The Bandstructure Problem
A one-dimensional model
(“easily generalized” to 3D!)
A One-Dimensional Bandstructure Model
• 1 e- Hamiltonian: H = (p)2/(2mo) + V(x)
p  -iħ(∂/∂x),
V(x)  V(x + a)
V(x)  Effective Potential. Repeat Distance = a.
• GOAL: Solve Schrödinger’s Equation:
Hψk(x) = Ekψk(x)
k  eigenvalue label
• First, it’s convenient to define
The Translation Operator  T.
• For any function, f(x), T is defined by:
T f(x)  f(x + a)
Translation Operator  T.
• For any f(x), T is defined by:
T f(x)  f(x + a)
• For example, if we take f(x) = ψk(x)
• ψk(x)  Wave function solution to the
Schrödinger Equation:
T ψk(x) = ψk(x + a) (1)
• Now, find the eigenvalues of T:
T ψk(x)  λkψk(x)
(2)
λk  An eigenvalue of T.
Translation Operator  T.
T ψk(x) = ψk(x + a) (1)
• The eigenvalues of T:
T ψk(x)  λkψk(x)
(2)
λk  An eigenvalue of T.
• Using (1) & (2) together, it is fairly easy to show:
λk  eika and ψk(x)  eikx uk(x)
With uk(x)  uk(x + a)
• For proof, see texts by BW or YC, as well as Kittel’s
Solid State Physics text & any number of other SS books
• This shows that the translation operator applied to an
eigenfunction of the Schrödinger Equation (of the
Hamiltonian H which includes a periodic potential) gives:
Tψk(x) = eika ψk(x)
 ψk(x) is also an eigenfunction of
the translation operator T!
• It also shows that the general form of ψk(x) must be:
ψk(x) = eikx uk(x) where
uk(x) is a periodic function with the
same period as the potential!
That is uk(x) = uk(x+a)
• In other words: For a periodic potential V(x),
with period a, ψk(x) is a simultaneous
eigenfunction of the translation operator T &
the Hamiltonian H
• From The Commutator Theorem of QM, this is
equivalent to [T,H] = 0. The commutator of T & H
vanishes so they commute!
 They share a set of eigenfunctions.
• In other words: The eigenfunction (the
electron wavefunction!) always has the form:
ψk(x) = eikx uk(x) with
uk(x) = uk(x+a)
 “Bloch’s Theorem”
Bloch’s Theorem
From translational symmetry
• For a periodic potential V(x), the eigenfunctions
of H (wavefunctions of e-) always have the form:
ψk(x) = eikx uk(x) with uk(x) = uk(x+a)
 “Bloch Functions”
• Recall, for a free e-, the wave functions have the form:
ψfk(x) = eikx (a plane wave)
 A Bloch function is the generalization of a
plane wave for an e- in periodic potential. It is
a plane wave modulated by a periodic function
uk(x) (with the same period as V(x)).
Bandstructure
A one dimensional model
• The wavefunctions of the e- have in a periodic crystal
Must always have the Bloch function form
ψk(x) = eikxuk(x), uk(x) = uk(x+a) (1)
• It is conventional to label the eigenfunctions &
eigenvalues (Ek) of H by the wavenumber k:
p = ħk  e- “quasi-momentum”
(rigorously, this is the e- momentum for a free e-, not for a Bloch e-)
• So, the Schrödinger Equation is
Hψk(x) = Ekψk(x)
where ψk(x) is given by (1) and
Ek  The Electronic “Bandstructure”.
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 (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 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 eparabola.
Free e- “bandstructure” in the 1d
extended zone scheme: Ek = (ħ2k2)/(2mo)
• Free e- “bandstructure” in the 1d reduced zone scheme:
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”
Bandstructure
To illustrate these concepts, we now discuss an EXACT
solution to a 1d model calculation (BW Ch. 2, S, Ch. 2)
The Krönig-Penney Model
• Developed in the 1930’s & is in MANY SS
& QM books
Why do this simple model?
• The process of solving it shares contain MANY
features of real, 3d bandstructure calculations.
• Results of it contain MANY features of real,
3d bandstructure results!
The Krönig-Penney Model
• Developed in the 1930’s & is in MANY SS & QM books
Why do this simple model?
• The results are “easily” understood & the math
can be done exactly. We won’t do this in class. It
is in the books!
A 21st Century Reason
to do this simple model!
It can be used as a prototype for the
understanding of artificial semiconductor
structures called SUPERLATTICES!
(Later in the course?)
A QM Review: The 1d (finite) Rectangular
Potential Well In most QM texts!!
• We want to solve the Schrödinger Equation for:
We want bound
states: ε < Vo
[-{ħ2/(2mo)}(d2/dx2) + V]ψ = εψ (ε  E)
V = 0, -(b/2) < x < (b/2); V = Vo otherwise
• Solve Schrödinger’s Equation:
[-{ħ2/(2mo)}(d2/dx2)
+ V]ψ = εψ
-(½)b
(½)b
(ε  E), V = 0, -(b/2) < x < (b/2)
V = Vo otherwise
The bound states are
in Region II
• In Region II:
ψ(x) is oscillatory
• In Regions I & III:
ψ(x) is exponentially
decaying
Vo
V= 0
The 1d (finite) Rectangular Potential Well
A brief math summary!
• Define: α2  (2moε)/(ħ2); β2  [2mo(ε - Vo)]/(ħ2)
• The Schrödinger Equation becomes:
(d2/dx2) ψ + α2ψ = 0, -(½)b < x < (½)b
(d2/dx2) ψ - β2ψ = 0, otherwise.
Solutions:
ψ = C exp(iαx) + D exp(-iαx), -(½)b < x < (½)b
ψ = A exp(βx),
x < -(½)b
ψ = A exp(-βx),
x > (½)b
Boundary Conditions:
 ψ & dψ/dx are continuous SO:
• Algebra (2 pages!) leads to:
(ε/Vo) =
2
2
(ħ α )/(2moVo)
ε, α, β are related to each other by
transcendental equations. For example:
tan(αb) = (2αβ)/(α 2- β2)
• Solve graphically or numerically.
• Get: Discrete energy levels in the well
(a finite number of finite well levels!)
• Even eigenfunction solutions (a finite number):
Circle, ξ2 + η2 = ρ2, crosses η = ξ tan(ξ)
Vo
o
o
b
• Odd eigenfunction solutions:
Circle, ξ2 + η2 = ρ2, crosses η = -ξ cot(ξ)
Vo
b o
o
b
|E2| < |E1|
The Krönig-Penney Model
Repeat distance a = b + c.
Periodic potential: V(x) = V(x + na)
n = integer
• Periodically Repeated Wells & Barriers.
The Schrödinger Equation is:
[-{ħ2/(2mo)}(d2/dx2) + V(x)]ψ = εψ
V(x) = Periodic potential
The Krönig-Penney Model
Repeat distance a = b + c.
Periodic potential: V(x) = V(x + na)
n = integer
• Periodically Repeated Wells & Barriers.
• The Schrödinger Equation is:
[-{ħ2/(2mo)}(d2/dx2) + V(x)]ψ = εψ
 The Wavefunction must have the Bloch form:
ψk(x) = eikx uk(x); uk(x) = uk(x+a)
Boundary conditions at x = 0, b: ψ, (dψ/dx) are continuous 
• The Math Manipulation is A MESS!
• But it’s doable EXACTLY!
• Instead of an explicit form for the bandstructure
εk or ε(k), we get:
k = k(ε) = (1/a) cos-1[L(ε/Vo)]
OR L = L(ε/Vo) = cos(ka)
WHERE L = L(ε/Vo) =
L = L(ε/Vo) = cos(ka)  -1< L< 1
• ε in this range are the allowed energies
(BANDS!)
• But also, L(ε/Vo) = messy function with no limit on L.
There is no real solution for ε in the range where |L| >1
 These are regions of forbidden energies
(GAPS!)
• No solutions exist there for real k
• Math solutions exist, but k is imaginary
• The wavefunctions have Bloch form for all k (& all L):
ψk(x) = eikx uk(x)
 For imaginary k, ψk(x) decays instead of propagating!
Krönig-Penney Results (For particular a, b, c, Vo)
• Each band has a
finite well level
“parent”.
L(ε/Vo) = cos(ka)
 -1< L< 1
• But also
L(ε/Vo) = messy
function with no
Limits. For ε in the
range -1 < L < 1
 Finite

Well
Levels


 Allowed energies (bands!) For ε when |L| > 1
 Forbidden energies (gaps!) (no solutions exist
for real k)
• Every band in the Krönig-Penney Model has a
finite well discrete level as its “parent”!
Evolution from the finite well to the periodic potential
 In its implementation, the Krönig-Penney model is
similar to the “almost free” e- approach, but the
results are similar to the tightbinding approach!
(As we’ll see). Each band is associated with an
“atomic” level from the well.
More on Krönig-Penney Solutions
L(ε/Vo) = cos(ka)  BANDS & GAPS!
• The Gap Size: Depends on the c/b ratio
• Within a band (see previous Figure) a good
approximation is that L ~ a linear function of ε. Use this
to simplify the results:
• For (say) the lowest band, let ε  ε1 (L = -1) & ε  ε2 (L = 1)
use the linear approximation for L(ε/Vo). Invert this & get:
ε-(k) = (½) (ε2+ ε1) - (½)(ε2 - ε1)cos(ka)
• For the next lowest band,
ε+(k) = (½) (ε4+ ε3) + (½)(ε4 – ε3)cos(ka)
• In this approximation, all bands are cosine
functions!!! This is identical, as we’ll see, to some
simple tightbinding results.
Lowest Krönig-Penney Bands
• In the linear
approximation
for L(ε/Vo): All
bands are cos(ka)
functions!
• Plotted in the
extended zone
scheme.
ε = (ħ2k2)/(2m0)
• Discontinuities at the BZ edges, at k = (nπ/a)
• Because of the periodicity of ε(k), the reduced
zone scheme (red) gives the same information as
the extended zone scheme (as is true in general).