Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/4-bandstr-DOS.pdfEnergy band theory P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure

P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids

http://folk.uio.no/ravi/CMT2015

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

Nadu, India

Evolution of Bands in Solids

1


2


3Schematic Illustration of Bands in Insulator,

Semiconductor and Metal


4

What are the energy levels?

2 2

( )2m

k

kSommerfeld:

Bloch: For a given band index n, has

no simple explicit form. The only general

property is periodicity in the reciprocal space:

( )n k

( ) ( )n n k G k


5

Bandgap and Evaluation of Bands in Na


6

Band-Structure Measurements


7


Metals and insulators

In full band containing 2N electrons all states within the first B. Z. are occupied.

A partially filled band can carry current, a filled band cannot

Insulators have an even integer numberof electrons per primitive unit cell.

With an even number of electrons perunit cell can still have metallic behaviourdue to band overlap.

Overlap in energy need not occurin the same k direction

EF

Metal due to overlapping

bands


Full Band

Empty Band

Energy Gap

Full Band

Partially Filled Band

Energy Gap

Part Filled Band

Part Filled Band

EF

INSULATOR METAL METAL

or SEMICONDUCTOR or SEMI-METAL

EF


Insulator -energy band theory


Solid state

N~1023 atoms/cm32 atoms 6 atoms

Energy band theory


Bound States in atoms

r4

qe = )r(V

o

2

Electrons in isolated

atoms occupy discrete

allowed energy levels

E0, E1, E2 etc. .

The potential energy of

an electron a distance r

from a positively charge

nucleus of charge q is

-8 -6 -4 -2 0 2 4 6 8

-5

-4

-3

-2

-1

0

F6

F7

F8

F9

r

V(r)E2

E1

E0

Increasing

Binding

Energy


Bound and “free” states in solids

-8 -6 -4 -2 0 2 4 6 8

-5

-4

-3

-2

-1

0

F6

F7

F8

F9

r-8 -6 -4 -2 0 2 4 6 8

-5

-4

-3

-2

-1

0

F6

F7

F8

F9

r

V(r)

E2

E1

E0

The 1D potential energy

of an electron due to an

array of nuclei of charge

q separated by a distance

R is

Where n = 0, +/-1, +/-2 etc.

This is shown as the

black line in the figure.

n o

2

nRr4

qe = )r(V

+ + + + +

RNuclear

positions

V(r) lower in solid (work

function).

V(r)

Solid


Metal – energy band theory


Each atomic orbital leads to a band of allowed

states in the solid

Band of allowed states



Gap: no allowed states

Gap: no allowed states


Band Theory of Solids

What happens in crystalline solids when we bring atoms so close together that

their valence electrons constitute only one electron each?

Band structure of diamond. http://home.att.net/~mopacmanual/node372.html

http://home.att.net/~mopacmanual/node372.html


Energy-level diagram for an isolated sodium atom.


Energy-level diagram for a hypothetical Nɑ4 molecule. The

four shared, outer orbital electrons are “split” into four slightly

different energy levels, as predicted by the Pauli exclusion

principle.


Energy-level diagram for solid sodium. The discrete 3s energy

level of Fig. has given way to a pseudocontinuous energy band

(half-filled). Again, the splitting of the 3s energy level is

predicted by the Pauli exclusion principle.


Band structure

In general the eigenvalues

are in a complicated

function of k:

E(k)=f(k)

Free electron

model:

The dispersion relation between the wave

vector and the energy eigenvalues


IV-Column materials


The energy levels of the overlapping

electron shells are all slightly altered.

The energy differences are very

small, but enough so that a large

number of electrons can be in close

proximity and still satisfy the Pauli

exclusion principle.

The result is the formation of

energy bands, consisting of many

states close together but slightly

split in energy.


The energy levels are so close together that for all practical purposes we can consider bands as a continuum of states, rather than discrete energy levels as we have in isolated atoms (and in the core electrons of atoms of metals).

A detailed analysis of energy bands shows that there are as many separate energy levels in each band as there are atoms in a crystal.

Suppose there are N atoms in a crystal. Two electrons can occupy each energy level (spin), so there are 2N possible quantum states in each band.

Let’s consider sodium as an example. Sodium has a single outer 3s electron.


When you bring two sodium

atoms together, the 3s energy

level splits into two separate

energy levels.

Things to note: 4 quantum

states but only 2 electrons.

You could minimize electron energy by putting both 3s electrons

in the lower energy level, one spin up and the other spin down.

There is an internuclear separation which minimizes electron

energy. If you bring the nuclei closer together, energy increases.


When you bring five sodium

atoms together, the 3s energy

level splits into five separate

energy levels.

The three new energy levels

fall in between the two for 2

sodiums.

There are now 5 electrons occupying these energy levels.

I’ve suggested one possible minimum-energy configuration. Notice

how the sodium-sodium internuclear distance must increase slightly.


When you bring N (some big

number) sodium atoms together,

the 3s energy level splits into N

separate energy levels.

The result is an energy band,

containing N very closely-spaced

energy levels.

There are now N electrons occupying this 3s band. They go into the lowest

energy levels they can find.

The shaded area represents available states, not filled states. At the selected

separation i.e. upto EF, these are the available states.


Now let’s take a closer look at the

energy levels in solid sodium.

Remember, the 3s is the outermost

occupied level.

When sodium atoms are brought

within about 1 nm of each other, the 3s

levels in the individual atoms overlap

enough to begin the formation of the 3s

band.

The 3s band broadens as the

separation further decreases.

3s band begins to form


Because only half the states in the 3s

band are occupied, the electron energy

decreases as the sodium-sodium

separation decreases below 1 nm.

At about 0.36 nm, two things happen:

the 3s energy levels start to go up and

the 2p levels start to form a band.

Further decrease in interatomic

separation results in a net increase of

energy.

3s electron energy is minimized


What about the 3p and 4s bands

shown in the figure?

Don’t worry about them—there are no

electrons available to occupy them!

Keep in mind, the bands do exist,

whether or not any electrons are in

them.

What about the 1s and 2s energy

levels, which are not shown in the

figure?

The sodium atoms do not get close

enough for them to form bands—they

remain as atomic states.


Energy levels for an actual crystal structure also

vary with different directions in space.

http://cmt.dur.ac.uk/sjc/thesis/thesis/node39.html, band structure of silicon

http://cmt.dur.ac.uk/sjc/thesis/thesis/node39.html


Figure shows energy bands in carbon

(and silicon) as a function of

interatomic separation.

At large separation, there is a filled 2s

band and a 1/3 filled 2p band.

But electron energy can be lowered if the carbon-carbon separation is reduced.

There is a range of carbon-carbon separations for which the 2s and 2p bands overlap

and form a hybrid band containing 8N states.


But the minimum total electron

energy occurs at this carbon

carbon separation.

At this separation there is a

valence band containing 4N

quantum states and occupied by

4N electrons.

The filled band is separated by a large gap from the empty conduction band.

The gap is 6 eV—remember, kT is about 0.025 eV at room temperature. The

gap is too large for ordinary electric fields to move an electron into the

conduction band. Carbon is an insulator.


Silicon has a similar band

structure. The forbidden gap is

about 1.1 eV.

The probability of a single electron

being excited across the gap is

small, proportional to

exp(-Egap/kT).

However, there are enough 3s+3p electrons in silicon that some of them might

make it into the conduction band. We need to consider such a special case.

P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids© Copyright 2005

Sharif University of Technology1st

Workshop on Photonic Crystals

Mashad, Iran, September 2005

Brillouin Zones

BZ # 1

a 23 2 3


Brillouin Zones

BZ #2

a 23 2 3


Brillouin Zones

Irreducible BZ

a 23 2 3


Bandstructure: E versus k

The “Extended Zone scheme”

A plot of Ek with no restriction on k

But note! ψk(x) = eikx uk(x) & uk(x) = uk(x+a)

Consider (for integer n): exp[i{k + (2πn/a)}a] exp[ika]

The label k & the label [k + (2πn/a)] give the same ψk(x) (&

the same energy)! In other words, the translational symmetry

in the lattice Translational symmetry in “k space”! So, we

can plot Ek vs. k & restrict k to the range

-(π/a) < k < (π/a)

“First Brillouin Zone” (BZ)

(k outside this range gives redundant information!)


Bandstructure: E versus k

“Reduced Zone scheme”

A plot of Ek with k restricted to the first BZ.

For this 1d model, this is -(π/a) < k < (π/a)

k outside this range is redundant & gives no new information!

Illustration of the Extended & Reduced Zone schemes in 1d with

the free electron energy:

Ek = (ħ2k2)/(2mo)

Note: These are not really bands! We superimpose the 1d lattice

symmetry (period a) onto the free e- parabola.


Free e- “bandstructure” in the 1d extended zone scheme:

Ek = (ħ2k2)/(2mo)


The free e- “bandstructure” in the 1d reduced zonescheme:

Ek = (ħ2k2)/(2mo)

For k outside the 1st BZ,

take Ek & translate it into

the 1st BZ by adding

(πn/a) to k

Use the translational

symmetry in k-space as

just discussed.

(πn/a)

“Reciprocal Lattice Vector”


Reduced Brillouin zone scheme

The only independent values of k are those in the first Brillouin zone.

Discard for

|k| > /a



How to Calculate DFT Band structure

I. Solve the Kohn-Sham

equations self-consistently to

determine the effective potential

using an even k-point sampling.

Bouckaert et al., Phys. Rev 50, 58 (1938).

II. Use the effective potential

while solving the Kohn-Sham

equations non self-consistently

along high symmetry lines in the

Brillouin zone


Example: Band structure of Cu

Cu has FCC structure.

High symmetry points in the Brillouin zone:

G=center of the Brillouin zone

L=mid point on the zone boundary plane in the {111}-directions

W=corner point on the hexagon of the {kikj}-plane

K=mid point on the edge between two hexagons {110}-direction

X= mid point on the zone boundary plane in the {100}-direction

Bouckaert et al., Phys. Rev 50, 58

(1938).


Band structure of Cu

Electronic configuration of Cu: 3d94s2

Bouckaert et al., Phys. Rev 50, 58

(1938).

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

Once SCF has been achieved, we compute the bands along

the high symmetry points in the first-Brillouin zone

Variables to plot the band structure

First-Brillouin zone of a FCC , with the high symmetry points

The band structure is dumped in a file called Al.bands


The bands cross the Fermi level

(metallic character)

Bands look like parabollas,

(Al resembles a free electron gas)

Al band structure - analyze your results


The Fermi energy lies in a gap

insulator

Theo. direct gap = 5.3 eV

Expt. Gap = 7.8 eV

(LDA band gap understimation)

MgO band structure analyze your results


Mg: 1s2 2s2 2p6 3s2

Mg loses two electrons

that are gained by O

O: 1s2 2s2 2p6

Mg2+

O2-

One would expect O (one s band and three p bands) bands completely full

and Mg bands completely empty

MgO analyze your results


MgO band structure analyze your r- Detail results

O 2s

O 2p

(three bands)

The Fermi energy lies in a gap

insulator

Theo. direct gap = 5.3 eV

Expt. Gap = 7.8 eV

(LDA band gap understimation)


The Electronic Configuration of a Magnesium Atom

n l ml ms

3 0 0 +1/2

3 0 0 -1/2

Mg: (Ne)3s2

1s

2s

3s2p

3p

Empty 3p

orbitals in Mg

valence shell

How it becomes metal?

P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids16a–52

Orbital energy

levels


A representation of the energy levels (bands)

in a magnesium crystal


Nomenclature

For most purposes, it is sufficient to know the En(k) curves - the

dispersion relations - along the major directions of the reciprocal

lattice.

This is exactly what is done when real band diagrams of crystals

are shown. Directions are chosen that lead from the center of

the Wigner-Seitz unit cell - or the Brillouin zones - to special

symmetry points. These points are labeled according to the

following rules:

• Points (and lines) inside the Brillouin zone are denoted with

Greek letters.

• Points on the surface of the Brillouin zone with Roman

letters.

• The center of the Wigner-Seitz cell is always denoted by a G


For cubic reciprocal lattices, the points with a high symmetry on the

Wigner-Seitz cell are the intersections of the Wigner Seitz cell with

the low-indexed directions in the cubic elementary cell.

Nomenclature

simple

cubic


Nomenclature

We use the following nomenclature: (red for fcc, blue for bcc):

The intersection point with the [100] direction is

called X (H)

The line G—X is called D.


called K (N)

The line G—K is called S.


called L (P)

The line G—L is called L.

Brillouin Zone for fcc is bcc

and vice versa.


We use the following nomenclature: (red for fcc, blue for bcc):


called X (H)

The line G—X is called D.


called K (N)

The line G—K is called S.


called L (P)

The line G—L is called L.

Nomenclature


Real crystals are three-dimensional and we must consider

their band structure in three dimensions, too.

Of course, we must consider the reciprocal lattice, and, as

always if we look at electronic properties, use the Wigner-

Seitz cell (identical to the 1st Brillouin zone) as the unit cell.

There is no way to express quantities that change as a

function of three coordinates graphically, so we look at a

two dimensional crystal first (which do exist in

semiconductor and nanoscale physics).

Electron Energy Bands in 3D


59

Bands in 3D

In 3D the band structure is

much more complicated than in

1D because crystals do not

have spherical symmetry.

The form of E(k) is dependent

upon the direction as well as the

magnitude of k.

Figure removed to reduce

file size

GermaniumGermanium


The lower part (the "cup") is

contained in the 1st Brillouin zone,

the upper part (the "top") comes

from the second BZ, but it is folded

back into the first one. It thus

would carry a different band index.

This could be continued to infinity;

but Brillouin zones with energies

well above the Fermi energy are of

no real interest.

These are tracings along major

directions. Evidently, they contain

most of the relevant information in

condensed form. It is clear that

this structure has no band gap.

Electron Energy Bands in 3D


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


65Band 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.


67

First Brillouin zone E vs. k band

diagram of zincblende semiconductors

One relevant conduction band isformed from s- like atomic orbitals

“unit cell” part of wavefunction isapproximately spherically symmetric.

The three upper valence bands areformed from (three) p- like orbitalsand the spin-orbit interaction splits offlowest, “split-off” hole (i. e., valence)band.

The remaining two hole bandshave the same energy (“degenerate”)at zone center, but their curvature isdifferent, forming a “heavy hole” (hh)band (narrower), and a “light hole” (lh)band (broad)


SnO2: band structure

VBM

CBM

Density of States


69


70

How to get conduction in Si?


71

Doping Silicon with Donors (n-type)


72

Doping Silicon with Acceptors (p-type)


73

Atomic Density

for Si


74

Summary of n- and p-type Silicon


75


76


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


78


79

The Bandgap Problem of DFT


80

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


81

Baandgap Error in Semiconductors from LDA


82Calculated Badgap values of Si from various level of Calculation


83GWA calculation of Bandgap of Semiconductors


84

Graphite band structure (Semi Metal)


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)

Documents

Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/4-bandstr-DOS.pdfEnergy band theory P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure