Photon-Induced Quantum Oscillations of the Terahertz Conductivity in

Letter pubs.acs.org/NanoLett

Photon-Induced Quantum Oscillations of the Terahertz Conductivity in Graphene R. Vega Monroy* and G. Salazar Cohen Facultad de Ciencias Básicas, Universidad del Atlántico, Km. 7, Via a Pto. Colombia, Barranquilla, Colombia S Supporting Information *

ABSTRACT: In this work, we present a theory that is able to explain the nonmonotonic decreasing behavior (observed in experimental data1−12) of the graphene terahertz conductivity with the increase of the field frequency. In this connection, the displacement of the structure of topological states inside the energy band gap, which appears in graphene due to the strong photon-electron coupling, and the narrowing of this gap, as result of electron transitions from bound photon-dressed electron states to extended states outside the energy gap driven by the field frequency, lead to a periodic change of singularities near the edge of the band gap, resulting in subtle quantum oscillations of the dynamical terahertz conductivity. This quantum contribution complements the Drude response, which fits the spectral range. On the other hand, the scattering processes by impurities favor interband transitions, suppressing this way intraband terahertz absorptions, which are related to optical transitions from inside to outside the gap. KEYWORDS: Graphene sheet, terahertz conductivity, quantum oscillations, photon-dressed electrons, topological states, energy gap

T

this manner, some topological properties emerge due the energy band filling and they remain in the system while the ac field is applied. These phenomena first were analyzed by Oka and Aoki in the Floquet formalism. They found that an illuminated two-dimensional Dirac system (honeycomb model for graphene) with intense circularly polarized light can change the topological properties of its quantum states.28,29 The Hamiltonian of a graphene sheet under intense polarized ac fields is defined by the expression27,30,31 Ĥ = vf(σ̂ · p̂) + εa(σ̂ · e)⃗ (â + â†). In this connection, mixed states between Dirac electrons and polarized photons are called “photon-dressed electrons” of which the dispersion relation can be written as follows27,30

he study of the ac dynamical conductivity of graphene has received considerable attention since this material experimentally was obtained. Theoretically, the effect on the dynamical conductivity, of two-dimensional Dirac Fermions and their interactions with phonons, disorder, and so forth has been analyzed .13−19 Some expressions for the dynamical conductivity of graphene, including the effect of disorder in a self-consistent Born approximation (SCBA), have been obtained by Peres et al.20 In this context, it has been predicted that the dynamical reaction of Dirac Fermions to applied ac fields holds all odd harmonics, implying high nonlinearity;21 likewise, the voltage and temperature dependences of the dynamical conductivity have been analyzed by Vasko and Ryzhii,22 using the Boltzmann approximation. On the other hand, it was pointed out that the population inversion, through interband transitions of electrons and holes near the Dirac points, leads to negative dynamical conductivity in graphene.23 Furthermore, many experimental data have proved the existence of the universal optical conductivity, which is related to interband transitions,24,25 whereas experimental studies of the intraband conductivity have been confined to few researches.26 Against this background, it is known that the strong coupling in the electron−photon interaction in graphene can introduce topological quantum states inside of an energy gap, transforming the energy spectrum of Dirac electrons, and thus changing the electromagnetic response of this material.27 This phenomenon induces quantum phases that manifest themselves through transport and optical properties out of equilibrium; in © 2016 American Chemical Society

ϵ2N = εN2 + ζ 2(k) = εa2N + ℏ2vf2k 2

(1)

where the previous terms are associated with N and k states respectively of photon-dressed electrons in graphene, with e ⃗ being the polarization unit vector, σ̂ as the Pauli matrices, â(â†) as the second quantization operators, εa = evf 2ℏ/ωϵoA , and the N parameter is the photon occupation number associated with coupled electron-photon states. As one can see from eq 1, in contrast to the Landau spectrum of graphene in static magnetic fields the quantization in eq 1 due to electron− photon interaction does not involve macroscopical degeneracy Received: June 17, 2016 Revised: October 24, 2016 Published: October 25, 2016 6797

DOI: 10.1021/acs.nanolett.6b02488 Nano Lett. 2016, 16, 6797−6801

Letter

Nano Letters on the wave vector k. Important to note is that we consider an intense quantizing electromagnetic field, and in this sense for a maximal N = No value, which is fixed by the field source, we define the top of the band gap as εg = εa No = vf eEo/ω. Then, any other N state with energy εN satisfies the relation εN ≤ εg. Therefore, a photon-dressed electron in the N state can be found only inside the band gap. In this context, the gap is formed by a structure of photon-dressed electron states of topological nature. On the other hand, the dressed electron states outside the band gap in the k < k* limit, satisfy the relation ⎛ ℏ2k 2 ⎞ ϵ = ±⎜εg + ⎟ 2m* ⎠ ⎝

σαβ =

Im Παβ(q, ω) (3)

ω

where Παβ(q,ω) is the current−current correlation function, which in the Matsubara formalism takes the form Παβ(q, iω) = −

g A

∫0

β

dτeiωτ⟨Tτjα† (q, τ )jβ (q, 0)⟩

(4)

Here j is the current operator, A is the graphene area, Tτ is the time-ordering operator and g = gsgv is the spin and valley degeneracy. If we neglect spatial dispersion of the ac field, that is, in the qvf ≪ ω limit, then j†α = jα and the off-diagonal elements of conductivity vanish.17,40 On the other hand, in the ℏω ≪ kBT limit the dynamical conductivity takes the form (see Appendix B in Supporting Information):

(2)

where k* = ℏω/2eEo is the threshold value for the wave-vector, Eo = 4πNoℏω/ϵoA is the amplitude for the classical electromagnetic field, and m*−1 ≈ v2f /εg the effective mass. The above expression in eq 2 evinces that the “photondressing” opens an energy gap, introducing an effective mass to Dirac electrons and leading to a parabolic energy dispersion near the band edge. These quasiparticle states outside the band gap are related to interband optical transitions, which determine the main contribution (Drude-like) to the optical conductivity in graphene and they are responsible for metal− insulator transitions as was shown by Kibis.27 Such type of topological states has been related to magneto-optical Franz− Keldysh effect,30 dissipationless electron transport without Joules heating,32 anomalous photon-assisted tunneling,33 as well as Bose−Einstein condensation34 and many other phenomena. In this stage, the goal of the present work is to show that the presence of photon-dressed electrons inside the gap favors subtle quantum oscillations of the dynamical conductivity in graphene. Such kind of behavior (not yet reported) can be observed in experiments of Terahertz (Thz) spectroscopy of graphene monolayers.1−12 Although the electron−photon coupling becomes strong even at weak fields,27 it is possible to describe the dynamical conductivity within the linear response regime for strong polarized ac fields. Oka and Aoki28 showed in the Floquet formalism that an intense circularly polarized ac field, A, deforms the single-body Hamiltonian, and each k point rotates in the Brillouin zone, forming a gap with topological states inside (this gap also can be opened for the case of linear polarization, which can be represented as a combination of clockwise and counterclockwise circularly polarized radiation)35 and the wave function of the system acquires a geometric Aharonov−Anandan (AA) phase independent of the Hamiltonian that introduces the deformation,36 leading to quantized transport, which can be described independently of the order of the ac field intensity in the coupling regime and the conductivity can be treated by the linear theory.37−39 With this background, in order to describe the dynamical conductivity in a graphene layer under intense polarized ac radiation we will use the standard Kubo formula without vertex corrections for two-dimensional systems, following the Oka’s idea, but in the context of photon-dressed electrons27,30 associated with the stationary solution of the time-independent Schrö dinger equation (see Appendix A in Supporting Information)

σ=g

e2 πA

∑ ∫ v 2(k)[2Im k ,N

⎛ ∂f ⎞ G R (N , k)]2 ⎜ − ⎟dϵ ⎝ ∂ϵ ⎠

(5)

where the complete retarded Green’s function is described by the following expression 1 1 + R ϵ − ϵN − Σ (ϵ) ϵ + ϵN − ΣR (ϵ)

G R (N , k ) =

(6)

Here ± ϵN are defined by eq 1 and Σ (ϵ) = Re Σ(ϵ) + i Im Σ(ϵ) is the complex retarded self-energy. The first and second terms in the right-hand of eq 6 are related to the retarded Green function in the conduction and valence bands, respectively. v = ∂ϵ/∂p is the electron velocity and f is the statistical Fermi function. Taking into account that the quadratic imaginary part of the retarded Green function can be expressed as [2 Im GR(k)]2 = G2R + G2A − 2GRGA, one can see that this expression has one first-order pole −2GRGA and one second-order pole G2R + G2A at the complex plane. Then, using the Poisson summation formula R

1 F(0) + 2

∞

∞

∑ F (N ) = ∑ N=1

r =−∞

∫0

∞

F(y)e 2πiry dy (7)

where r ∈ Z; after some integral calculations, we easily obtain an expression for the sum of the quadratic imaginary part of the Green’s function (see Appendix C in Supporting Information):

∑ [2Im GR (N , k)]2 = N ∞

∑ r =−∞

2π i | r | ( 4π ϵ RDe 2 εa |Im Σ|

ϵ2 − ζ 2 ⎛ ) εa2 ⎜1

⎝

+ 8π |r |

|Im Σ|ϵ ⎞ ⎟ εa2 ⎠

(8)

In the previous equation, we took into account that the electron self-energy arises from electron dispersion on short-range impurities and, therefore, it is an energy-dependent complex quantity; then, for simplicity one can neglect the quantity Re Σ because it leads only to a small correction for the chemical potential and this way it does not affect in essence the oscillating effect. In contrast, the imaginary part Im Σ plays a fundamental role labeling the relaxation of the electron momentum. Replacing the expression 8 in eq 5, the dynamical conductivity for the graphene layer can be rewritten in a compact form as follows σ= 6798

g ℏe 2 πAm

∞

∑ r =−∞

⎛ ∂f ⎞

∫ |Imϵ Σ| M(ϵ, r)e2πir ϵ /ε ⎜⎝− ∂ϵ ⎟⎠dϵ 2

2 a

(9)


Letter

Nano Letters ∞

Here we have introduced the parameter ⎛ |Im Σ|ϵ ⎞ 4π ⎟ ⎜ ϵ ϵ + π | | R ( , r ) N ( , r ) 1 8 r D εa2 εa2 ⎠ ⎝

M (ϵ , r ) =

|Im Σ| = gn iVo2

r =−∞

ϵ = εa

(10)

where N(ϵ,r) is defined by the integral ϵ

N (ϵ , r ) =

∫−∞ dζ ϱ(ζ)ζe−2πirζ

2

σD =

σQ =

4g ℏe

(11)

Amεa2

4g ℏe 2 Amεa2

∫

ϵ ⎛ ∂f ⎞ ⎜ − ⎟S( δ , λ ) dϵN (ϵ, 0) |Im Σ| ⎝ ∂ϵ ⎠

σ= (12)

(13)

The S(δ,λ) factor is defined by the summation ∞

S(δ , λ ) =

∑

e−|r| λ cos rδ =

r =−∞

sinh λ cosh λ − cos δ

(14)

Here δ = (2πε2g/ε2a )(1 − 2ϵ/εg) and the disorder parameter λ = πϵ|Im Σ|/ε2a . On the other hand, as we have indicated above, the scatterassisted contribution to the transport properties is expressed via the electron scattering time, which can be defined in terms of the imaginary part of the self-energy through the relation 1/ τ(ϵ) = |Im Σ|, taking into account that in an electron gas most scattering processes are inelastic, and the electron loses energy each time it scatters. We will consider electron scattering on short-range impurities, and for the sake of simplicity we do not attempt to evaluate the self-energy dependence of the quantum number N, so we will look at this quantity as Σ(N,ϵ) ≈ Σ(ϵ). In order to understand the behavior of photon-induced quantum oscillations, we will limit us to the framework of the simple point impurity model V(r − R) = Voδ(r − R), where R is the position vector of randomly distributed impurities and Vo is the strength of the impurity potential. It is important to note that the SCBA is valid not only for pointlike impurities but also for smooth random potentials. The expression for Σ(ϵ) is reduced in the Dyson equation to the summation of subsequent Feynman diagrams without self-intersections, which are not important when the concentration of impurities is small and so one get the self-consistent equation for the self-energy as follows41 Σ(ϵ) = n iVo[1 + gVo ∑ N

2

2 a

ϵ

∫−∞ dζ ϱS(δ , λ)

(16)

4ℏe 2τo Amεa

∫

⎛ ∂f ⎞⎡ ∂S(δ , λ) ⎤ dϵN (ϵ, 0)⎜ − ⎟⎢1 − Γ ⎥ ⎝ ∂ϵ ⎠⎣ ∂λ ⎦

(17)

Figure 1. Conductivity versus frequency for T = 359 K, μ = 43 meV, and τ = 46 fs. σo = e2/4ℏ.

∞

∫−∞ dζ ϱ(ζ)GR (N , k)]

∞

∫−∞ dζ ϱe−2πirζ /ε

where Γ = g8πϵ2/ε3a τo. The N(ϵ,0) factor in eq 17, which is related to electron dispersion along extended states for photondressed electrons (see eq 11), contributes to the explicit form of the dynamical conductivity in two different ways. In the monotonic Drude term σD (first term in eq 17), which is associated with interband absorptions, the N(ϵ,0) factor determines the condition for which interband transitions can take place and at the same time defines the minimal conductivity in graphene. This aspect establishes the main behavior of the terahertz conductivity in this material as it is extensively known from numerously works. On the other hand, in the quantum term σQ in eq 13, which is linked to a more subtle effect (not yet reported), leading to an oscillating dependence of the dynamical conductivity on the frequency of the ac field, the N(ϵ,0) factor contributes to this effect decreasing the amplitude of the oscillations. In Figure 1, we show the conductivity as a function of the frequency from eq 17. From this picture, one evinces the

⎛ ∂f ⎞

∫ dϵN(ϵ, 0) |Imϵ Σ| ⎜⎝ ∂ϵ ⎟⎠λ ∂S(∂δλ, λ)

2 2 ϵ R e 2πir ϵ / εa 2 D εa

Thus, as we have seen from eqs 12, 13, and 16, the calculation of the conductivity is reduced to the solution of the self-energy equation. Using the sum in eq 14 and the fact that the S(δ,λ) parameter weakly depends on ζ(k) and that the relation ∫ ϱ(ζ)dζ ≈ AεgEf/2πℏ2vf2, then eq 16 becomes τo ≈ gϵS(δ,λ)τ(ϵ)/εa. Here the τo = ℏvf /2An i Vo2 πneNo parameter is the scattering time on a single impurity and Ef = ℏvf πne ≈ 40 meV is the Fermi energy of the system without ac field. Keeping the previous expression for τo in mind, the total dynamical conductivity acquires the compact form

/ εa2

and in addition in the presence of impurity scattering, the RD(ϵ,r) = exp(−4πrϵ|Im Σ|/εa2) energy-dependent Dingle factor appears after integration as an intrinsic consequence of the energy-dependent self-energy. Then, with the help of eqs 11 and 1 and performing the summation in eq 9 by the r parameter the dynamical conductivity can be written as a sum of the Drude and quantum terms correspondingly σ = σD + σQ, where 2

∑

modulated behavior of the terahertz conductivity with the rising of the frequency. This quantum effect is stationary because photon-dressed states in eq 1 appear as a solution to the timeindependent Schrodinger equation.27,30 From experimental data of irradiated graphene,1−12 which can support our theoretical results, one observes that the Drude contribution fits the spectral range as normally is known, but it

(15)

We have performed the summation over k in eq 15 by an integration over ζ according to eq 1. Now separating the imaginary and the real parts for the self-energy in eq 15 and using again the Poisson summation formula one obtains 6799


Letter

Nano Letters

parameters but for ϵ = Ef. In this connection, δf is the parameter that characterizes the deviation of ζf from the Ef value at T = 0. The shifting of the singularities, driven by the field frequency, leads to periodic change of the density of states (DOS) (which is related to the Green function in eq 8) for photon-dressed electrons42 and at the same time to the rising of the Fermi wave-vector kf. In other words, as one can see by definition of the εg quantity, it depends inversely on the field frequency; therefore if the frequency increases, then εg decreases and this way decreases the density of photon-dressed electrons inside the gap. The previous explanation can be understood on the basis of electron transitions from inside to outside the gap with the increasing field frequency and basically this effect is responsible for the quantum oscillations reported in the present work. In Figure 4, we show the frequency dependence of the

is obvious that the terahertz conductivity does not fall with the increasing frequency in a smooth monotonic way but rather in an oscillating manner, likewise as we display in Figure 1 (for more details see the behavior of the THz conductivity in Figures 1c and 3b of ref 1, Figure 1c of ref 2, Figure 6 of ref 3, and so forth). In Figure 2 we picture the same dependence as in

Figure 2. Conductivity versus frequency for the same values of parameters as in Figure 1 and temperatures T = 336 K (lower curve), 347 K (middle curve), and 359 K (upper curve).

Figure 1 but for different temperatures, taking into account that − ∂f/∂ϵ = [4T cosh2(ϵ/2T)]−1. Likewise, experiments show a strong dependence of the conductivity on the field intensity. In Figure 3, we show the terahertz conductivity versus the field Figure 4. Conductivity versus frequency for the same values of parameters as in Figure 1; τ = 30 fs (black line) and τ = 44 fs (gray line).

terahertz conductivity for different values of the scattering time. As we see from this curve, the impurities reduce the amplitude, destroying the quantum oscillations and driving the system to a completely Drude behavior. In conclusion, the diminution of the spectrum of topological states inside the energy gap, which is induced by intense ac fields and is driven by the field frequency, leads to a periodic change of electron singularities of the DOS near the band gap edge, resulting in electron transitions, from bound photondressed electron states to extended states outside the energy gap and finally driving to quantum oscillating phenomenon. This effect can be evidenced in the nonmonotonic behavior of the terahertz conductivity by the increase of the field frequency, as is shown by experimental data. On the other hand, it is relevant that the effect of electron scattering processes by impurities favoring interband transitions and at the same time suppressing intraband THz absorptions, which are related to optical transitions from inside to outside the gap. This phenomenon is evinced from Figure 4 with the decreasing scattering time, leading to a decreasing modulation and this way diminishing the oscillatory behavior of the conductivity. Note that the scattering mechanism analyzed in the present work, which corresponds to short-range scattering due to neutral pointlike impurities, is one of the most important in the range of temperatures that we have considered in Figure 2 (300 < T < 400 K). The problem of electron dispersions by acoustic and optical phonons, which are important at high temperatures, will be addressed elsewhere.

Figure 3. Conductivity versus field amplitude for the same values of parameters as in Figure 1. E* = (2ℏω/ϵoA)1/2.

amplitude for the same values of parameters as in Figure 1. The inset of Figure 3 evinces in a clear way the oscillating behavior of the conductivity. In order to show the nature of such kind of oscillations, let us analyze, for sake of simplicity, the system at T = 0 temperature, then − ∂f/∂ϵ ≈ δ(ϵ − Ef), and the dynamical conductivity acquires a very compact form σ=

∂S(δf , λf ) ⎤ 4e 2 τo 2⎡ ζf ⎢1 − Γf ⎥ h εa ⎣ ∂λf ⎦

(18)

The ζf = ℏvfkf quantity is defined by the deviation of the Fermi energy Ef (measured from the top of the band gap) due to nonzero ac fields, which introduce topological states inside the gap. The parameter kf = [2m*|Ef − εg|]1/2/ℏ is the Fermi wave-vector in the presence of ac fields in concordance with eq 1 and for edge states in the effective mass approximation according to eq 2. Γf, δf, and λf are the previously defined 6800


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Nano Letters

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(22) Vasko, F. T.; Ryzhii, V. Phys. Rev. B: Condens. Matter Mater. Phys. 2007, 76, 233404. (23) Satou, A.; Vasko, F. T.; Ryzhii, V. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78, 115431. (24) Nair, R. R.; Blake, P.; Grigorenko, A. N.; Novoselov, K. S.; Booth, T. J.; Stauber, T.; Peres, N. M. R.; Geim, A. K. Science 2008, 320, 1308. (25) Mak, K. F.; Sfeir, M. Y.; Wu, Y.; Lui, C. H.; Misewich, J. A.; Heinz, T. F. Phys. Rev. Lett. 2008, 101, 196405. (26) Liu, W.; Valdés Aguilar, R.; Hao, Y.; Ruoff, R. S.; Armitage, N. P. J. Appl. Phys. 2011, 110, 083510. (27) Kibis, O. Phys. Rev. B: Condens. Matter Mater. Phys. 2010, 81, 165433. (28) Oka, T.; Aoki, H. Phys. Rev. B: Condens. Matter Mater. Phys. 2009, 79, 081406. (29) Physics of Graphene; Aoki, H., Dresselhaus, M., Eds.; Springer International Publishing: Switzerland, 2014. (30) Vega Monroy, R.; Mera Acosta, C. Phys. Rev. B: Condens. Matter Mater. Phys. 2012, 85, 235442. (31) Roslyak, O.; Gumbs, G.; Mukamel, S. J. Chem. Phys. 2012, 136, 194106. (32) Kibis, O. Phys. Rev. Lett. 2011, 107, 106802. (33) Iurov, A.; Gumbs, G.; Roslyak, O.; Huang, D. J. Phys.: Condens. Matter 2012, 24, 015303. (34) Vega-Monroy, R. Phys. E 2014, 63, 134. (35) Syzranov, S.; Fistul, M.; Efetov, K. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78, 045407. (36) Aharonov, Y.; Anandan, J. Phys. Rev. Lett. 1987, 58, 1593. (37) Zhou, Y.; Wu, M. W. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 245436. (38) Yudin, D.; Eriksson, O.; Katsnelson, M. I. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 91, 075419. (39) Kibis, O.; Morina, S.; Dini, K.; Shelykh, I. Phys. Rev. B: Condens. Matter Mater. Phys. 2016, 93, 115420. (40) Falkovsky, L. A.; Varlamov, A. A. Eur. Phys. J. B 2007, 56, 281. (41) Mahan, G. Many-Particle Physics, 2nd ed.; Plenum Press: New York, 1990. (42) Vega-Monroy, R.; Martinez-Castro, O.; Salazar-Cohen, G. Phys. Lett. A 2015, 379, 1169.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.6b02488. Appendices with additional equations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +57 (5) 3197010−1104. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Universidad del Atlántico for the total financial support in developing this work. REFERENCES

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Photon-Induced Quantum Oscillations of the Terahertz Conductivity in

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