Chap 8 Nearly free and tightly bound electrons
• (<<) Nearly-free electron
• lattice perturbation
• empty lattice approximation
• Fermi surface
• (>>) Tightly bound electron
• Linear combination of atomic orbitals
• Wannier function
Special topic:
Geometric phase in crystalline solid
Dept of Phys
M.C. Chang
Nearly-free electron
Lattice perturbation to plane wave
(ε
0
k −G
)
− ε k Ck (G ) + ∑ U G ' −G Ck (G ') = 0
G'
2 2
⎛ 0
⎞
k
U
ε
≡
>>
⎜ k
G ⎟
m
2
⎝
⎠
• O-th order:
ε
By iteration,
• Let G=G1,
(0)
k
(ε
=ε
0
k −G
0
k −G1
⎧⎪1 when G = G1
C (G ) = ⎨
⎪⎩0 when G ≠ G1
(0)
k
;
)
− ε k Ck (G ) + U G −G Ck0 (G1 ) = 0
⇒ ε
(1)
k
=ε
1
0
k −G1
• Let G≠G1,
⇒ C (G ) =
(1)
k
+ U 0 + O (U )
2
U G −G
1
ε k0−G − ε k0−G
1st order energy
correction
1st order state
correction
1
• If, for G=G2≠G1,
ε k0−G ≅ ε k0−G , then the perturbation above fails.
2
1
Nearly-free electron (for illustration, consider 1D)
i ( k −G ) x
• The Bloch state ψ nk ( x) = ∑ Cnk (G )e
G
is a superposition of … exp[i(k-g)x], exp[ikx], exp[i(k+g)x] …
ε
Free electron:
-g
exp[i(k+g)x] exp[ikx] exp[i(k-g)x]
0
g
k
Under a weak perturbation:
• If k～0, then the most significant component of ψ1k(x) (at low energy)
is exp[ikx] (little superposition from other plane waves)
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• If k～g/2, then the most significant components of ψ1k(x) and ψ2k(x)
(at low energy) are exp[i(k-g)x] and exp[ikx], others can be neglected.
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Degenerate perturbation
• If {G1, G2,… Gm} give similar energy ε0k-G
(and are away from other energy levels), then
• for G≠ {G1, G2,… Gm}, one has
m
∑U
C (G ) =
i =1
(1)
k
0
C
(Gi )
k
G −G
i
ε k0−G − ε k0−G
, G ≠ G1
i
• for G= {G1, G2,… Gm}, one has
(ε
0
k −Gi
)
m
− ε k Ck (Gi ) + ∑ U G
j =1
j − Gi
Ck (G j ) +
∑U
G '≠ Gi
G ' −Gi
Ck (G ') = 0
or
⎛ ε k0−G − ε k
1
⎜
⎜ U G1 −G2
⎜
⎜
⎜
⎜ U G −G
m
1
⎝
U G −G
2
⎞ ⎛ C (G ) ⎞
⎟⎜ k 1 ⎟
⎟⎜
⎟
⎟⎜
⎟=0
U G −G ⎟
m
m −1
⎜
⎟
⎟
⎜
ε k0−Gm − ε k ⎟⎠ ⎝ Ck (Gm ) ⎟⎠
UG
m − G1
1
UG
m −1 − Gm
→ 1st order eigen-energy and 0-th order eigen-states
For example, m=2
near |k|=|k-G|,
⎛ ε k0 − ε k
⎜
⎜ U
⎝ − G2
⎞ ⎛ Ck (0) ⎞
⎟⎜
⎟⎟ = 0
0
⎜
⎟
(
)
C
G
ε k −G − ε k ⎠ ⎝ k
⎠
UG
21
• Energy eigenvalues
ε k0 + ε k0−G
⎛ ε k0 − ε k0−G
(1)
εk± =
± ⎜
⎜
2
2
⎝
G
→ ε k0 = ε k0−G when k ⋅ Gˆ =
2
2
⎞
⎟⎟ + U G
⎠
∴ for a k near a Bragg plane, need to use
degenerate perturbation and the energy
correction is of order U
2
Back to the example with m=2,
• Bloch states with q on the Bragg plane
ε k(1)± = ε k0 ± | U G |
⎛ ∓U G
⇒⎜ *
⎜ UG
⎝ 2
⎞ ⎛ Ck (0) ⎞
⎟⎟ ⎜
⎟=0
Ck (G ) ⎠
2 ⎠⎝
UG
∓U G
• From inversion symmetry, UG is real, then
⎛ Ck (0) ⎞ 1 ⎛ 1 ⎞
⎜
⎟=
⎜ ⎟
C
G
(
)
2
⎝ ±1⎠
⎝ k
⎠
ψ k(0)± (r ) = Ck ± (0)eik ⋅r + Ck ± (G )ei ( k −G )⋅r
⎧
2 ⎛ G⋅r ⎞
2
cos
⎪
⎜
⎟
2
2
⎪
⎝
⎠
,
⇒ ψ k(0)± (r ) = ⎨
⎪
2 ⎛ G⋅r ⎞
2sin
⎜
⎟
⎪
2
⎝
⎠
⎩
Bragg reflection at BZB forms two standing wave
with a finite energy difference (energy gap)
Higher Brillouin zones
Reduced zone scheme
1
2
2
2
2
3
3
3 3
3 3
3
3
Same area
• At zone boundary, k points to the plane bi-secting the G vector,
thus satisfying the Laue condition
G
k ⋅ Gˆ =
2
G
k
• Bragg reflection at zone boundaries produce energy gaps (Peierls, 1930)
Beyond the 1st
Brillouin zone
BCC crystal
FCC crystal
“Empty lattice” in 2D
2D square lattice
Free electron in vacuum:
2
k2
εk =
2m
Free electron in empty lattice:
ε k = ε nk ′ =
2
(k′ + G )
2
2m
k = k′ + G
k ′ ∈1st BZ
M
2π/a
• How to fold a parabolic “surface” back to the first BZ?
Γ
X
Folded parabola along ΓX (reduced zone scheme)
z For U≠0, there are
energy gaps at BZ
boundaries
M
Γ
2π/a
X
Empty FCC lattice
Energy bands for empty FCC lattice
along the Γ-X direction.
Comparison with real band structure
The energy bands for
“empty” FCC lattice
Actual band structure for
copper (FCC, 3d104s1)
d bands
From Dr. J. Yates’s ppt
Fermi surface for (2D) empty lattice
For a monovalent element,
the Fermi wave vector
3
2
k F = 2π a
For a divalent element
k F = 4π a
For a trivalent element
k F = 6π a
Distortion due to lattice potential
1
Fermi surface of alkali metals (monovalent, BCC lattice)
kF = (3π2n)1/3
n = 2/a3
→ kF = (3/4π)1/3(2π/a)
ΓN=(2π/a)[(1/2)2+(1/2)2]1/2
∴ kF = 0.877 ΓN
Fermi spheres of alkali metals
Percent deviation of k from the
free electron value
Fermi surface of noble metals (monovalent, FCC lattice)
Band structure
(empty lattice)
kF = (3π2n)1/3,
n = 4/a3
→ kF = (3/2π)1/3(2π/a)
ΓL= ___
kF = ___ ΓL
Fermi surface
(a cross-section)
Fermi surfaces of noble metals
Periodic zone scheme
Fermi surface of Al (trivalent, FCC lattice)
1st BZ
• Empty lattice approximation
2nd BZ
• Actual Fermi surface
Tightly bound electron
Tight binding model:
Energy bands as an extension of atomic orbitals
• Covalent solid
• d-electrons in
transition metals
• Alkali metal
• noble metal
"We have the rather curious result that not only is it possible to obtain
conduction with bound electrons, but it is also possible to obtain nonconduction with free electrons.“ A. Wilson
important
Tight binding method (Bloch, 1928)
Let am(r) be the eigenstate of an electron in the potential Uat(r)
of an isolated atom.
H at am (r ) = ε am (r )
at
m
atomic
orbital
Consider a crystal with N atoms at lattice sites R,
• A wave function with translation symmetry
(but still not an energy eigenstate)
ϕmk (r ) = ∑ d k ( R)am (r − R)
R
=
Check:
1
N
∑ eik ⋅R am (r − R)
R
Linear combination of
atomic orbitals (LCAO)
1
eik ⋅R ' am (r − ( R '− R))
∑
N R'
1
eik ⋅R ' am (r − R ') = eik ⋅Rϕ mk (r )
=eik ⋅R
∑
N R'
ϕmk (r + R) =
important
An energy eigenstate (Bloch state)
ψ nk (r ) = ∑ Cmnϕ mk (r )
define
Schrödinger equation
S mm '
( )
(k ) =
H mm ' k = ϕ mk H ϕm ' k
m
p2
+ U (r )
H=
2m
H ψ nk = ε n ψ nk
then
∑(H
mm '
ϕmk ( H − ε n ) ψ nk = 0
m'
n
m'k
1
N
∑
amR H am ' R ' e
(
− ik ⋅ R − R '
R,R '
where r am ' R ' ≡ am ' (r − R ').
⇒
mk
− ε n Smm ' )Cmn ' = 0
m'
i H mm ' =
∑ ϕ (H −ε ) ϕ
ϕmk ϕm ' k
C =0
n
m'
i Smm '
1
=
N
∑
amR am ' R ' e
(
− ik ⋅ R − R '
)
R,R '
=δ mm ' +∑ amR am '0 e
− ik ⋅ R
R≠0
≡ α mm ' ( R)
Overlap
integral
)
,
important
△U(r)
p2
H=
+ U at + (U − U at )
2m
= H at + ΔU
H mm ' =
1
N
∑
amR H am ' R ' e
am(r)
U(r)
(
− ik ⋅ R − R '
)
R,R '
= am 0 H am '0 + ∑ amR H am '0 e
− ik ⋅ R
R ≠0
i
am 0 H am '0 = am 0 H at + ΔU am '0
=δ mm 'ε mat + am 0 ΔU am '0
≡ β mm '
i
amR H am '0 =ε mat amR am '0 + amR ΔU am '0
=ε α mm ' ( R) + γ mm ' ( R)
at
m
⇒ H mm ' = δ mm 'ε mat + β mm ' + ε matα mm ' (k ) + γ mm ' (k )
energy shift due to the
potential of neighboring
atoms. (U in Marder’s)
inter atomic matrix element
between nearby atoms.
(t in Marder’s)
n
H
ε
S
C
−
(
)
∑ mm ' n mm ' m ' = 0
∼ same status as the central eq. in NFE model
m'
⇓
∑ ⎡⎣ε (δ
at
m
m'
mm '
)
(
)
+ α mm ' (k ) + β mm ' + γ mm ' (k ) ⎤Cmn ' = ε n ∑ δ mm ' + α mm ' (k ) Cmn '
⎦
m'
i.e.
AC = ε n BC ⇒
(B A) C = ε C
−1
n
: an eigenvalue problem
i so far no approximation has been used!
Approximation 1:
Approximation 2:
The ranks of A, B depend on
the number of atomic orbitals am
Keep only a few overlap integrals
(e.g. for NN and NNN)
• s-orbital, m=1
α mm ' ( R) = amR am '0
• p-orbital, m=1…3
γ mm ' ( R) = amR ΔU am '0
• d-orbital, m=1…5
• s-p mixing, m=1…4 etc
β mm ' = am 0 ΔU am '0
(no R-dependence)
Example: s-band from the s-orbital (m=1)
(
)
(
)
⎡ε sat 1 + α (k ) + β + γ (k ) ⎤ C n = ε n 1 + α (k ) C n
⎣
⎦
⇒ ε n = ε sat +
β + γ (k )
1 + α (k )
α ( k ) = ∑ α ( R )e
− ik ⋅ R
R≠0
; α ( R ) = ∫ d 3 r as* (r − R )as (r )
β = ∫ d 3 r as* (r )ΔUas (r )
γ ( k ) = ∑ γ ( R )e
− ik ⋅ R
R≠0
; γ ( R ) = ∫ d 3 r as* (r − R )ΔUas (r )
• If we keep only the NN integrals, then
(α (− R) = α (− R); γ (− R) = γ (− R) have been used )
α (k ) = 2
α 0 cos k ⋅ R , α 0 = α ( RNN )
∑
γ0
half of NN
γ (k ) = 2
( )
cos ( k ⋅ R ) ,
∑
half of NN
γ 0 = γ ( RNN )
Square lattice
• 3D
εn
ε sat + β + γ (k )
=2
∑
(
)
γ 0 cos k ⋅ R +const.
Energy contours
half of NN
=2γ 0 ( cos k x a + cos k y a + cos k z a )
• 2D
ε n =2γ 0 ( cos k x a + cos k y a )
• 1D
ε n =2γ 0 cos ka
Density of states
From Dr. P. Young’s at UCSC
Wannier function (1937)
1
i ψ nk (r ) =
eik ⋅R Cmn am (r − R)
∑∑
N R m
let wn (r − R) = ∑ Cmn am (r − R)
m
1
N
1
⇔ wn (r − R) =
N
i.e.
ψ nk (r ) =
∑e
ik ⋅ R
wn (r − R)
R
∑
k ∈1st BZ
e − ik ⋅Rψ nk (r )
ψ nk ψ n ' k ' = δ nn 'δ k ,k ' ⇔
(localized)
wnR wn ' R ' = δ nn 'δ R , R '
An orthonormal set
Comparison
Bloch state
Wannier function
• Energy eigenstate
not an energy eigenstate
i TRψ nk = eik ⋅Rψ nk
( Pn rPn ) wnR = RwnR
• Extended function
localized function
• orthonormal basis
orthonormal basis
Kivelson, PRB, ‘82
(for 1D)
Wannier function for the Kronig-Penny model
Pedersen et al, PRB 1991
Tight-binding model (TBM)
• As a basis, Wannier functions are
better than atomic orbitals:
H=
∑
wnR
wnR wn ' R ' = δ nn 'δ R , R '
amR am ' R ' ≠ δ mm 'δ R , R '
p2
, H=
+ U (r )
2m
wnR H wn ' R ' wn ' R '
nR , n ' R '
• One-band approx. (omit n)
H TB = ∑ wR H R , R ' wR ' ,
H R , R ' ≡ wnR H wnR '
R,R '
≈ ∑ U R wR
R
wR + ∑ t R ,δ wR
R ,δ
wR +δ , U R ≡ H R , R ; t R ,δ = H R , R +δ
• For a uniform system, U, t are indep of R,
then by
1
N
wR =
∑
k ∈1st BZ
On-site
energy
(usually are treated
as parameters)
e − ik ⋅R ψ k
⇒ H TB = ∑ ε k ψ k ψ k ,
k
Hopping
amplitude
ε k =U + t ∑ eik ⋅δ
δ
Cf: spectrum from LCAO
Geometric phase
(aka Berry phase)
Brief introduction of the Berry phase
Adiabatic evolution of a quantum system
• Energy spectrum:
H ( r , p; λ )
• After a cyclic evolution
E(λ(t))
n+1
x
n
x
λ (T ) = λ (0)
ψ n , λ (T )
n-1
0
λ(t)
• Phases of the snapshot states at different λ’s are
independent and can be arbitrarily assigned
ψ n ,λ (t ) → eiγ
n (λ )
ψ n ,λ ( t )
• Do we need to worry about this phase?
i
T
dt ' En ( t ')
∫
0
=e
ψ n ,λ ( 0 )
−
Dynamical phase
No!
• Fock, Z. Phys 1928
• Schiff, Quantum Mechanics (3rd ed.) p.290
Pf :
Consider the n-th level,
Ψ λ (t ) = e
iγ n ( λ )
t
i
dt ' En ( t ')
∫
0
e
ψ n ,λ
−
H Ψ λ (t ) = i
∂
Ψ λ (t )
∂t
γ n = i ψ n ,λ
∂
ψ n ,λ ⋅ λ ≠ 0
∂λ
Stationary,
snapshot state
Hψ n ,λ = Enψ n ,λ
≣An(λ)
• Redefine the phase, (gauge transformation)
ψ 'n ,λ = eiφ
n (λ )
ψ n ,λ
An’(λ) = An(λ) − ∂φn
∂λ
• Choose a φ (λ) such that,
An’(λ)=0
Thus removing the extra phase.
One problem:
∇λφ = A(λ )
does not always have a
well-defined (global) solution
A
Vector flow
Vector flow
Contour of φ
A
φ is not
defined here
Contour of φ
C
C
γ (T ) − γ (0 ) =
∫
C
A⋅dλ = 0
∫
C
A⋅dλ ≠ 0
M. Berry, 1984 :
Parameter-dependent phase NOT always removable!
T
i
dt ' E ( t ')
∫
0
=e e
ψ λ (0)
iγ C
ψ λ (T )
−
Index n neglected
• Berry phase (gauge independent, path dependent)
γC =
∫
C
A(λ ) ⋅ d λ
• Berry connection (or Berry potential)
A(λ ) ≡ i ψ λ ∇ λ ψ λ
(“R” in Marder’s)
λ3
• Berry curvature (or Berry field)
λ (t)
Ω(λ ) ≡ ∇ λ × A(λ ) = i ∇ λψ λ × ∇ λψ λ
S
• Stokes theorem (3-dim here, can be higher)
γC =
∫
C
A ⋅ d λ = ∫ Ω ⋅ da
S
λ1
C
λ2
Berry phase in crystalline solid (for the n-th band)
H (k )unk = ε nk unk
⎛∇
⎞
where H (k ) =
k
+
⎜
⎟ + U (r )
2m ⎝ i
⎠
2
2
• Berry phase
γn =
∫
C
unk i
∂
unk ⋅ dk
∂k
• Berry connection
An (k ) = unk i∇ k unk
kz
• Berry curvature
k
Ω n (k ) = ∇ k × An (k ) = i ∇ k unk × ∇ k unk
• Stokes theorem
γn =
∫
C
S
kx
An ⋅ dk = ∫ ∇ k × An ⋅ d 2 k
S
C
ky
‹ Symmetry and Berry curvature
For non-degenerate band:
• Space inversion
symmetry
Ω n (−k ) = Ω n (k )
both symmetries
• Time reversal
symmetry
Ω n (−k ) = −Ω n (k )
Ω n (k ) = 0, ∀ k
‹ When could we see nonzero Berry curvature?
Ω n (k ) ≠ 0
• SI symmetry is broken
← electric polarization
• TR symmetry is broken
← QHE
• band crossing
← “monopole”
For more, see Xiao D. et al, Rev Mod Phys 2010
Berry phase (crossing the BZ) in one dimension
Dirac comb model
g1=5 g2=4
…
…
0
b
a
• For a 1D lattice with inversion symm,
Berry phase can only be 0 or π
Lowest
energy
band: γ1
← g2=0
γ1=π
r =b/a
Rave and Kerr, EPJ B 2005
Realistic Berry curvature for BCC Fe
→ (intrinsic) anomalous Hall effect
From Dr. J. Yates’s ppt
(Karplus and Luttinger, 1954)
• In addition to ε n(k), there is a 2nd fundamental quantity Ω n(k)