Projection operators II: Modified group projectors

In this section I again find the E_1g phonon eigenvectors of armchair tubes, but with the help of the modified group projectors. In Fig. 2.8 I show schematically the six projection steps,

Ä

000010

100 030

|añ |nñ |mñ

S

_int

H

^{( m*)}

|m ñ^*

|añ |nñ |mñ

|añ |nñ |mñ

|añ |nñ |mñ

...

a) b) c)

=

=

H

_aux⁰

|añÄ|m ñ^*

fixed point

|m1ñ, m2ñ,..., m ñ| | f_m

|m1ñ⁰⁰⁰

|m1ñ⁰¹⁰

|m1ñ¹⁰⁰

... }

H

_aux^big

partial scalar product

|m1, ñm ám^*m|

d) e) f)

|nñÄ

|m^*ñ

|mñÄ|m ñ^*

Figure 2.8: The modified group pro-jector technique: a) The total space is divided into interior spaces of the atoms b) The interior S_int is spanned by |αi⁰,|νi⁰,|µi⁰, . . . c) To construct the fixed point in the auxiliary space the functions in S_int are multiplied by

|µ^∗i. d) The auxiliary space H⁰_aux of the representing atom contains the fixed points |µ1i⁰,|µ2i⁰, . . . ,|µ f_µi⁰ e) A fixed point |µt_µi of the total auxiliary spaceH^big_auxis induced with the operators switching between the interior space. f) The symmetry adapted basis|µt_µ,miis the partial scalar producthµ^∗m|µt_µi.

which I first describe in general and then apply to my example. Fig. 2.8a) depicts the total space H of the physical problem. The Hilbert space is divided into an infinite number of interior spaces (rectangles) each belonging to a different atom in the nanotube (or any other system). The interior spaceS_int of the representing atom is singled out in Fig. 2.8b). Since nanotubes are single orbit systems one interior space represents the total space. For, e.g., phonon eigenvectors the interior is a three-dimensional vector space. S_int is spanned by the functions|α_i⁰,|ν_i⁰,|µ_i⁰. . . transforming as the irreducible representationsα,ν,µ. . . of the nanotube symmetry group. With the modified group projector I want to find the functions|µ_itransforming as µ. I first construct the auxiliary space H⁰

aux =S_int⊗H^(µ∗) with the functions|α_i⁰_{⊗ |}µ^∗_i,|ν_i⁰_{⊗ |}µ^∗_i,|µ_i⁰_{⊗ |}µ^∗_i, . . .as shown in Fig. 2.8c) and d).

Only|µ_{i ⊗ |}µ^∗_iis left invariant under any symmetry operation of the group. The generally reducible representation inH⁰

auxconstructed from the representation D_δ in the interior space is

γ^µ(SSS) =D_δ(SSS)⊗D^(µ∗)(GGG↓SSS) (2.21) for the example of achiral nanotubes and the vector representation

γ^µ(C_1h) =Dvec(C_1h)⊗D^(µ∗)(GGG↓C_1h). (2.22) SSS is the stabilizer of the atoms equal to C_1h ={E,σ_} in armchair or zig-zag nanotubes.

The symbol (GGG↓SSS) denotes that the full symmetry group GGG is restricted to SSS, i.e., only those symmetry operations belonging to SSS are considered. The trace of γ^µ(SSS) is equal to the frequency number f_µ, the number of times the irreducible representationµ ^{appears in} the reducible representation D_δ. The eigenvectors ofγ^µ(SSS)to the eigenvalue 1 are the fixed points|µ¹i⁰,|µ²i⁰, . . . ,|µ^fµi⁰we are looking for.

The fixed points|µ^t_µi⁰of the interior auxiliary space H⁰

aux of the representing atom 0 are now expanded to the total auxiliary spaceH^big_aux. A fixed point|µ^t_µi^inHbigauxis obtained with the transformation operator|µ^tµi=B^µ|µ^tµi⁰, where

B^µ= 1

√Z

∑

t

E₀^t⊗βt^µ= 1

√Z

∑

t

E₀^t⊗I_δ⊗D^(µ∗)(Z_t). (2.23) The sum in Eq. (2.23) is over all elements of the transversal. The transversal is the group of symmetry operations Z_t transforming the atoms of the same orbit into each other; Z is the order of the transversal. The operator E₀^t⊗β_t^µ transforms the fixed point|µ^t_µi ^{of the} representing atom into the fixed point|µ^tµi^tof another atom t. E₀^t literally switches between the atoms; it is a matrix with the 0t’th element equal to 1, all other are zero. βt^µ =I_δ⊗ D^(µ∗)(Z_t)takes care of the symmetry in the auxiliary Hilbert space. The identity matrix I_δ

of the interior space is used only to enlarge the dimension. The sum over the fixed point in all interior spaces t is the fixed point in the total auxiliary space H^big_aux. The action of B^µ is shown in Fig. 2.8e). Finally, in Fig. 2.8f) the symmetry adapted basis in the original Hilbert spaceHis extracted with the help of the partial scalar product

hµ^∗m|µt_µi=hµ^∗m|(|µt_µni ⊗ |µ^∗mi) =|µt_µmi. (2.24) The vectors|µ^tµmiin the total Hilbert spaceHis the m’th component of the t_µ’th eigen-vector transforming according to the irreducible representationµ^.

After discussing the modified group projectors in a general way I now apply the method to the problem of finding the phonon eigenvectors transforming as E_1g in a (5,5) armchair nanotube. As already mentioned the interior space for this physical problem is the three-dimensional vector space, the reducible representation is the vector representation, and the stabilizer in armchair nanotubes is C_1h. With the standard basis of E_1g (xz,yz) Eq. (2.21) reads

γ^E1g(C1h) = D_vec(C1h)⊗D^(E1g)(G↓C_1h) (2.25)

= 1 2

"

 1

1 1





| {z }

Dvec(E)

⊗ 1

1

| {z }

D_E1g(E)

+



 1

1

−1





| {z }

Dvec(σh)

⊗ −1

−1

| {z }

D_E1g(σh)

#

= 1 2









 0

0 0

0 2

2











(2.26)

The trace Tr[γ^E1g(C1h)] =2, i.e., two phonons in armchair tubes are of E_1g symmetry as already obtained in Section 2.4.1. The two eigenvectors (0,0,0,0,1,0) and (0,0,0,0,0,1) are the fixed points in the auxiliary space H⁰

aux. In the following I consider only the first fixed point for simplicity; its vector can be expressed as

|E_1g1i⁰=



 0 0 1



⊗ 1

0

+



 0 0 0



⊗ 0

1

=z₀⊗ 1

0

, (2.27)

where z₀stands for the z displacement component of the representing atom. The two vectors (1,0)and(0,1)are a basis of E_1g in the nanotube’s symmetry group. Note that we already found that E_1g modes are always axial in armchair nanotubes. Only the z component in the interior space of the representing atom transforms as E_1g. Since no line group symmetry

operation transforms the principal axis in the x or y axis, the eigenvector for the whole tube will be axially as well. I did not specify the particular armchair nanotube up to now. The finding of only axial E_1g eigenvectors is therefore valid for any armchair nanotube.

With the fixed point in the interior space of the representing atom I have to induce the fixed point in the total auxiliary space H^big_aux. The transversal for armchair nanotubes is TTT^w_qDDD_n= T

TT_2nDDD_n. To apply the transformation operator B^E1g to the first fixed point |E_1g1i⁰I need an E_1g matrix representation for the generators of the transversal and its combinations

C_n=

cos 2π/n sin 2π/n

−sin 2π/n cos 2π/n

(C2n|^a2)^k=0=

cosπ/n sinπ/n

−sinπ/n cosπ/n

U =

−1 0 0 1

(C_n^sC^l_2nU^u|l^a₂)^k=0=

"

(−1)^ucosπ^2s+l_n ^sinπ^2s+l_n

−(−1)^usinπ^2s+l_n ^cosπ^2s+l_n

#

. (2.28)

In Eq. (2.27) I separated the eigenvector|E_1g1i⁰into a sum of two direct products. Therefore we can skip the identity representation I_δ appearing inβ_t^µ in Eq. (2.23)

|E_1g1i=

E₀⁰⁰⁰⊗β₀₀₀^E1g+E₀⁰¹⁰⊗β₀₁₀^E1g+E₀¹⁰⁰⊗β₁₀₀^E1g+E₀⁰³⁰⊗β₀₃₀^E1g+E₀²⁰⁰⊗β₂₀₀^E1g+ +E₀⁰⁵⁰⊗β₀₅₀^E1g+···+E₀⁰⁰¹⊗β₀₀₁^E1g+E₀¹⁰¹⊗β₁₀₁^E1g. . .

z0⊗ ¹0

with(A⊗B)(A⁰⊗B⁰) = (AA⁰)⊗(BB⁰)and c l =cos(l·2π/n),s l=sin(l·2π/n)

=z₀₀₀⊗(¹₀) +z₀₁₀⊗(₋^{c 1}_{s 1}) +z₁₀₀⊗(₋^{c 2}_{s 2}) +z₀₃₀⊗(₋^{c 3}_{s 3}) +z₂₀₀⊗(₋^{c 4}_{s 4}) +z₀₅₀⊗(₋^{c 5}_{s 5}) +···+ (−z001)⊗(⁻¹₀) + (−z101)⊗(^{−c 1}_{s 1}). . . (2.29) The interior coordinate system is transformed together with the atoms. Therefore the z com-ponent of the displacement changes sign under the U operation, which I indicated by the minus sign at z for the symmetry operations in Eq. (2.29) containing U (the last two). The vector in Eq. (2.29) is the fixed point in the total auxiliary Hilbert space H_aux^big. To finally find the phonon eigenvectors |µ^1m_i=|E_1g1,1i and |E_1g1,2i transforming as E_1g I take the partial scalar product and obtain

|E1g1,1i=hE_1g^∗ 1|E1g1i=z000+c 1·z010+

+c 2·z₁₀₀+c 3·z₀₃₀+c 4·z₂₀₀+c 5·z₀₅₀+. . .z₀₀₁+c 1·z₁₀₁. . .

|E_1g1,2i=hE_1g^∗ 2|E_1g1i=−s 1·z₀₁₀−s 2·z₁₀₀−

−s 3·z030−s 4·z200−s 5·z₀₅₀+···+s 1·z101. . . (2.30) Fig. 2.9 shows the two phonon eigenvectors|E_1g1,1i(left) and|E_1g1,2i(right), which are degenerate. The displacement pattern to the left can already be found on page 31 in the

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Ÿ Ÿ Ÿ

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Ÿ Ä

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000 010 030 100

200

050 100

101

| E

_1g

1,1ñ | E

_1g

1,2ñ

Figure 2.9: Degenerate E_1g phonon eigenvectors in armchair nanotubes obtained with the modified group pro-jector technique. For the transforma-tion explicitly written in Eq. (2.30) the atoms were labeled correspondingly in the left hand tube.

upper right corner of Fig. 2.6, where the same eigenvector was projected with the help of the graphical projectors. It is a mode with two nodes around the circumference and an in-phase displacement of the two graphene sublattices. In an isolated, finite molecule these eigenvectors are a rotation around the x and y axis. The projection of the second fixed point ofγ^E1g(C_1h)yields the two degenerate high-energy modes of E_1g symmetry. Similarly, the other results of Section 2.4.2. can also be obtained with the modified group projector tech-nique. In chiral nanotubes group projectors per se are not suited for finding the eigenvectors, because of the trivial stabilizer in this tubes. After introducing the technique in this section I now turn to the projection of the electronic states in carbon nanotubes.

Im Dokument Carbon Nanotubes: Vibrational and electronic properties (Seite 42-46)