evolution of bands in solids - universitetet i oslofolk.uio.no/ravi/cutn/ccmp/4b-bandstr-dos.pdfthe...

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 19 June 2015 Evolution of Band structure of Solids

http://folk.uio.no/ravi/MENA9520-15

Prof.P. Ravindran, Department of Physics, Central University of Tamil

Nadu, India

&Center for Materials Science and Nanotechnology,

University of Oslo, Norway

Evolution of Bands in Solids

1


Band Structure: KCl

We first depict the band structure of an ionic crystal, KCl. The bands are very

narrow, almost like atomic ones. The band gap is large around 9 eV. For alkali

halides they are generally in the range 7-14 eV.


Band Structure: silver (fcc)


Band Structure: tungsten (bcc)


Empirical pseudopotential method

Energy band of Si, Ge and Sn

Empirical pseudopotential

method

Si Ge Sn


6Band structure of semiconductor

Calculated energy band structure of

Silicon

Calculated energy band structure of

GaAs

- Interband transitions : The excitation or relaxation of electrons between

subbands

- Indirect gap : The bottom of the conduction band and the top of the

valence band do not occur at the same k

- Direct gap : The bottom of the conduction band and the top of the

valence band occur at the same k


DOS is Nanomaterials

In bulk (a), layered (b) and wire (c) materials, there are always states populated which do not contribute to gain. These are parasitic states and contribute to inefficiency.

In quantum dot (d) materials, the DOS is a set of discrete states. Theory predicts this type of material is ideal for the gain region of a laser because fewer parasitic states are occupied.


8

First Brillouin zone E vs. k band

diagram of zincblende semiconductors

One relevant conduction band is

formed from s- like atomic orbitals“unit cell” part of wavefunction is

approximately spherically symmetric.

The three upper valence bands are

formed from (three) p- like orbitalsand the spin-orbit interaction splits off

lowest, “split-off” hole (i. e., valence)

band. The remaining two hole bands

have the same energy (“degenerate”)at zone center, but their curvature is

different, forming a “heavy hole” (hh)

band (broad), and a “light hole” (lh)band (narrower)


SnO2: band structure

VBM

CBM

Density of States


10


11

How to get conduction in Si?


12

Doping Silicon with Donors (n-type)


13

Doping Silicon with Acceptors (p-type)


14

Atomic Density

for Si


15

Summary of n- and p-type Silicon


16


17


Introduction to Silicon

6

The appearance of Band Gap, separating CB and VB

The 6 CB minima are not located at the center of 1st

Brillouin zone, INDIRECT GAP

CB VB-H VB-L

1st Brillouin zone of Diamond

lattice

CB

VB

Anisotropy in surface of E


19


20

The Bandgap Problem of DFT


21

The Bandgap problem [Sham,Schluter, PRL, 51, 1888 (1983).


22

Baandgap Error in Semiconductors from LDA


23Calculated Badgap values of Si from various level of Calculation


24GWA calculation of Bandgap of Semiconductors


25

Graphite band structure (Semi Metal)

P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues

Fermi level

As Fermions are added to an energy band, they will fill the

available states in an energy band just like water fills a

bucket.

The states with the lowest energy are filled first, followed by

the next higher ones.

At absolute zero temperature (T = 0 K), the energy levels are

all filled up to a maximum energy, which we call the Fermi

level. No states above the Fermi level are filled.


22/3

2

( ) 3( )

8F

hc NE

mc V

FERMI ENERGY


Fermi surface sampling for metallic systems

The determination of the Fermi level might be delicate for metallic systems

Slightly different choices of k-points can lead to bands

entering or exiting the sum, depending if a given

eigenvalue is above or below the Fermi level.

Band structure of bulk Al

For this k-point, three

bands are occupied

For this k-point, two

bands are occupied

For this k-point, one

band is occupied

For a sufficiently dense Brillouin zone sampling, this should not be a problem


For the k-points close to the Fermi surface, the

highest occupied bands can enter or exit the sums

from one iterative step to the next, just because the

adjustement of the Fermi energy

Difficulties in the convergence of the self-consistence procedure

with metals: smearing the Fermi surface

Instability of the self-consistent procedure

Solution 1: Use small self-consistent mixing coefficients

Solution 2: Smear the Fermi surface introducing a distribution of occupation number

The occupations are not any longer 1 (if below EF) or 0 (if above EF)

Gaussians

Fermi functions

C. –L. Fu and K. –M. Ho, Phys. Rev. B 28, 5480 (1989)


Smearing the Fermi surface: the Electronic Temperature

is a broadening energy parameter that is adjusted to avoid instabilities

in the convergence of the self-consistent procedure. It is a technical

issue. Due to its analogy with the Fermi distribution, this parameter is

called the Electronic Temperature

For a finite , the BZ integrals converge faster but to incorrect values. After

self-consistency has been obtained for a relatively large value of Tc , this has to

be reduced until the energy becomes independent of it.


Comparing energies of structures having

different symmetries: take care of BZ samplings

The BZ sampling of all the structures must be sampled

with the same accuracy

Since for unit cells of different shapes it is not possible to

choose exactly the same k-point sampling, a usual strategy is

to try and maintain the same density of k-points


CONTACT POTENTIAL

1 2contactV

e


BZ integration, “FERMI”-methods Replace the “integral” of the BZ by a finite summation on a mesh of “k-points”

weights wk,n depend on k and bandindex n (occupation)

– for full “bands” the weight is given by “symmetry”

w(G)=1, w(x)=2, w(D)=4, w(k)=8

shifted “Monkhorst-Pack” mesh

– for partially filled bands (metals) one must find the

Fermi-energy (integration up to NE) and determine

the weights for each state Ek,n

linear tetrahedron method (TETRA, eval=999)

linear tetrahedron method + “Bloechl” corrections (TETRA)

“broadening methods”

– gauss-broadening (GAUSS 0.005)

– temperature broadening (TEMP/TEMPS 0.005)

– broadening useful to damp scf oszillations, but dangerous (magnetic moment)

kk

nk

nknknk

EE

n

wkdrFn

*

,

,

3

,

*

,)(

G D X


Relativistic treatment

Valence states

– Scalar relativistic

mass-velocity

Darwin s-shift

– Spin orbit coupling on demand by

second variational treatment

Semi-core states

– Scalar relativistic

– on demand

spin orbit coupling by second

variational treatment

Additional local orbital (see Th-6p1/2)

Core states

– Fully relativistic

Dirac equation

For example: Ti


Relativistic semi-core states in fcc Th

additional local orbitals for

6p1/2 orbital in Th

Spin-orbit (2nd variational method)

J.Kuneš, P.Novak, R.Schmid, P.Blaha, K.Schwarz,

Phys.Rev.B. 64, 153102 (2001)

evolution of bands in solids - universitetet i oslofolk.uio.no/ravi/cutn/ccmp/4b-bandstr-dos.pdfthe...

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