All Optical Measurement Proposed for the Photovoltaic Hall Effect
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l All Optical Measurement Proposed for thePhotovoltaic Hall Effect
Takashi Oka, and Hideo Aoki
Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, JAPANE-mail: [email protected]
Abstract.
We propose an all optical way to measure the recently proposed gphotovoltaicHall effecth, i.e., a DC Hall effect induced by a circularly polarized light in the absence ofstatic magnetic fields. For this, we have calculated the Faraday rotation angle induced by thephotovoltaic Hall effect with the Kubo formula extended for photovoltaic optical response inthe presence of strong AC electric fields treated with the Floquet formalism. We also pointout the possibility of observing the effect in three-dimensional graphite, and more generally inmulti-band systems such as materials described by the dp-model.
1. Introduction
There are increasing fascinations with new types of Hall effect besides the quantum Hall effectin magnetic fields, where a notable example is the spin Hall effect in topological insulators. Wehave previously proposed, as another new type of Hall effect, the gphotovoltaic Hall effecth,which should take place when we shine a circularly polarized light to graphene[1]. The effectis interesting since this Hall effect occurs in the absence of uniform magnetic fields. Physically,the photovoltaic Hall current originates from the Aharonov-Anandan phase (a non-adiabaticgeneralization of the geometric (Berry) phase) that an electron acquires during its circular motionaround the Dirac points in k-space in an AC field, and bears, in this sense, a topological originsimilar to the quantum Hall effect. However, unlike the quantum Hall effect, the photovoltaicHall conductivity is in general not quantized.We have originally proposed an experimental setup to measure the photovoltaic Hall currentthrough the gate electrodes attached to the sample. In the present report we propose a second,all-optical setup, which employs the Faraday rotation as in the optical Hall effect discussed inrefs. [2, 3], which becomes here non-linear optical measurements, since this is a “pump-probe”type experiment, where one optically probes the Hall effect that is induced in a system pumpedby a circularly polarized light from a continous laser source. This effect should grow with theintensity of the applied circularly polarized light. We also point out the possibility of observingthe effect not only in graphene, but also in three-dimensional graphite, and more generally inmulti-band systems such as materials described by the dp -lattice.
2. Kubo-formula extended for optical responses in a strong background light
So let us first propose a method to measure the photovoltaic Hall effect with non-linear opticalmeasurements. The experimental setup is schematically displayed in Fig.1(a), where the sampleis subject to a strong circularly polarized light with strength F and frequency (photon energy) Ω.he strong external AC field changes the properties of the electronic state and the photo-inducedband, or the Floquet band to be more precise, can acquire a non-trivial topological nature. Thiswas discussed in ref. [1] as an emergence of the photovoltaic Berry’s curvature. Optical repsonsecan be drastically change in the presence of the photovoltaic Berry’s curvature, which we showhere to be captured through an optical Faraday (or Kerr) rotation measurement. Our startingpoint to calculate the optical response is the Floquet expression for the Kubo formula extendendto incorporate photovoltaic transports [1], σ ab ( A ac ) = i Z d k (2 π ) d X α,β = α [ f β ( k ) − f α ( k )] ε β ( k ) − ε α ( k ) hh Φ α ( k ) | J b | Φ β ( k ) iihh Φ β ( k ) | J a | Φ α ( k ) ii ε β ( k ) − ε α ( k ) + iη (1)with a, b = x or y . The DC expression can be extended to AC responses, which reads σ xy ( ω ; A ac ) = − i Z d k (2 π ) d X α,β = α f β ( k ) ε β ( k ) − ε α ( k ) (2) × " hh Φ α ( k ) | J x | Φ β ( k ) iihh Φ β ( k ) | J y | Φ α ( k ) ii ε β ( k ) − ε α ( k ) + ω + iη − hh Φ α ( k ) | J y | Φ β ( k ) iihh Φ β ( k ) | J x | Φ α ( k ) ii ε β ( k ) − ε α ( k ) − ω + iη , for the xy -component andRe σ xx ( ω ; A ac ) = πω Z d k (2 π ) d X α<β [ f α ( k ) − f β ( k )] |hh Φ β ( k ) | J x | Φ α ( k ) ii| δ ( ε β ( k ) − ε α ( k ) − ω ) (3)for the xx -component. Here | Φ α ( t ) i is the α -th Floquet state, the double brakets includes atime average, f α ( k ) the non-equilibrium distribution (occupation fraction) of the α -th Floquetstate, and η a positive infinitesimal. Strictly speaking, we must take into account the effect ofrelaxation (phonons, electrodes, etc.) to determine the non-equilibrium distribution function asa detailed balanced state with photo-absorbtion, photo-emission and the relaxations. Here weassume for simplicity that the non-equilibrium distribution function is given by f α ( k ) = X i |hh ψ i | Φ α ( k ) ii| f FD ( E i , µ ; β eff ) , (4)where f FD ( E i , µ ; β eff ) = 1 / [exp( β eff ( E i − µ ))+1] is the equilibrium Fermi-Dirac distribution withan effective inverse temperature β eff . This corresponds to employing a sudden approximation. Aproper treatment of the relaxation can be done using the Keldysh Green’s function as in ref. [1]. In discussing the experimental feasibility, we can start from the relation between the inducedconductivities σ xx ( ω ) , σ xy ( ω ) and the Faraday-rotation angle Θ H [2],Θ H = 12 arg " n + n s + ( σ xx + iσ xy ) / ( cε ) n + n s + ( σ xx − iσ xy ) / ( cε ) (5)= 1( n + n s ) cε σ xy ( ω ) ∼ ( σ xy in units of e h ) × . (6)This gives an estimate for the experimental precision demanded to observe the photovoltaicHall effect through optical measurements. For example, in Fig.1 below, we estimate the photo-induced response σ xy ( ω ; A ) for a single-layered graphene to be around 0 .
2. This corresponds toa Faraday-rotation angle of Θ H ∼ . × . ω I m σ ( ω ) xy F =0.00 F =0.01 F =0.04 F =0.03 F =0.02 F =0.05 energy of CPL R e σ ( ω ) xx F =0.00 F =0.01 F =0.04 F =0.03 F =0.02 F =0.05 ω R e σ ( ω ) xy F =0.00 F =0.01 F =0.04 F =0.03 F =0.02 F =0.05 ω (b) (d)(c) Faraday rotation circularly polarized lightprobepump F ; strength Ω : energy (a) Figure 1. (a) Schematic, all-optical experimental setup to measure the photovoltaic Hall effectvia non-linear optical responses. (b)(c)(d) Photovoltaic optical response in the honeycomb lattice(single-layer graphene). The optical absorption spectrum Re σ xx ( ω ; A ) (b) and the “photovoltaicoptical Hall coefficient” σ xy ( ω ; A ) (c,d) are plotted for various values of strength F of thecircularly polarized light. Note the different scales between (b) and (c,d). Here the energy oflight is fixed to Ω = 1 . t ∼ β eff = 100.
3. Photovoltaic optical response
Using the extended Kubo formula (eqns. (2),(3)), we calculate the non-linear optical responsein the presence of a strong circularly-polarized light in the honeycomb lattice. The opticalabsorption spectrum in Fig. 1 (b) exhibits the following: (i) At the energy of the circularly-polarized light (CPL), there is a dip in the absorption ( σ xx ( ω ), which is an analogue of holeburning. (ii) Near the DC limit ( ω → Hall coefficient are plotted in Fig. 1 (c,d), which grow in a low-frequencyregion. The peak position of the real part shifts to higher frequency as the strength of the CPLis increased. This is important, since the experimental detection is easier for higher photonenergies.
Photovoltaic Hall effect is not restricted to systems having a Dirac cone as in the single-layergraphene. To show this, we calculate the photovoltaic optical response in bilayer graphene (see[4] for notation and refs) described by the Hamiltonian, H = k x − ik y t p k x + ik y t p k x + ik y k x − ik y . (7)We plot the photovoltaic optical response in Fig. 2 for this effective model. The basic featuresare similar to those in the monolayer graphene. The result suggests the possibility of observingphotovoiltaic Hall effect in mulilayer graphene and graphite, since one can block-diagonalize theHamiltonian for a multilayer system into sub-blocks of single and bi-layer graphene componentsas was shown in ref. [5].
4. Photovoltaic Berry’s curvature in the dp -lattice We finally explore the possibility of more generally observing the photovoltaic Hall effect inmaterials other than graphene/graphite. For a typical multiband model, we consider here the dp -lattice, which is a well-studied lattice in connection with the high-Tc cuprates[7]. Here .0 0.2 0.4 0.6 0.80.060.080.100.120.140.160.18 F =0.000 F =0.001 F =0.004 F =0.003 F =0.002 F =0.005 ω energy of CPL R e σ ( ω ) xx (a) I m σ ( ω ) xy ω F =0.000 F =0.001 F =0.004 F =0.003 F =0.002 F =0.005 (c) R e σ ( ω ) xy F =0.000 F =0.001 F =0.004 F =0.003 F =0.002 F =0.005 ω (b) Figure 2.
Photovoltaic optical response in a bilayer graphene for Ω = 1 . , t p = 0 . Figure 3. (a) dp -lattice. (b) The band dispersion in the dp model. (c)(d)(e) PhotovoltaicBerry’s curvature in the three bands (top d-band (c), and middle (d), bottom (e) p-bands) for t dp = 0 . , ε d − ε p = 4 , F = 0 . , Ω = 3 . .we neglect the electron-electron interaction to consentrate on the one-particle properties. TheHamiltonian is given by H = ε d − ε p − t sin k x − t sin k y − t sin k x t dp sin k x sin k y − t sin k y t dp sin k x sin k y , (8)where ε d − ε p is the potential difference between the d and p bands. The photovoltaic DC Hallconductivity is expressed as Berry’s curvature as [1, 6] σ xy ( A ac ) = e Z d k (2 π ) d X α f α ( k ) (cid:2) ∇ k × A α ( k ) (cid:3) z , (9)in terms of a gauge field A α ( k ) ≡ − i hh Φ α ( k ) |∇ k | Φ α ( k ) ii . Note that this expression reduces tothe TKNN formula [8] in the adiabatic limit.The band dispersion of the dp -lattice is shown in Fig. 3 (b) and the photovoltaic Berry’scurvature for each band is plotted in (c)-(e). A peak in Berry’s curvature emerges at the centerof the Brilliouin zone in the d -band, while the curvatue has complex structures in the p -bands.his is because the two p -bands intersect with each other, for which the ac field induces a bandmixing, while the d -band is separated from the p -bands with the separation larger than thephoton energy Ω considered in the figure. Figure 4.
Photovoltaic Berry’s curvature in the d -band of the dp -lattice for severa values ofthel photon energy Ω for t dp = 0 . , ε d − ε p = 4 , F = 0 . dp -lattice when compared to a system with a massless Dirac cone as in graphene? In orderto clarify this, we have calculated the photovoltaic Berry’s curvature for several values of Ω inFig. 4. When Ω is smaller than the gap, the photovoltaic Berry curvature increases as the valueof Ω approaches the band gap. When Ω exceeds the gap, when a direct photo-transition becomespossible, the Berry’s curvature changes into a concentric form with a negative part (not apparentin the figure). In the massless Dirac case, both the cone-like contribution and the concentricform coexist[1, 6], while in the dp -lattice they appear separately. Another important obervationis that the magnitude of the photovoltaic Hall effect becomes large in multiband systems whenΩ is slightly below the band gap, whereas in the massless Dirac case small Ω is better.To conclude, we have studied the possibility of observing the photovoltaic Hall effect in anall-optical fashion. We have also shown that the effect is not restricted to lattices with a masslessDirac cone but is a universal phenomenon in multi-band systems. HA has been supported inpart by a Grant-in-Aid for Scientific Research No.20340098 from JSPS, TO by a Grant-in-Aidfor Young Scientists (B) from MEXT and by Scientific Research on Priority Area “New Frontierof Materials Science Opened by Molecular Degrees of Freedom”. References [1] T. Oka, and H. Aoki,
Phys. Rev.
B 79 , 081406 (R) (2009); ibid , 169901(E) (2009).[2] T. Morimoto, Y. Hatsugai, and H. Aoki, Phys. Rev. Lett. , 116803 (2009).[3] Y. Ikebe, T. Morimoto, R. Masutomi, T. Okamoto, H. Aoki, and R. Shimano,
Phys. Rev. Lett. , 256802(2010).[4] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim,
Rev. Mod. Phys. , 109(2009).[5] M. Koshino and T. Ando, Phys. Rev. B , 085425 (2007).[6] T. Oka and H. Aoki, J. Phys.: Conf. Ser. , 062017 (2010).[7] F. C. Zhang and T. M. Rice,
Phys. Rev.
B 37 , 3759 (1988).[8] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs,
Phys. Rev. Lett.49