Pseudopotential Method - Solid State Physics - Lecture Slides, Slides of Solid State Physics

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The Pseudopotential Method

Builds on all of this.

-^ Given

O ψ (r),^ k^ we want to solve an

Effective Schrödinger

Equation

for the valence e

-^ alone (for the bands

E):k

O H ψ (r) = Ek

O ψ (r)kk

OIn ψ (r)k^ now replace

ikr^ ewith a more general expression

f ψ (r):k^

O ψ (r) =k

f ψ (r) +k

∑β (k)n

ψ (r)n

Put this into (1) & manipulate. This involves^ H ψ (r)n

^ E^ ψ n

(r)^ n

(2) is the

- Core e Schrödinger Equation. -^ Core eenergies & wavefunctions

E^ &^ ψ (r)n^ n^

are assumed to be known. (^2) H = (p) /(2m^ ) + V(r)o^ V(r)^ ^ True Crystal Potential

-^ (3) is an Effective Schrödinger Equation

^ The Pseudo-Schrödinger Equation for the smooth part of the valence e

-^ wavefunction

(& for^ E^ ):k

H´ ψ (r) = Ek

ψ (r)^ kk

(The^ f^ superscript on

f ψ (r)^ has been dropped). So we finally get ak^

Pseudo-Hamiltonian:

H´^ 

H + V´

or^ H´= (p)

2 /(2m^ ) + [V(r) + V´]o^ or^ H´= (p)

2 /(2m^ ) + Vo^

(r), whereps

V^ (r) = V(r)ps

+ V´

^ The “Pseudopotential”

Now, we want to solve

The Pseudo-Schrödinger Equation^2 [(p)^ /(2m

)+V^ ops

(r)] ψ k

(r) = E

ψ (r)kk^

Of course we put

p = -i ħ 

In principle, we could use the formal expression for V(r)^ (a “smooth”, “small” potential),ps^

including the messy

sum over core states from

V´.

BUT,^

this is^

almost NEVER

done!

A Typical Real Space

Pseudopotential

(In the Direct Lattice)

A Typical k-Space

Pseudopotential

(In the Reciprocal Lattice)

a 11 docsity.com

• Pseudopotential Form FactorsUsed as fitting parameters in theempirical pseudopotential methods V
  • s V
  • s V V
  • a V
  • a V

Pseudopotential Effective Masses( Γ -point)

Compared to experiment! Ge^ GaAs

InP^

InAs^ GaSb

InSb^

CdTe docsity.com

Pseudopotential Bands of GaAs & ZnSe

GaAs^

ZnSe

( Direct^ bandgap)

( Direct^ bandgap)

^ Eg

^ E^ g

Recall that^

our^ GOALS

were that

after this chapter, you should:

1. Understand the underlying Physics behindthe existence of bands & gaps.2. Understand how to interpret a

bandstructure diagram.

3. Have a rough, general idea about howrealistic bands are calculated.4. Be able to calculate the energy bands forsome simple models of a solid