Tuning of optical phonons by fermi level in graphene

Physica E 43 (2011) 645–650

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Physica E journal homepage: www.elsevier.com/locate/physe

Tuning of optical phonons by fermi level in graphene Tsuneya Ando Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

a r t i c l e in fo

abstract

Available online 2 August 2010

A brief review is given on recent theoretical study on continuum models of optical phonons as well as zone-boundary phonons and effects of electron–phonon interaction in monolayer and bilayer graphene, resulting in strong dependence of the frequency and broadening on the electron density. & 2010 Elsevier B.V. All rights reserved.

1. Introduction Recently, monolayer graphene was fabricated using the so-called scotch-tape technique [1] and the magnetotransport was measured including the integer quantum Hall effect [2,3]. Since then the graphene became the subject of extensive theoretical and experimental study [4,5]. The carbon nanotube is graphene rolled into a cylindrical form, discovered and synthesized earlier than graphene [6]. The purpose of this paper is to give a brief review on optical phonons as well as zoneboundary phonons and effects of electron–phonon interaction in monolayer and bilayer graphene and in nanotubes.

2. Monolayer graphene and nanotube In a monolayer graphene the conduction and valence bands consisting of p orbitals cross at K and Ku points of the Brillouin zone, where the Fermi level is located [7,8]. Electronic states near a K point are described by a k  p equation equivalent to Weyl’s equation for a neutrino or a Dirac equation with vanishing rest mass [6,9–14]. In the vicinity of the K point, in particular, we have ! FA ðrÞ ^ ~  kÞFðrÞ ¼ eFðrÞ, FðrÞ ¼ gðs , ð1Þ FB ðrÞ where FA and FB describe the amplitude at sublattice points A and B, respectively, g is a band parameter, k^ ¼ ðk^ x , k^ y Þ is the wave~ ¼ ðsx , sy Þ is the Pauli matrix. The equation vector operator, and s ~ with s ~ in the of motion for the Ku point is obtained by replacing s above equation. Electronic states in a carbon nanotube (CN) are obtained by imposing generalized periodic boundary condition Fðr þLÞ ¼ expð 82pni=3ÞFðrÞ (upper sign for K and lower for Ku) in the circumference direction specified by chiral vector L with n ¼ 0 or E-mail address: [email protected] 1386-9477/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2010.07.021

71 determined by the CN structure. We have n ¼ 0 for a metallic CN and n ¼ 71 for a semiconducting CN.

3. Long-wavelength optical phonon Long-wavelength optical phonons are known to be measured directly by the Raman scattering [15,16]. Usually, they are described perfectly well in a continuum model. Such a model was developed and the Hamiltonian for electron–phonon interactions was derived [17], and effects of electron–phonon interaction on optical phonons were recently studied in graphene [18,19] and in nanotubes [17,20]. Optical phonons are represented by the relative displacement of two sub-lattice atoms A and B, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ‘ uðrÞ ¼ ðb þby Þe ðqÞeiqr , ð2Þ 2NM o0 qm qm m q, m where N is the number of unit cells, M is the mass of a carbon atom,

o0 is the phonon frequency at the G point, q ¼ ðqx ,qy Þ is the wave vector, m denotes the modes (‘t’ for transverse and ‘l’ for longitudinal), and byqm and bqm are the creation and destruction operators, respectively. For qx ¼ qcosjq and qy ¼ qsinjq with q ¼ jqj, we have el ðqÞ ¼ iðcosjq ,sinjq Þ and et ðqÞ ¼ iðsinjq ,cosjq Þ. The interaction between optical phonons and an electron in the vicinity of the K and Ku points is given by [17] pffiffiffi b g ~  uðrÞ, HKint ¼  2 G2 s b

ð3Þ

where b is the bond length, bG ¼ dlng0 ðbÞ=dlnb with g0 being the hopping integral in a nearest-neighbor tight-binding model, and vector product for vectors p¼(px, py) and q ¼(qx, qy) in two dimensions is defined by p  q ¼ px qy py qx . We have bG  2 in conventional semiconductors [21] and bG may be slightly larger in carbon systems. The lattice distortion gives rise to a shift in the origin of the wave vector or an effective vector potential, i.e., ux in the

T. Ando / Physica E 43 (2011) 645–650

y direction and uy in the x direction. The Hamiltonian for the Ku ~ by s ~ . The interaction strength point is obtained by replacing s is characterized by the dimensionless coupling parameter pffiffiffi  2 2 36 3 ‘ 1 bG lG ¼ : ð4Þ p 2Ma2 ‘o0 2 For M¼1.993  10  23 g and ‘o0 ¼ 0:196 eV, we have lG  3  103 ðbG =2Þ2 . This shows that the interaction is not strong and therefore the lowest order perturbation gives sufficiently accurate results. The phonon Green’s function is written as 2‘o0 ð‘oÞ2 ð‘o0 Þ2 2‘o0 Pm ðq, oÞ

:

ð5Þ

The phonon frequency is determined by the pole of Dm ðq, oÞ. In the case of weak interaction, the shift of the phonon frequency, Dom , and the broadening, Gm , are given by

Dom ¼

1

‘

RePm ðq, o0 Þ,

1

Gm ¼  ImPm ðq, o0 Þ: ‘

ð6Þ

When we calculate the self-energy of optical phonons starting with the known phonon modes in graphene, its direct evaluation causes a problem of double counting [22]. In fact, if we apply the above formula to the case of vanishing Fermi energy, we get the frequency shift due to virtual excitations of all electrons in the p bands. However, this contribution is already included in the definition of the frequency o0 . In order to avoid such a problem, we have to subtract the contribution in the undoped graphene for o ¼ 0 corresponding to the adiabatic approximation. Fig. 1 shows the frequency shift and broadening for various values of 1=o0 t. For nonzero d or 1=o0 t, the logarithmic singularity of the frequency shift and the sharp drop in the broadening at eF ¼ ‘o0 =2 disappear, but the corresponding features remain for 1=o0 t 5 1, where eF is the Fermi energy. Similar results were reported independently [23] and experiments giving qualitatively similar results were reported [24,25]. The calculation can easily be extended to the case in the presence of magnetic field B, where discrete Landau levels are formed and oscillations due to resonant interactions appear

10

5

0

-1 -2

-5 0.0

+2 +1

0 +1 -1 0

0.5 1.0 Magnetic Energy: hωB (units of hω0)

1.5

10

2.0

εF0/ hω0=0.250 δ/hω0=0.05 Frequency Shift

1/ω0τ 0.5 0.2 0.1 0.0

1.5

Spectral Function (units of 1/hω0)

Frequency Shift and Broadening (units of λω0)

εF0/hω0=0.250 δ/hω0=0.10 0.05 0.02 Frequency Shift Broadening

1.0

0.5

1.5

1.0 5

0.5

Magnetic Energy: hωB (units of hω0)

Dm ðq, oÞ ¼

in the frequency shift and the broadening [19]. The Landau-level pffiffiffiffiffiffi energy is given by en ¼ sgnðnÞ jnj‘oB ðn p¼ ffiffiffi 0, 71, . . .Þ, where sgn(n) denotes the sign of n and ‘ o ¼ 2g=l with magnetic B pffiffiffiffiffiffiffiffiffiffiffiffiffi length l ¼ c‘=eB. Fig. 2 shows calculated frequency shift and broadening when e0F =‘o0 ¼ 0:25 and the corresponding phonon spectral function, where e0F is the Fermi energy in the absence of a magnetic field. All resonant transitions from  n to n+ 1 and from  n  1 to n with n 4 0 appear at the field where their energy difference becomes equal to ‘o0 . At resonances, the phonon spectrum exhibits characteristic behavior. This magneto-phonon resonance was studied also in Ref. [26] and recently observed in Raman experiments [27]. The same tuning of the optical-phonon frequency and broadening due to change in the Fermi level is also possible in carbon nanotubes. In fact, effects of the electron–phonon interaction on the optical phonon in carbon nanotubes were theoretically studied earlier than in graphene [17]. In nanotubes, the modes

Frequency Shift and Broadening (units of λω0)

646

0.0 Frequency Shift

-5

-0.5 0.0

0.5 1.0 1.5 Fermi Energy (units of hω0)

0.0

0

Broadening

2.0

Fig. 1. The frequency shift and broadening of optical phonons in monolayer graphene as a function of the Fermi energy. t is a phenomenological relaxation time characterizing the level broadening effect due to disorder.

0

5

Frequency (units of λω0) Fig. 2. (a) The frequency shift and broadening of optical phonons in monolayer graphene as a function of effective magnetic energy ‘oB . Thin vertical lines show resonance magnetic fields: e0F =‘o0 ¼ 0:25. The results for d=‘o0 ¼ 0:1, 0.05, and 0.02 are shown. (b) The phonon spectral function for d=‘o0 ¼ 0:05. The dotted line shows the peak position as a function of ‘oB .

T. Ando / Physica E 43 (2011) 645–650

are classified into longitudinal and transverse, depending on their displacement in the axis or circumference direction. Fig. 3 shows the results of the similar calculations in carbon nanotubes [20]. In semiconducting nanotubes, the imaginary part vanishes identically because of the presence of a gap. The frequency of the longitudinal and transverse modes is both shifted to higher frequency side and the shift is smaller for the longitudinal mode for small kF. The behavior of two modes as a function of kF is similar to that of ‘‘level crossing’’. In metallic nanotubes, the transverse mode is not affected by the doping at all. For the longitudinal mode, the energy shift exhibits a downward shift and considerable broadening [17]. For nonzero kF, the self-energy has a logarithmic divergence at gkF ¼ ‘o0 =2 and increases logarithmically with kF for gkF 4 ‘o0 =2 for vanishing d. This behavior in CN, the same as in graphene theoretically predicted [18,19,23,28] and experimentally observed [24,25], was observed in Raman experiments [29,30] and discussed theoretically [31,32].

Frequency Shift (units of α (L) ω0)

2.5 Semiconducting Longitudinal Transverse

2.0

1.5

1.0

0.5

0.0 0.0

0.1

0.2

0.3

0.4

0.5

Fermi Wave Vector (units of 2π/L)

Frequency Shift and Broadening (units of α (L) ω0)

4 δ/hω0 0.500 0.200 0.100 0.010

3

2

Longitudinal Eφ /hω0=0.00 Shift Broadening

-2 0.0

0.5

1.0

Phonons near the K and Ku point, called zone-boundary phonons, can play important roles in intervalley scattering between the K and Ku points. In general, there exist four independent eigen-modes for each wave vector. However, after straightforward calculations, we can see that only one mode with the highest frequency contributes to the electron-phonon interaction [33]. This mode is known as a Kekule´ type distortion generating only bond-length changes. The interaction Hamiltonian is given by ! 0 o1 DðrÞsy b g Hint ¼ 2 K2 , ð7Þ b oDy ðrÞsy 0 where o ¼ e2pi=3 and bK is another appropriate parameter, which is equal to bG for the tight-binding model. In the second quantized form, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ‘ ðbKq þ byKuq Þeiqr , DðrÞ ¼ ð8Þ 2NMoK q where oK is the frequency of the Kekule´ mode. It is worth noting that D cannot be given by a simple summation over the K and Ku modes. We should take a proper linear combination of the K and Ku modes in order to make the lattice displacement a real variable. We can easily understand the operator form of D and Dy in the interaction Hamiltonian by considering the momentum conservation with the fact that 2KKu and K2Ku are reciprocal lattice vectors, where K and Ku are the wave vectors at the K and Ku point. The dimensionless coupling parameter, lK , is given by the same expression as Eq. (4) except that o0 is replaced with oK and bG with bK . For ‘oK ¼ 161:2 meV, we have lK ¼ 3:5  103 ðbK =2Þ2 . The lifetime of an electron with energy e is given by the scattering probability from the initial state to possible final states via emission and absorption of one phonon. For the zone-center phonon, the summation of the contributions of longitudinal and transverse modes gives isotropic scattering probability in each of the K and Ku points. For the zone-boundary phonon, an electron around the K point can be scattered to the Ku point accompanied by absorption of one phonon around the K point or by the emission of one phonon around the Ku point. Further, backward scattering is stronger than forward scattering, suggesting that the zone-boundary phonon is more effective in reducing the electron velocity in high-field transport. In graphene, the calculated scattering probabilities for both phonons are given by the same formula,

t

0

-1

4. Zone-boundary phonon

‘

1

1.5

2.0

647

¼ pla je‘oa j,

ð9Þ

where a represents G or K and we have neglected the phonon occupation due to large oa at room temperature. This simply shows that the electron lifetime is inversely proportional to the coupling parameter la and to the density of states at the energy of the final state. The phonon emission is possible only when the energy of the initial electron is larger than that of the phonon to be emitted. Otherwise, the final states are fully occupied at zero temperature and the phonon emission never takes place. In this sense, the zone-boundary phonon has another advantage over the zone-center phonon. Therefore, the zone-boundary phonon gives dominant scattering for high-field transport in graphene and in nanotube owing to its smaller frequency, larger coupling constant, and large backward scattering.

Fermi Energy (units of hω0)

5. Bilayer graphene Fig. 3. (a) The frequency shift in a semiconducting nanotube as a function of the Fermi wave vector kF. The solid and dashed lines represent the longitudinal and transverse modes, respectively. (b) The frequency shift and broadening of the longitudinal mode in metallic nanotubes as a function of the Fermi energy.

We consider a bilayer graphene which is arranged in the AB (Bernal) stacking. The upper layer is denoted as 1 and the lower

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T. Ando / Physica E 43 (2011) 645–650

layer denoted as 2. In each layer, the unit cell contains two carbon atoms denoted by A1 and B1 in layer 1 and A2 and B2 in layer 2. For the inter-layer coupling, we include only the coupling between vertically neighboring atoms. Then, electronic states are described by the k  p Hamiltonian [34,35]: B1

A1

0 B ^þ k g H¼ B B B B 0 @

A2

B2

gk^ 

0

0

D

D

0 ^ gk þ

0

0

0

1

C 0 C C C, gk^  C A 0

ð10Þ

Frequency Shift and Broadening (units of λω0)

where D ð  0:4 eVÞ represents the inter-layer coupling between sites B1 and A2. Let us define sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 D D ¼ eðkÞcosc, eðkÞ ¼ þ ðgkÞ2 , gk ¼ eðkÞsinc, ð11Þ 2 2

e 7 1 ðkÞ ¼ 7 2eðkÞsin2 ðc=2Þ, e 7 2 ðkÞ ¼ 72eðkÞ cos2 ðc=2Þ:

2.5 Symmetric Δ/hω0 =2.0

6

2.0 δ/hω0 0.500 0.200 0.100 0.010

1.5

1.0

0.5

0.0 Shift Broadening

-0.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

5

Δ/hω0 =2.0 εF0 /hω0 =0.25

4

Shift Broadening Symmetric

3 2 1 0 -1 -2 δ/hω0 = 0.10 0.05 0.02

-3

Fermi Energy (units of hω0)

0.0

Frequency Shift and Broadening (units of λω0)

ð12Þ

The band e þ 1 ðkÞ represents the lowest conduction band which touches the highest valence band e1 ðkÞ at k¼ 0. The bands e 7 2 ðkÞ are the excited conduction and valence bands and e þ 2 ðkÞe þ 1 ðkÞ ¼ d independent of k. In the vicinity of e ¼ 0, the Hamiltonian is reduced to a (2,2) form for the basis set (A1, B2) and the energy dispersion 2  becomes eðkÞ ¼ p 7ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ‘2 k2 =2m with m ¼ ‘ D=2g2 . The Landau levels ffi become 7 ‘oc nðn þ 1Þ ðn ¼ 0, 7 1, . . .Þ with oc ¼ eB=m c. In bilayer graphene, optical phonons are classified into symmetric and antisymmetric modes in which the displacement of the top and bottom layers are in-phase and out-of-phase, respectively. They are affected by electron–phonon interactions in a different manner [28]. The symmetric mode causes interband transitions between e 7 1 ðkÞ and therefore exhibits logarithmic singularity in a manner same as monolayer graphene when eF ¼ ‘o0 =2. On the other hand, this transition is not allowed for

Frequency Shift and Broadening (units of λω0)

0

where c vanishes for k¼0 and approaches p=2 with increasing k. Then, the eigen-energies are given by

1.5 Antisymmetric Δ/hω0=2.0

0.5 1.0 Magnetic Energy (units of hω0)

1.5

Fig. 5. Calculated frequency shift and broadening of the symmetric mode in a bilayer graphene as a function of the effective magnetic energy ‘oc . The thin vertical straight lines denote the field where the energy difference of optically allowed Landau levels become equal to the phonon energy: D=‘o0 ¼ 2 and e0F =‘o0 ¼ 0:25.

1.0

0.5

0.0 δ/hω0 0.500 0.200 0.100 0.010

-0.5

-1.0

Shift Broadening

-1.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Fermi Energy (units of hω 0) Fig. 4. Some examples of calculated frequency shift and broadening of the symmetric (a) and antisymmetric mode (b) at the G point in a bilayer graphene: D=‘o0 ¼ 2. The amount of the broadening due to disorder is denoted by d.

Fig. 6. A schematic illustration of the bilayer graphene with a top gate and a bottom gate and the potential energy diagram. The distance between the layers is given by d, the potential difference by eFd, and Fext represents the field due to the top gate.

T. Ando / Physica E 43 (2011) 645–650

the antisymmetric mode but interband transitions between e þ 1 ðkÞ and e þ 2 ðkÞ contribute to the phonon renormalization. Fig. 4 shows calculated frequency shift and broadening for two phonon modes. Fig. 5 shows an example of results in the presence of a magnetic field. One important feature is that the band structure can be strongly modified due to opening-up of a band gap by applied electric field [36]. Fig. 6 shows a schematic illustration of the device structure, where eFd represents the potential difference between layers 1 and 2 (F is the effective electric field and d¼0.334 nm is the interlayer distance). The effective

649

Hamiltonian becomes 0

eFd=2 B B gk^ þ B H¼B B 0 @ 0

1

gk^ 

0

0

eFd=2

D

D

eFd=2 gk^ þ

0 gk^ 

0

C C C C: C A

ð13Þ

eFd=2

2 Frequency Shift (units of λω0)

1.5 eFd / Δ 0.0 0.2 0.5

0 -1

1.0 0.5

Monolayer

0.0

-0.5 -1.5

Broadening (units of λω0)

1.5

-1.0

-0.5 0.0 0.5 Wave Vector (units of Δ /γ)

1.0

1.5

Intensity

Energy (units of Δ)

1.0

1

1.0 0.5 0.0 1.0 0.5 0.0

-1.5

Fig. 7. The energy dispersion for varying values of the potential difference eFd. The dot-dot-dashed lines show that of a monolayer graphene.

0.5

0.0

0.5

1.0

1.5

Δ2/2πγ 2)

Frequency Shift (units of λω0)

1 0 -1

Broadening (units of λω0)

1.5

0.0

-0.5

-1.0

-1.5

-1.0

-0.5

0.0

0.5

Electron Density (units of

1.0

1.5

1.0

Δ/hω0=2 eFext d / hω0=0.0 δ/hω0=0.1

0.5 0.0 1.0 0.5 0.0

κ=2

-1.5 -2.0

-0.5

2

eFextd/Δ 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5

Intensity

Energy Difference: eFd (units of Δ)

1.0

-1.0

Electron Density (units of gvgs

2.0

1.5

Δ/hω0=2 eFext d / hω0=0.0 δ/hω0=0.1

-1.5

2.0

gvgsΔ2/2πγ 2)

Fig. 8. Calculated energy difference eFd as a function of the electron concentration for various values of eFextd. The static dielectric constant of the environment is chosen as k ¼ 2.

-1.0

-0.5

0.0

0.5

Electron Density (units of gvgs

1.0

1.5

Δ2/2πγ 2)

Fig. 9. Calculated frequency shift, broadening, and strength of the symmetric component. The solid and dashed lines denote the high- and low-frequency modes, respectively. The thin dotted lines in the top panel show the frequencies for symmetric and antisymmetric modes calculated without inclusion of their mixing: D=‘o0 ¼ 2, d=‘o0 ¼ 0:1. (a) eF ext d=‘o0 ¼ 0 and (b) 0.5.

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T. Ando / Physica E 43 (2011) 645–650

Fig. 7 shows the energy bands obtained by the diagonalization of this Hamiltonian. The band gap appears and the minimum gap is located at a nonzero value of k depending on the field. The effective field is determined in a self-consistent manner [36] because the electron density distribution between two layers becomes different, giving rise to interlayer potential difference. Some examples of the results of such calculations within this model Hamiltonian are shown in Fig. 8 [37]. The unit n0s ¼ gv gs D2 =2pg2 is the electron concentration at eF ¼ D for eFd¼0. We have n0s  2:5  1013 cm2 for D  0:4 eV. We can see eFd can become as large as  D=2 although being dependent on eF ext d. Fig. 9(a) shows the frequency shift (top panel), broadening (middle panel), and the spectral intensity of the symmetric component (bottom panel) as a function of the electron concentration for eF ext d ¼ 0 [38]. We have assumed D=‘o0 ¼ 2 corresponding to D  0:4 eV and ‘o0  0:2 eV and d=‘o0 ¼ 0:1. The symmetric component shown in the figure describes the relative intensity of the Raman scattering. At ns ¼0 with eFd¼0, the optical phonons are exactly classified into symmetric and antisymmetric modes. With the increase of ns, they become mixed with each other, which becomes particularly important when they cross each other. Fig. 9(b) shows results for eF ext d=‘o0 ¼ 0:5, for which the asymmetry is enhanced in the electron side and reduced in the hole side. The appearance of two peaks at sufficiently high electron concentration, where the potential asymmetry is significant, is in agreement with the feature of recent experiments showing double peaks in a highly doped bilayer graphene [39]. In order to make detailed comparison with experiments, we may have to consider a small frequency splitting of symmetric and antisymmetric modes, which can be present even in the symmetric bilayer and is independent of the change in the electron concentration.

Acknowledgements The author acknowledges the collaboration with Drs. H. Suzuura and M. Koshino. This work was supported in part by Grant-in-Aid for Scientific Research on Priority Area ‘‘Carbon Nanotube Nanoelectronics’’, by Grant-in-Aid for Scientific Research, and by Global Center of Excellence Program at Tokyo Tech ‘‘Nanoscience and Quantum Physics’’ from Ministry of Education, Culture, Sports, Science and Technology Japan.

Note added in proof After submission, more experiments were reported concerning observation of shift and splitting of optical phonons of doped bilayer graphene in Raman scattering [40–42] and infrared absorption [43,44].

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