A force-constant model of graphene for conductivity calculations

(1)

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M K

Frequencycm1 30x30 smear 630K

60x60 smear 315K

A force-‐constant model of graphene for conduc4vity calcula4ons

Henrique Miranda

⁽¹⁾

and Ludger Wirtz

⁽¹⁾

(1)

Physics and Materials Science Research Unit, University of Luxembourg, Luxembourg

Mo=va=on

[1] O. Dubay and G. Kresse, Phys. Rev. B 67, 035401 (2003).

[2] T. Sohier, M. Calandra, C.-‐H. Park, N. Bonini, N. Marzari, and F. Mauri, Phys. Rev. B 90, 125414 (2014).

[3] J. L. Mañes, Phys. Rev. B 76, 045430 (2007).

[4] R. A. Jishi, L. Venkataraman, M. S. Dresselhaus, and G. Dresselhaus, Chemical Physics LeXers 209, 77 (1993).

[5] L. Wirtz and A. Rubio, Solid State CommunicaZons 131, 141 (2004).

[6] Y. N. Gartstein, Physics LeXers A 327, 83 (2004).

[7] S. Piscanec, M. Lazzeri, F. Mauri, A. C. Ferrari, and J. Robertson, Phys. Rev. LeX. 93, 185503 (2004).

[8] S. Pisana, M. Lazzeri, C. Casiraghi, K. S. Novoselov, A. K. Geim, A. C. Ferrari, and F. Mauri, Nat. Mater. 6, 198 (2007).

[8] J. Li, H. P. C. Miranda, Y. Niquet, C. Delerue, L. Genovese, I. Duchemin, and L. Wirtz (submiXed)

Graphene has some interes=ng proper=es:

•  high charge-‐carrier mobility (limited by electron-‐phonon interacZon)

•  InteresZng for high frequency applicaZons (control of mobility is important) Aim of the project:

•  Create a 4 nearest-‐neighbor (4NN) force-‐constant model of graphene that reproduces the DFT calculated phonon frequencies and modes.

•  The model will be used for calculaZng the phonon-‐limited electrical conducZvity of graphene and carbon nanotubes.^[8]

•  Incorporate long-‐range interacZons in the model.

Ab-‐ini4o phonon dispersion of graphene

Force constant model

The force constants are second derivatives of the total energy with respect to the positions of two atoms:

They depend on:

•  Types of atoms interacting

•  Relative position vector

Transformation from local to global coordinates:

Build the dynamical matrix:

Calculate the phonon frequencies:

References

Electron-‐Phonon coupling

Comparison of 4NN models

–  The off-diagonal elements of the force constant matrix are VERY important!

•  They can be modeled as angular springs between the carbon atoms ^[6].

•  4NN diagonal model enough to reproduce the dispersion but gives bad phonon eigenvectors

•  4NN model with 2 off-diagonal parameters: phonon eigenvectors and eigenvalues OK!

–  Simple model with few parameters (14 parameters instead of 12 ).

–  Strain dependent model by fitting to strained graphene.

–  Local to global coordinates transformation is delicate with off-diagonal terms.

–  The same analysis and model can be done for other materials (BN)

Conclusions

•  Ground-‐state: LDA norm-‐conserving pseudopotenZal, 35 Ha plane-‐wave cut-‐oﬀ, 4.65 Bohr ladce constant (relaxed).

•  Phonon dispersion: DFPT sampling of 30x30 q-‐points.

C_ij = @²E

@x_i@x_j Global reference frame C_ij⁰ = @²E

@x⁰_i@x⁰_j Local reference frame

C = R

^T

C

⁰

R

Long-‐range force constants

D

_ab

(~q) = X

s

C

^s_ab

e

^i~^{q.( ~}^r^a ^r^~^s⁾

✏^l/ti₁ = ✏^l/ti₃ = 0

✏^ti/l₁ = ✏^ti/l₃ = 0

✏^l/ti₂ = ✏^ti/l₂

✏^l/ti₄ = ✏^ti/l₄

C⁰_n = 0 B@

ln ✏^l/ti_n 0

✏^ti/l_n ^ti_n 0

0 0 ^to_n

1 CA

Oﬀ diagonal terms:

ogen not included ^[4]

Phonon displacements

det |D

^ab

(~q) !(~q)

²

| = 0

The canonical phonon modes are deﬁned using two rules^[2]:

•  Eigenvector of a longitudinal (transverse) mode is parallel (perpendicular) to the phonon’s momentum

•  Phase diﬀerences between atoms is for acousZc and for opZcal modes.

The DFT calculated phonon modes tend to the canonical ones in the long wavelength limit.

At ﬁnite momentum there is an acousZc/opZcal mixing linear with |q|^[2-‐3].

Can we capture this dependence correctly in a 4NN model? Why is it important?

•  Real space cut-‐oﬀ the force constants up to 4NN.

•  Kohn anomalies are suppressed

•  Out-‐of-‐plane mode frequencies become imaginary (unstable system)

•  The 4NN model does not reproduce totally the phonon dispersion.

•  Can we add an analyZcal correcZon?

e

^i~^{q.( ~}^r¹ ^r^~²⁾

e

^i~^{q.( ~}^r¹ ^r^~²⁾

•  We generate the dynamical matrix using 4NN and compare to the ab-‐ini&o dynamical matrix and/or the eigenvalues using the quality funcZon and ﬁnd the best ﬁt using a simplex method:

•  We calculate from the phonon eigenvectors calculated with the diﬀerent models:

•  Good agreement with reported value ^[2]

e^a_{q, ˜}_LA = 1

p2e^iq^·(R+r^a⁾ q

|q|

e^a_{q, ˜}_{T A} = 1

p2e^iq^·(R+r^a⁾ q ?

|q ?|

e^a_{q, ˜}_LO = _a 1

p2e^iq^·(R+r^a⁾ q

|q|

e^a_{q, ˜}_{T O} = _a 1

p2e^iq^·(R+r^a⁾ q ?

|q ?|

TA LA

ZA ZO

TO

LO •  Kohn anomalies at Γ and K points that

connect the Fermi surface (non-‐analyZc phonon dispersion).

•  We can’t describe the full dispersion with a ﬁnite set of force constants

(Fourier interpolaZon fails).

•  The electronic temperature changes the slope of the phonon dispersion.

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Frequencycm1

ab-initio ref. [4]

Diagonal force-‐constant tensor

l ti

to

x_j x_i

Phonon dispersion of doped graphene

Phonon dispersion of compressed/stretched graphene

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Frequencycm1

ab-initio ref. [5]

Oﬀ-‐diagonal for 2NN

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ab-initio New fit

Oﬀ-‐diagonal for 2NN and 4NN

The electron-‐phonon coupling matrix for graphene has the form^[2-‐3]:

Where and are related to the acousZc-‐opZcal mixing in the phonon modes:

Where and are calculated using the canonical phonon modes.

To get the correct electron-‐phonon coupling the model should reproduce the correct value of .

•  Fit 4NN force constants to

strained and stretched graphene.

•  When doping graphene we

change the Fermi energy and the Kohn anomaly is found now on a circle in the Brillouin zone

CalculaZons within the adiabaZc approximaZon

e_q,LA =p

1 ↵²|q|²e_{q, ˜}_LA |q|[sin(3✓q)e_{q, ˜}_LO + cos(3✓_q)e_{q, ˜}_{T O}] e_{q,T A} =p

1 ↵²|q|²e_{q, ˜}_{T A} |q|[cos(3✓^q)e_{q, ˜}_LO sin(3✓_q)e_{q, ˜}_{T O}] e_q,LO =p

1 ↵²|q|²e_{q, ˜}_LO |q|[sin(3✓^q)e_{q, ˜}_LA + cos(3✓_q)e_{q, ˜}_{T A}] e_{q,T O} =p

1 ↵²|q|²e_{q, ˜}_{T O} |q|[cos(3✓q)e_{q, ˜}_LA sin(3✓_q)e_{q, ˜}_{T A}]

Canonical modes Modes with Acous=c/Op=cal mixing

˜_O

˜_A

Acous=c phonons Op=cal phonons

A O

A

⇡ ˜

^A

˜

_O _O

= ˜

_O

n 1 2 3 4

ln 40.905 7.402 -1.643 -0.609

tin 16.685 -4.051 3.267 0.424

ton 9.616 -0.841 0.603 -0.501

✏^l/tin 0.000 0.632 0.000 -1.092

✏^ti/ln 0.000 -0.632 0.000 -1.092

[1]

Model Phonon

frequencies Phonon displacements

near Γ

Ref [4] ✗ ✗

Ref [5] ✓ ✗

New Fit ✓ ✓

Table 2. FiXed force constants

-10 0 10 20 30 40 50 60 70

1 1.5 2 2.5 3 3.5 4

Force(dyn/cm)

Distance (˚A) long.

trans. in-plane trans. out-of-plane

-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2

M K

Energy(ev)

ef

AcousZc-‐opZcal mixing

Table 1. Comparison of the diﬀerent models

Ref. [4] Ref. [5] New Fit

Fermi energy

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ef +.0168

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nn 4

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Frequencycm1

-5%

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Frequencycm1

5%

compressed stretched

H_q^e-ph = i|q|

 2↵(q)QLA Ae^2i✓^q(QLA + iQT A)

Ae ^2i✓^q(Q_LA iQ_{T A}) 2↵(q)Q_LA + i

 0 _Oe^i✓^q(Q_LO + iQ_{T O})

Oe ^i✓^q(QLO iQT O) 0

Q_⌫ = e_q · e^q,⌫

ProjecZon on the canonical modes

⇡ 0.10˚ A

= ↵_eig X

q=Nq

X

n

h!₀(~q)_n !(~q)^ab_n i2

+ ↵_dyn X

q=Nq

X

ij

X

a,b

hD₀(~q)^ab_ij D(~q)^ab_ij i2

-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3

ref [4] ref [5]

New fit

ab-in

itio LD A

neighbor

A force-constant model of graphene for conductivity calculations

A force-­‐constant model of graphene for conduc4vity calcula4ons

Henrique Miranda

and Ludger Wirtz

Physics and Materials Science Research Unit, University of Luxembourg, Luxembourg

Mo=va=on

Ab-­‐ini4o phonon dispersion of graphene

Force constant model

References

Electron-­‐Phonon coupling

Comparison of 4NN models

Conclusions

C = R

C

R

Long-­‐range force constants

D

(~q) = X

C

e

Phonon displacements

det |D

(~q) !(~q)

| = 0

e

e

⇡ ˜

˜

= ˜

⇡ 0.10˚ A

A force-‐constant model of graphene for conduc4vity calcula4ons

Ab-‐ini4o phonon dispersion of graphene

Electron-‐Phonon coupling

Long-‐range force constants