LectureReview of quantum mechanics, statistical physics, and solid stateBand structure of materialsSemiconductor band structureSemiconductor nanostructures

Ref. Ihn Ch. 3, Yu&Cardona Ch. 2

Reminder from previous lectures

The existence of this periodic potential is the cause of the bands & the gaps!

Gaps: occur at the k where the electron waves (incident on atoms & scattered from atoms) undergo constructive interference (Bragg reflections!)

Realistic Bandstructures for SemiconductorsBandstructure Theory Methods are

Highly Computational.Calculational methods fall into 2 general categories which have their roots in 2 qualitatively very different physical pictures for e-

in solids:

“Physicist’s View”: Start from the “almost free” e- & add a periodic potential in a highly sophisticated, self-consistent

manner. Pseudopotential Methods“Chemist’s View”: Start with the atomic picture & build up

the periodic solid from atomic e- in a highly sophisticated, self-consistent manner.

Tightbinding/LCAO methods

Qualitative Picture #1“A Physicist’s viewpoint”- The solid is looked at “collectively”

• Almost Free ElectronsFor free electrons,

E(k) = (p)2/2mo = (ħk)2/2mo

• Almost Free Electrons: Start with the free electron E(k), add small (weakly perturbing) periodic potential V.– This breaks up E(k) into bands (allowed energies) &

gaps (forbidden energy regions).

Qualitative Picture #1Forms the basis for REALISTIC bandstructure computational methods!

• Starting from the almost free electron viewpoint & adding a high degree of sophistication & theoretical + computational rigor:

Results in a method that works VERY WELL forcalculating E(k) for metals & semiconductors!An “alphabet soup” of computational techniques:

• OPW: Orthogonalized Plane Wave method• APW: Augmented Plane Wave method• ASW: Antisymmetric Spherical Wave method• Many, many others

The Pseudopotential Method(the modern method of choice!)

• Atomic / Molecular ElectronsAtoms (with discrete energy levels) come together to form the solid

– Interactions between the electrons on neighboring atoms cause the atomic energy levels to split, hybridize, & broaden. (Quantum Chemistry!) First approximation: Small interaction V!

– Occurs in a periodic fashion (the interaction V is periodic). – Groups of levels come together to form bands (& also gaps).– The bands E(k) retain much of the character of their “parent”

atomic levels (s-like and p-like bands, etc.)

Qualitative Picture #2“A Chemist’s viewpoint”- The solid is looked at as a collection of atoms & molecules.

• Starting from the atomic / molecular electron viewpoint & adding a high degree of sophistication & theoretical & computational rigor

Results in a method that works VERY WELL forcalculating E(k) (mainly the valence bands) for insulators &

semiconductors! (Materials with covalent bonding!) An “alphabet soup” of computational techniques:

• LCAO: Linear Combination of Atomic Orbitals method• LCMO: Linear Combination of Molecular Orbitals method

The “Tightbinding” method & many others.

Qualitative Picture #2Forms the basis for REALISTIC bandstructure computational methods!

Theories of Bandstructures in Crystalline Solids

Pseudopotential Method Tightbinding (LCAO) Method

Semiconductors,Insulators

Electronic Interaction with lattice

Metals

Almost Free Electrons

Molecular Electrons

Free Electrons

Isolated Atom, Atomic Electrons

Method #1 (Qualitative Physical Picture #1): “Physicists View”:Start with free e- & add periodic potential.

The “Almost Free” e- Approximation• First, it’s instructive to start even simpler, with FREE electrons. Superimpose

the symmetry of the diamond & zincblende lattices on the free electron energies:

“The Empty Lattice Approximation”Diamond & Zincblende BZ symmetry superimposed on the free e- “bands”.

This is the limit where the periodic potential V 0But, the symmetry of BZ (lattice periodicity) is preserved.

Why do this?It will (hopefully!) teach us some physics!!

• The Free Electron Energy is:E(k) = ħ2[(kx)2 +(ky)2 +(kz)2]/(2mo)

• So, superimpose the BZ symmetry (for diamond/zincblende lattices) on this energy.

• Then, plot the results in the reduced zone scheme

Free Electron “Bandstructures”“The Empty Lattice Approximation”

Free Electrons: ψk(r) = eikr

• Superimpose the diamond & zincblende BZ symmetry on the ψk(r). This symmetry reduces the number of k’s needing to be considered. For example, from the BZ, a “family” of equivalent k’s along (1,1,1) is:

(2π/a)(1, 1, 1)• All of these points map the Γ point = (0,0,0) to

equivalent centers of neighboring BZ’s. The ψk(r) for these k are degenerate (they

have the same energy).

Diamond “Empty Lattice” Bands(Reduced Zone Scheme)

E(k) = ħ2[(kx)2 +(ky)2 +(kz)2]/(2mo)

(111) (100)These E(k) show some features of real bandstructures.If a finite potential is added: Gaps will open up at the BZ edge

Pseudopotential Bandstructures• A highly computational version of this (V is

not treated as perturbation!)

Pseudopotential Method

For more details, see many pages in BW, Ch. 3 & YC, Ch. 2.

Pseudopotential Method (Overview)• Use Si as an example (could be any material, of course).• Electronic structure of an isolated Si atom:

1s22s22p63s23p2

• Core electrons: 1s22s22p6

– Do not affect electronic & bonding properties of solid! Do not affect the bands of interest.

• Valence electrons: 3s23p2

– They control bonding & all electronic properties of solid. These form the bands of interest!

Si Crystallizes in the tetrahedral, diamond structure.The 4 valence electrons Hybridize & form 4 sp3 bonds with

the 4 nearest neighbors. (Quantum) CHEMISTRY!!!!!!

The Pseudopotential Bandstructure of Si Note the qualitative similarities of these to the bands of

the empty lattice approximation.

Empty lattice

Eg

Pseudopotential Bands of Si & Ge

Si GeBoth have indirect bandgaps

Eg Eg

Pseudopotential Bands of GaAs & ZnSe

GaAs ZnSe(Direct bandgap) (Direct bandgap)

Eg Eg

Spin-Orbit CouplingFirst Some General Comments

• An Important (in some cases) effect we’ve left out! • We’ll discuss it mainly for terminology & general

physics effects only.• The Spin-Orbit Coupling term in the Hamiltonian:

Comes from relativistic corrections to the Schrödinger Equation.

• It’s explicit form isHso = [(ħ2)/(4mo

2c2)][V(r) p]σV(r) The crystal potential

p = - iħ The electron (quasi-) momentumσ the Pauli Spin Vector

• The cartesian components of the Pauli Spin Vector σ are 2 2 matrices in spin space:

σx = ( ) σy = ( ) σz = ( )Hso has a small effect on electronic bands.

It is most important for materials made of heavieratoms (from down in periodic table).

• It is usually written

Hso = λLSThis can be derived from the previous form

with some manipulation!

0 11 0

0 -ii 0

1 00 -1

• The Spin-Orbit Coupling Hamiltonian:Hso = λLSλ A constant

“The Spin-Orbit Coupling Parameter”.Sometimes, in bandstructure theory, this parameter is called .

L orbital angular momentum operator for the e-.S spin angular momentum operator for the e-.

Hso adds to the Hamiltonian from before, & is used to solve the Schrödinger Equation. The new H is:

H = (p)2/(2mo) + Vps(r) + λLS

Now, solve the Schrödinger Equation with this H. Usepseudopotential or other methods & get bandstructures as before.

Hso = λLSSpin-Orbit Coupling’s most

important & prominent effect is:Near band minima or maxima at high symmetry points in BZ:

Hso Splits the Orbital Degeneracy.

• The most important of these effects occurs near the valence band maximum at the BZ center at

Γ = (0,0,0)

Hso = λ LS• The most important effect occurs at the top of

the valence band at Γ = (0,0,0). In the absence of Hso, the bands there are

p-like & triply degenerate. • Hso partially splits that degeneracy. It gives rise

to the “Spin-Orbit Split-Off” band, or simply the “Split-Off” band.

• Also, there are “heavy hole” & “light hole” bands at the top of valence band at Γ. (YC use the kp method & group theory to discuss this in detail.)

Schematic Diagram of the bands of a Direct Gap material near the Γpoint, showing Heavy Hole, Light hole, & Split-Off valence bands.

Calculated bands of Si and Ge near the Γ point, showing Heavy Hole, Light Hole, & Split-Offvalence bands.

Si Ge

Experimental bands of Si and Ge near the Γ point, showing Heavy Hole, Light Hole, & Split-Offvalence bands.

Si Ge

Calculated bands of GaAs near the Γpoint, showing Heavy Hole, Light Hole, & Split-Offvalence bands.

The Tightbindingor

LCAO Approach to Bandstructure Theory

Bandstructures

REMINDER: Theories fall into 2 general categories, which have their roots in 2 qualitatively very different physical pictures for e- in solids:“Physicist’s View”: Start from an “almost free” e- & add the periodic potential in a highly sophisticated, self-consistent manner. Pseudopotential Methods“Chemist’s View”: Start with atomic e- & build up the periodic solid in a highly sophisticated, self-consistent manner. Tightbinding or LCAO Methods

Now, we’ll focus on the 2nd method.

Method #2 (Qualitative Physical Picture #2)“A Chemists Viewpoint”

• Start with the atomic/molecular picture of a solid.• The atomic energy levels merge to form molecular

levels, & merge to form bands as periodic interatomic interaction V turns on.

Tightbindingor

Linear Combination of Atomic Orbitals (LCAO) method.

• This method gives good bands, especially valence bands!• The valence bands are ~ almost the same as those from the

pseudopotential method! Conduction bands are not so good!

QUESTIONHow can 2 (seemingly) completely differentapproaches (pseudopotential & tightbinding) lead to essentially the same bands? (Excellent agreement with valence bands; conduction bands are not too good!).

ANSWER(partial, from YC):

The electrons in the conduction bands are ~ “free” & delocalized. The electrons in the valence bands are ~ in the bonds in r space. The valence electrons in the bonds have atomic-like character. (So, LCAO is a “natural” approximation for these).

The Tightbinding Method

• The Tightbinding / LCAO method gives a much clearer physical picture (than pseudopotential method does) of the causes of the bands & the gaps.

• In this method, the periodic potential V is discussed as in terms of an Overlap Interaction of the electrons on neighboring atoms.

• As we’ll see, we can define these interactions in terms of a small number of PHYSICALLY APPEALING parameters.

First: Qualitative diatomic molecule discussionConsider a 2 atom molecule ABwith one valence e- per atom, & a strongcovalent bond. Assume that the atomicorbitals for A & B, ψA & ψB, are known.Now, solve the Molecular SchrödingerEquation as a function of the A-Bseparation. The Results are:

A Bonding State(filled, 2 e-. Spin-up & Spin-down ) &

An Antibonding State (empty)

Bonding StateΨ- = (ψA + ψ B)/(2)½

Antibonding StateΨ+ = (ψA - ψ B)/(2)½

Bond Center (Equilibrium Position)

Tightbinding Method• “Jump” from 2 atoms to 1023 atoms!

The bonding & antibonding states broaden to become bands.

• A gap opens up between the bonding & the antibonding states (due to the crystal structure & the atom valence).

Valence bands: Occupied Correspond to bonding levels in the molecular picture.

Conduction bands: Unoccupied Correspond to antibonding levels in the molecular picture.

Schematic: Atomic Levels Broadening into Bands

In the limit asa , the atomic levels forthe isolated atomscome back

a0

s-like AntibondingStates

s-like Bonding States

p-like Bonding States

p-like AntibondingStates

a0 material lattice constant

Schematic: Evolution of Atomic-Molecular Levels into Bands

p antibonding

Fermi Energy, EF

s antibonding

s bonding Molecule

Isolated Atoms & p Orbital Energies

Solid (Semiconductor) Bands The Fundamental Gap is on

both sides of EF!

p antibonding

p bonding

Fermi Energy, EF

Schematic Evolution of s & p Levels into Bands at the BZ Center (Si)

Atom Solid

EG

Lowest Conduction

Band

Fermi Energy Highest

Valence Band

The Tightbinding(LCAO) Method

A Realistic Treatment of Semiconductor Materials.

Tightbinding MethodRealistic Treatment for Semiconductor Materials!

• For most of the materials of interest, in the isolated atom, the valence electrons are in s & p orbitals.

• Before looking at the bands in the solid, lets first briefly &QUALITATIVELY

look at the molecular orbitals for the bonding & antibonding states.

• A Quantitative treatment would require us to solve theMolecular Schrödinger Equation

That is, it would require us to do some CHEMISTRY!!

• Now, a mostly qualitative review of elementary molecular physics.

Shapes of charge (& probability) densities |ψ|2 for atomic s & p orbitals:

s orbitals are sphericallysymmetric!

p orbitals have directional lobes!

The px lobe is along the x-axis

The py lobe is along the y-axis

The pz lobe is along the z-axis

Wavefunctions Ψ and energy levels ε formolecular orbitals in Diatomic Molecule AB

An s-electron on atom A bonding with an s-electron on atom B.

Result: A bonding orbital (occupied; symmetric on exchange of A & B)

Ψ = (ψsA+ ψsB)/(2)½

A antibonding orbital (unoccupied; antisymmetric on exchange of A & B)

Ψ = (ψsA - ψsB)/(2)½

Ψ for a σ antibonding orbital

ε for a σ antibonding orbitalε for atomic

s electrons

For a homopolar molecule(A = B)

ψsA ψsB

Ψ for a σ bonding orbital

ε for a σ bonding orbital

Wavefunctions Ψ & energy levels ε for molecular orbitals in a Diatomic Molecule AB

An s-electron on atom A bondingwith an s-electron on atom B.

For a heteropolar molecule(A B)

ε for atomic s electrons on atoms A & B

Ψ for σ bonding orbital

Ψ for σ antibonding orbital

ε for σ antibonding orbital

ε for σ bonding orbitalResult: A bonding orbital (occupied; symmetric on exchange of A & B)

Ψ = (ψsA+ ψsB)/(2)½

A antibonding orbital (unoccupied; antisymmetric on exchange of A & B)

Ψ = (ψsA - ψsB)/(2)½

Charge (probability) densities |Ψ|2 for molecular orbitals in a Diatomic Molecule AB

An s-electron on atom Abonding with an s-electron

on atom B to get

bonding (+) & antibonding (-)

molecular orbitals.

bonding orbital:Ψ = (ψsA+ ψsB)/(2)½

(occupied; symmetric on exchange of A & B)

antibonding orbitalΨ = (ψsA - ψsB)/(2)½

(unoccupied; antisymmetric on exchange of A & B)

Simple examplesof the

Tightbinding (LCAO) Method

Tightbinding: 1 Dimensional Model #1• Consider an Infinite Linear Chain of identical atoms, with 1

s-orbital valence e- per atom & interatomic spacing = a

• Approximation: Only Nearest-Neighbor interactions.(Interactions between atoms further apart than a are ~ 0).

This model is called the “Monatomic Chain”.

Each atom has s electron orbitals only!Near-neighbor interaction only means that the s orbital on site

n interacts with the s orbitals on sites n – 1 & n + 1 only!

n = Atomic Label a

n = -3 -2 -1 0 1 2 3 4

• The periodic potential V(x) for this Monatomic Linear Chain of atoms looks qualitatively like this:

n = -4 -3 -2 -1 0 1 2 3

a

V(x) = V(x + a)

• The localized atomic orbitals on each site for this Monatomic Linear Chain of atoms look qualitatively like this:

n = -4 -3 -2 -1 0 1 2 3

The spherically symmetric s orbitals on each site overlap slightly with those of their neighbors, as shown. This allows the electron on site n to interact with its nearest-neighbors on sites n – 1 & n + 1!

a

The True Hamiltonian in the solid is:H = (p)2/(2mo) + V(x), with V(x) = V(x + a).

• Instead, approximate it as H ∑n Hat(n) + ∑n,nU(n,n)where, Hat(n) Atomic Hamiltonian for atom n.U(n,n) Interaction Energy between atoms n & n.

Use the assumption of only nearest-neighbor interactions: U(n,n) = 0 unless n = n -1 or n = n +1

• With this assumption, the Approximate Hamiltonian is

H ∑n [Hat(n) + U(n,n -1) + U(n,n + 1)]

H ∑n [Hat(n) + U(n,n -1) + U(n,n + 1)] • Goal: Calculate the bandstructure Ek by solving the

Schrödinger Equation:HΨk(x) = Ek Ψk(x)

• Use the LCAO (Tightbinding) Assumptions:1. H is as above.2. Solutions to the atomic Schrödinger Equation are known:

Hat(n)ψn(x) = Enψn(x) 3. In our simple case of 1 s-orbital/atom:

En= ε = the energy of the atomic e- (known)4. ψn(x) is very localized around atom n5. The Crucial (LCAO) assumption is:

Ψk(x) ∑neiknaψn(x) That is, the Bloch Functions are linear combinations of atomic orbitals!

• Dirac notation: Ek Ψk|H|Ψk(This Matrix Element is shorthand for a spatial integral!)

• Using the assumptions for H & Ψk(x) already listed: Ek = Ψk|∑n Hat(n) |Ψk + Ψk|[∑nU(n,n-1) + U(n,n-1)]|Ψk

also note that Hat(n)|ψn = ε|ψn

• The LCAO assumption is: |Ψk ∑neikna|ψn• Assume orthogonality of the atomic orbitals:

ψn |ψn = δn,n (= 1, n = n; = 0, n n)

• Nearest-neighbor interaction assumption: There is nearest-neighbor overlap energy only! (α = constant)

ψn|U(n,n 1)|ψn - α; (n = n, & n = n 1)

ψn|U(n,n 1)|ψn = 0, otherwiseIt can be shown that for α > 0, this must be negative!

“Energy band” of this model is:Ek= ε - 2αcos(ka) or Ek = ε - 2α + 4α sin2[(½)ka]

• A trig identity was used to get last form. ε & α are usually taken as parameters in the theory, instead of actually calculating them from the atomic ψn

The “Bandstructure” for this monatomic chain with nearest-neighbor interactions only looks like (assuming 2α < ε ): (ET Ek - ε + 2α)

It’s interesting to note that:The form Ek= ε - 2αcos(ka)is similar to Krönig-Penney model results in the linear approximation for the messy transcendental function! There, we got:

Ek = A - Bcos(ka) where A & B were constants.

ET

4α

n = -1 0 1

Tightbinding: 1 Dimensional Model #2A 1-dimensional “semiconductor”!

• Consider an Infinite Linear Chain consisting of 2 atom types, A& B (a crystal with 2-atom unit cells), 1 s-orbital valence e- per atom & unit cell repeat distance = a.

• Approximation: Only Nearest-Neighbor interactions.(Interactions between atoms further apart than ~ (½ )a are ~ 0).

This model is called the “Diatomic Chain”.

A B A B A B A

a

The True Hamiltonian in the solid is:H = (p)2/(2mo) + V(x), with V(x) = V(x + a).

• Instead, approximate it (with γ = A or =B) asH ∑γnHat(γ,n) + ∑γn,γnU(γ,n;γ,n)

where, Hat(γ,n) Atomic Hamiltonian for atom γ in cell n.U(γ,n;γ,n) Interaction Energy between atom of type γ

in cell n & atom of type γ in cell n.Use the assumption of only nearest-neighbor interactions:

The only non-zero U(γ,n;γ,n) are U(A,n;B,n-1) = U(B,n;A,n+1) U(n,n-1) U(n,n+1)

• With this assumption, the Approximate Hamiltonian is:H ∑γnHat(γ,n) + ∑n[U(n,n -1) + U(n,n + 1)]

H ∑γnHat(γ,n) + ∑n[U(n,n -1) + U(n,n + 1)] • Goal: Calculate the bandstructure Ek by solving the Schrödinger Equation:

HΨk(x) = EkΨk(x)• Use the LCAO (Tightbinding) Assumptions:

1. H is as above.2. Solutions to the atomic Schrödinger Equation are known:

Hat(γ,n)ψγn(x) = Eγnψγn(x)3. In our simple case of 1 s-orbital/atom:

EAn= εA = the energy of the atomic e- on atom AEBn= εB = the energy of the atomic e- on atom B

4. ψγn(x) is very localized near cell n5. The Crucial (LCAO) assumption is:

Ψk(x) ∑neikna∑γCγψγn(x) That is, the Bloch Functions are linear combinations of atomic orbitals!

Note!! The Cγ’s are unknown

• Dirac Notation: Schrödinger Equation: Ek Ψk|H|ΨkψAn|H|Ψk = EkψAn|H|Ψk (1)

• Manipulation of (1), using LCAO assumptions, gives (student exercise):εAeiknaCA+ μ[eik(n-1)a + eik(n+1)a]CB = EkeiknaCA (1a)

• Similarly: ψBn|H|Ψk = EkψBn|H|Ψk (2) • Manipulation of (2), using LCAO assumptions, gives (student exercise):

εBeiknaCB+ μ[eik(n-1)a + eik(n+1)a]CA = EkeiknaCA (2a)

Here, μ ψAn|U(n,n-1)|ψB,n-1 ψBn|U(n,n+1)|ψA,n+1= constant (nearest-neighbor overlap energy) analogous to α in the previous 1d model

• Student exercise to show that these simplify to:0 = (εA - Ek)CA + 2μcos(ka)CB , (3)

and0 = 2μcos(ka)CA + (εB - Ek)CB, (4)

• εA, εB , μ are usually taken as parameters in the theory, instead of computing them from the atomic ψγn

• (3) & (4) are linear, homogeneous algebraic equations for CA & CB

22 determinant of coefficients = 0• This gives: (εA - Ek)(εB - Ek) - 4 μ2[cos(ka)]2 = 0

A quadratic equation for Ek! 2 solutions: a “valence” band

& a “conduction” band!

• Results:“Bandstructure” of the Diatomic Linear Chain (2 bands):

E(k) = (½)(εA + εB) [(¼)(εA - εB)2 + 4μ2 {cos(ka)}2]• This gives a k = 0 bandgap of

EG= E+(0) - E-(0) = 2[(¼)(εA - εB)2 + 4μ2]½

• For simplicity, plot in the case

4μ2 << (¼)(εA - εB)2 & εB > εA

Expand the [ ….]½ part of E(k) & keep the lowest order term

E+(k) εB + A[cos(ka)]2, E-(k) εA - A[cos(ka)]2

EG(0) εA – εB + 2A , where A (4μ2)/|εA - εB|

“Bandstructure” of a 1-dimensional “semiconductor”:

Tightbinding Method: 3 Dimensional Model• Model: Consider a monatomic solid, 3d, with only

nearest-neighbor interactions. Hamiltonian: H = (p)2/(2mo) + V(r)

V(r) = crystal potential, with the full lattice symmetry & periodicity.

• Assume (R,R = lattice sites):H ∑RHat(R) + ∑R,RU(R,R)

Hat(R) Atomic Hamiltonian for atom at RU(R,R) Interaction Potential between atoms at R & R

Near-neighbor interactions only! U(R,R) = 0 unless R & R are nearest-neighbors

• Goal:Calculate the bandstructure Ek by solving the Schrödinger Equation:

HΨk(r) = EkΨk(r)• Use the LCAO (Tightbinding) Assumptions:

1. H is as on previous page. 2. Solutions to the atomic Schrödinger Equation are known: Hat(R)ψn(R) = Enψn(R), n = Orbital Label (s, p, d,..),

En= Atomic energy of the e- in orbital n3. ψn(R) is very localized around R4. The Crucial (LCAO) assumption is:

Ψk(r) = ∑ReikR∑nbnψn(r-R) (bn to be determined)ψn(R): The atomic functions are orthogonal for different n & RThat is, the Bloch Functions are linear combinations ofatomic orbitals!