln.uniroma2.it
• There are several available codes which can be
  used directly on the web portal as a web
  application
• http//www.nanohub.org
• Many codes (also for educational purposes). See
  NEMO 3D.

http//www.wsi.tu-muenchen.de/T33/index.htm
Several codes