Space group symmetry So far we have covered point group symmetry and translation symmetry seperately. We need to combine them → space group. While there are an infinite number of point groups, there are only a finite number that are consistent with translation → 32 crystallographic point groups We focused on a few point groups (all crystallographic) in our examples: C i , C 3v , C 4v . Highest symmetry point group is O h (cubic group, h = 48). There are 230 space groups (http://www.cryst.ehu.es/). Space group symmetry elements have a point and translation component: { ˆ R i , t + τ } (where τ is additional shift). 11/12/2020 at 14:24:48 slide 1
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Space group symmetry - Chris Marianetti...Space group symmetry elements have a point and translation component: fR^i;t + ˝g(where ˝is additional shift). 11/21/2019 at 14:00:12 slide
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Space group symmetry
So far we have covered point group symmetry and translationsymmetry seperately.
We need to combine them → space group.
While there are an infinite number of point groups, there areonly a finite number that are consistent with translation→ 32 crystallographic point groups
We focused on a few point groups (all crystallographic) in ourexamples: Ci ,C3v ,C4v .
Highest symmetry point group is Oh (cubic group, h = 48).
There are 230 space groups (http://www.cryst.ehu.es/).
Space group symmetry elements have a point and translationcomponent: {Ri , t + τ} (where τ is additional shift).
11/12/2020 at 14:24:48 slide 1
Symmorphic space group
We will only concern ourselves with symmorhphic, as opposedto non-symmorhphic, space groups (73/230 are symm).
Non-symmorhphic space groups (e.g. diamond structure) areslightly more complicated and can be left for another day.
Symmorphic are those with all operations of the form: {Ri , t}→ no glide or screw operations.
A sufficient condition to be symmorhphic is:
Only one species, all on lattice sites.If more than one species, each must form a lattice (e.g. NaCl).At least one point in space has full point symmetry of lattice.
Basic strategy is to start with irreps of translation (ie.k-points) and then use whatever point symmetry remains.
11/12/2020 at 14:24:48 slide 2
Some notes on vector notation for real vectors
Consider usual column vectors vi , and orthogonal operationsO, where Ovi is some new column vector.
When taking a dot product, we can explicitly use matrixnotation, or the dot symbol will account for transpose.
v1 · v2 = v2 · v1 = vᵀ1 v2 = vᵀ
2 v1 = |v1||v2| cos(θ)
We can insert identity 1 = OᵀO in different notations.
vᵀ1 v2 = vᵀ
1 OᵀOv2 = (Ov1)ᵀOv2 ↔ v1 · v2 = (Ov1) · (Ov2)
We have chosen most quantities to be row vectors, such as t,T , k , K , etc, and so k · t = ktᵀ, etc.
Recall equality
K · T = (kb) · (ta) = KT ᵀ = kbaᵀtᵀ = 2πktᵀ = 2πk · t
11/12/2020 at 14:24:48 slide 3
Recall 2d transformations of Cartesian space for C4v
Recall all operations that leave square invariant.a
b
x
y
Below are matrices which transform Cartesian space.
1 =
[1 00 1
]Rπ/2 =
[0 1−1 0
]Rπ =
[−1 00 −1
]R3π/2 =
[0 −11 0
]
Rma =
[0 11 0
]Rmb =
[0 −1−1 0
]Rmx =
[1 00 −1
]Rmy =
[−1 00 1
]
These matrices can be used to build any representation.
Same symbol (e.g. Rmx) used for different reps (e.g.triangluar, square, hexagon molecule).
11/12/2020 at 14:24:48 slide 4
Final points on notation
Given point operations in usual column format, then we haveRT ᵀ; changing column vector to row format gives TRᵀ.
A column operation R operators on a row vector as TRᵀ.
We normally prefer to use lattice coordinates, not Cartesian.
Recall that Ti = ti a, all row ordered. Then we have:
RjT ᵀ = Rj aᵀtᵀ = rjtᵀ rj = Rj aᵀ
We have introduced notation r for operation transforminglattice coordinate.
A column operation r operates on row vector as trᵀj .
11/12/2020 at 14:24:48 slide 5
Point operations on irreps on translation
Recall vectors that transform like irreps of translation
|uαk 〉 =∑
te i2πk·t |uαt 〉 =
∑
t1t2
e2πi(k1t1+k2t2)|uα(t1,t2)〉
〈uαk | =∑
te−i2πk·t〈uαt | =
∑
t1t2
e−2πi(k1t1+k2t2)〈uα(t1,t2)|
Recall that k and t are row vectors.Effect of point group operation centered at some point:
R|uαk 〉 =∑
te i2πkt
ᵀR|uαt 〉 =
∑
te i2πkrᵀ r tᵀRTtR−1RT−1
t |uαt 〉
=∑
te i2π(r kᵀ)ᵀ r tᵀRTtR−1R|uα0 〉 =
∑
te i2π(r kᵀ)ᵀ r tᵀTr tᵀR|uα0 〉
=∑
t′e i2π(krᵀ)·t′Tt′R|uα0 〉 =
∑
t′e i2π(krᵀ)·t′Tt′R|uα0 〉
There are two possible scenarios:
krᵀ = k + G krᵀ = k ′11/12/2020 at 14:24:48 slide 6
Point operations on irreps on translation
If krᵀ = k + G , then we know that the irrep of translationgroup is unchanged:
In the latter case, we actually transform the vector to anentirely different k-point:
R|uαk 〉 =∑
β
Γ(R)α,β|uβk ′〉
In any case, given that [R, V ] = 0, we know that:
V |φnk〉 = mω2n,k |φnk〉 → V R|φnk〉 = mω2
n,k R|φnk〉 = mω2n,k |φnk ′〉 → ωn,k = ωn,krᵀ
11/12/2020 at 14:24:48 slide 7
Implications of point symmetry
We see that if R maps the k-point to a distinct one, it must
have the same Eigenvalue: ω(n)k = ω
(n)krᵀ = ω
(n)k ′
Is this a degeneracy? Yes, as the Eigenfunctions are related bya symmetry operation (see Cornwell, 1997).
Let us return to our 1d chain example (point group Ci )ω
k12−1
2
2√
γm
Given that σkᵀ = −kᵀ, we find that the mirror impliesωk = ωk
11/12/2020 at 14:24:48 slide 8
Implications of point symmetry
The maximum degeneracy can be the order of the pointgroup, in the case that the given k-vector is mapped todistinct ones.
In practice, this is not normally called a degeneracy, eventhough it is. Dresselhaus goes so far as to say that it is not.
The important takeaway message is that we only need tocompute a subset of k-point in FBZ that are not related by apoint group operation.
This gives rise to the Irreducible Brilluoin Zone (IBZ)
Once Eigenfunctions in IBZ are known, the rest of FBZ canbe constructed using point operations.
11/12/2020 at 14:24:48 slide 9
Star of the wave vector k and the IBZ
Given a wave vector k , we can form a set by acting with allpoint operations → “identity star of wave vector”.
This is also sometimes called an “identity orbit”.
Consider two stars for the 1d chain (point group Ci ).
In the second case, there is ony one wavevector in the stargiven that the other is equivalent by a reciprocal lattice vector.
The IBZ (green) is simply a collection of one member of allpossible stars in the FBZ (light blue).
The smaller the star, the higher the point symmetry of thatk-point. i.e. the more Ri where krᵀi = k + GAll Ri that leave k invariant is known as “little group” ofk-point. Always subgroup of crystal PG in symmorphic case.
11/12/2020 at 14:24:48 slide 10
IBZ for square lattice and the “little group”
Γ
Σ
R
X∆
Shaded green region is IBZ.
All points in FBZ can be generated frompoints in IBZ using an element of C4v
Must find point symmetry for every pointin IBZ (aka “little group” of k-point) .
Most generally, this info given by “Wycoffpositions” (real or reciprocal).
Why do we care about the little group? These operationsleave k-vector invariant so we may further block diagonalizeour Hamiltonian.
Little group of k is group of Dk (ie. elements that commute)
11/12/2020 at 14:24:48 slide 11
Symmetry of k-points in IBZ
Γ
Σ
R
X∆
Which elements R of C4v
satisfy krᵀ = k + GΓ→ C4v
R → C4v
X → {1,Rπ, σx , σy} → C2v
∆→ {1, σx} → Ci
Σ→ {1, σx} → Ci
All other points → {1}
http://www.cryst.ehu.es/
All of this information haspreviously been tabulated.
11/12/2020 at 14:24:48 slide 12
Square Lattice - Dispersion
We can already understand origin of degeneracy.
Only C4v has multidimensional irreps.
0
0.5
1
1.5
2
2.5
ω
γd=0.1
γσ=1
γπ=1/3
Γ ΓX R
11/12/2020 at 14:24:48 slide 13
Square Lattice - Dispersion
How do |uxk〉 and |uyk〉 transform at each k-point?
Γ(C4v ),∆(Ci ),X (C2v ),R(C4v ),Σ(Ci )
0
0.5
1
1.5
2
2.5
ω
γd=0.1
γσ=1
γπ=1/3
Γ ΓX R
D(k1,k2) = 2
γσ(1− cos 2πk1) + γπ(1− cos 2πk2)+γd [2− cos 2π(k1 + k2)− cos 2π(k1 − k2)]
γd [− cos 2π(k1 + k2) + cos 2π(k1 − k2)]
γd [− cos 2π(k1 + k2) + cos 2π(k1 − k2)]γπ(1− cos 2πk1) + γσ(1− cos 2πk2)+γd [2− cos 2π(k1 + k2)− cos 2π(k1 − k2)]
11/12/2020 at 14:24:48 slide 14
Square Lattice - Dispersion
0
0.5
1
1.5
2
2.5
ω
γd=0.1
γσ=1
γπ=1/3
Γ ΓX R
D(k1,k2) = 2
γσ(1− cos 2πk1) + γπ(1− cos 2πk2)+γd [2− cos 2π(k1 + k2)− cos 2π(k1 − k2)]
γd [− cos 2π(k1 + k2) + cos 2π(k1 − k2)]
γd [− cos 2π(k1 + k2) + cos 2π(k1 − k2)]γπ(1− cos 2πk1) + γσ(1− cos 2πk2)+γd [2− cos 2π(k1 + k2)− cos 2π(k1 − k2)]
Left panel: |uxΓ〉; Right panel: |uyΓ〉; modulation e0
11/12/2020 at 14:24:48 slide 17
Visualizing the R-point of square lattice
Left panel: |uxR〉; Right panel: |uyR〉; modulation e iπ(t1+t2)
11/12/2020 at 14:24:48 slide 18
Visualizing the X-point of square lattice
Left panel: |uxX 〉; Right panel: |uyX 〉; modulation e iπt1
11/12/2020 at 14:24:48 slide 19
Visualizing the Y-point of square lattice
Left panel: |uxY 〉; Right panel: |uyY 〉; modulation e iπt2
11/12/2020 at 14:24:48 slide 20
Visualizing the ∆ = (1/3, 0) of square lattice
Left panel: |ux( 1
3,0)〉; Right panel: |uy
( 13,0)〉; modulation e i
2π3t1
11/12/2020 at 14:24:48 slide 21
Visualizing the Σ = (1/3, 1/3) of square lattice
Left panel: |ux( 1
3, 1
3)〉; Right panel: |uy
( 13, 1
3)〉; mod. e i
2π3
(t1+t2)
11/12/2020 at 14:24:48 slide 22
Visualizing the Σ = (1/3, 1/3) (irrep) of square lattice
Left panel: |uAk 〉; Right panel: |uBk 〉
11/12/2020 at 14:24:48 slide 23
Graphene
Let us return to graphene to do a major example.
Triangular lattice point symmetry is C6v .
How do we know graphene is symmorphic?
a1
a2
b1
b2
M
∆
K
Σ
Γ
Consider K = ( 23 ,
13 ); what is LG?
At K-point, Krᵀ = K + G for all R in C3v
Symmetrize in-plane displacements at K-point.
11/12/2020 at 14:24:48 slide 24
Recall in-plane dispersion relation
0
500
1000
1500
cm
-1
Γ ΓM K
b1
b2
M
∆
K
Σ
Γ
Symmetry will explain degeneracy at K -point.
Recall definition of translationally symmetrized basis:
|u(a,α)(k1,k2)〉 =
∑
t1t2
e i2π(k1t1+k2t2)|u(a,α)(t1,t2)〉
|u(a,α)
( 23, 1
3)〉 =
∑
t1t2
e i2π3
(2t1+t2)|u(a,α)(t1,t2)〉
11/12/2020 at 14:24:48 slide 25
Recall D( 23 ,
13 ) for NN and NNN springs in graphene
d1 and d2 are linear combinations of γσ, γπ, γ′σ, γ′π
D( 23, 1
3) =
d1 0 −d2 00 d1 0 d2
−d2 0 d1 00 d2 0 d1
+ i
0 0 0 −d2
0 0 −d2 00 d2 0 0d2 0 0 0
d1 =3
2
(γσ + γπ + 3γ′σ + 3γ′π
), d2 =
3
4(γσ − γπ)
d1 = 65.5950 eV/A2 and d2 = 12.8213 eV/A2
Taking the Eigenvals of this we get
λ =(39.9525, 65.595, 65.595, 91.2375
)
divide by Carbon mass, take square root, and convert to cm-1:
ω =(951.1, 1218.6, 1218.6, 1437.2
)
Prove why we get a 2-fold and two 1-folds.
Are zeros in above D( 23, 1
3) required by symmetry?
11/12/2020 at 14:24:48 slide 26
Irreducible representation vectors of little group
Need to keep big picture in mind.
Already symmetrized according to translation (i.e. |u(a,α)k 〉)
Now we know that Dk is invariant to little group.
Symmetrize |u(a,α)k 〉 according to little group.
How? The same recipe as always.
Represent little group elements in basis of choice.
Find composition by tracing representation.
Project irrep vectors using projection operator.
11/12/2020 at 14:24:48 slide 27
Decompose basis at K-point: little group C3v
Recall the character table of C3v
C3v 1 R2π/3R4π/3 σy σaσb
A1 1 1 1A2 1 1 −1E 2 −1 0
We have a 4-d representation for each R:
|uA,xK 〉, |uA,yK 〉, |uB,xK 〉, |uB,yK 〉
〈u(a,α)K |R|u(b,β)
K 〉 =1
N2
∑
t1t2t′1t′2
e i2π[ 23
(t1−t′1)+ 13
(t2−t′2)]〈u(a,α)(t′1,t
′2)|T(t1,t2)R|u(b,β)
(0,0) 〉
Let us quickly find the composition. We know tr(1) = 4, solet us make R2π/3 and σy .I will use the C3v subgroup of C6v , located at a1.Only need to know following for R ∈ {R2π/3, σy}: