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Calculation of divergent photon absorption in ultra-thin lms of a topological insulator

Jing Wang, 1 Hideo Mabuchi, 2 and Xiao-Liang Qi 1

1 Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA2 Department of Applied Physics, Stanford University, Stanford, California 94305-4045, USA

(Dated: November 18, 2013)

We perform linear and nonlinear photon absorption calculations in ultra-thin lms of a topologicalinsulator on a substrate. Due to the unique band structure of the coupled topological surface states,novel features are observed for suitable photon frequencies, including a divergent edge singularityin one-photon absorption process and a signicant enhancement in two-photon absorption process.The resonant frequencies can be controlled by tuning the energy difference and coupling of the topand bottom surface states. Such unique linear and nonlinear optical properties make ultra-thin lmsof topological insulators promising material building blocks for tunable high-efficiency nanophotonicdevices.

PACS numbers: 78.20.Bh 73.20.-r 78.20.-e 78.40.-q

I. INTRODUCTION

Time-reversal invariant topological insulators (TIs) arenew states of quantum matter characterized by an insu-lating bulk state and gapless Dirac-type surface states. 1–4A range of compounds have been found to be three-dimensional (3D) TIs, 5–9 among which layered Bi 2Se3is demonstrated to be a prototype 3D TI with a largeinsulating bulk gap of about 0 .3 eV and metallic sur-face states with a single Dirac cone. 5,6 A thin layerof TI is expected to be a promising material for high-performance optoelectronic devices such as photodetec-tors 10 and transparent electrodes 11 due to its spin mo-mentum locked massless Dirac surface state, which istopologically protected against time-reversal invariantperturbations.

Two-photon absorption (TPA) is a primary process of

interest in various emergent photonics applications.12–16

For application purposes, a good TPA material must dis-play large absorptive nonlinearities tuned within specicspectral regions. 17,18 To gain insight into the origin of large (degenerate) TPA coefficients β , we consider the ex-pression for β in second-order perturbation theory, whichis proportional to the transition dipole moments and jointdensity of states (JDOS):

β (ω) ≡ 2 ωW 2

I 2=

2 ω

I 22π

k i

ψc|H1 |ψi ψi |H1 |ψv

E i (k i ) −E v (k v ) − ω

2

×δ (E c(k c) −E v (k v ) −2 ω) , (1)where ψc , ψi and ψv are Bloch wavefunctions of theelectrons in conduction, intermediate and valence bands,whose energies are E c(k c), E i (k i ) and E v (k v ) and mo-menta are k c , k i and k v . ω is the frequency of light.The delta function expresses energy conservation require-ments for optical transition, and the summations overi extend over all possible intermediate states. The k

summation is over the entire rst Brillouin zone, H1 isthe electron-photon interaction Hamiltonian and I is the

light irradiance. In general, one can get large β whenreaching the resonant condition ( E i −E v = 2 ω). More-over, sharp peaks in the frequency dependence of theTPA coefficient should occur at critical points of the

JDOS, such as Van Hove singularities. Two-dimensional(2D) systems may offer a novel avenue for creating usefulTPA materials, unlike in 3D, Van Hove singularities in2D may induce divergent JDOS.

In this paper we show that thin lms of a TI couldprovide a powerful setting, in which both linear and non-linear optical processes of interest are greatly enhancedand are also highly tunable. A key feature of TIs is theexistence of robust topological surface states, in whichelectrons propagate as massless relativistic fermions asshown in Fig. 1(a). In an ultra-thin lm (5 nm or thin-ner for Bi2Se3) of TIs the top and bottom surface statesare coupled, giving rise to an energy gap. 19–22 The cou-

pling strength is controlled by the lm thickness20

andthe energy difference between the two surface states canbe controlled by substrate or electrical gating. Such tun-ability makes the TI thin lm a unique 2D electron sys-tem. Due to the coupling of surface states, the conduc-tion band minima and valence band maxima occur at thesame nonzero wave-vector, leading to a divergent JDOSand thus a divergent one-photon absorption (OPA) atthe gap frequency, illustrated by optical process α1 inFig. 1(b). Furthermore, by tuning the relative ampli-tude between the gap and the top-bottom surface energydifference, one can achieve a band structure in which atwo-photon process [ β 1 and β 2 in Fig. 1(b)] is greatly en-hanced due to the existence of an intermediate band atthe resonance frequency and the almost divergent JDOSof the initial and nal states [ δ in Fig. 1(b)]. With suchproperties, thin lms of Bi 2Se3 (and similar materialsBi2Te 3 and Sb 2Te 3) are unique material building blocksfor new nanophotonic devices.

The organization of this paper is as follows. After thisintroductory section, Sec. II describes the model for thethin lm of a TI and perturbative approach to calculat-ing the linear and nonlinear optical absorption. SectionIII presents the results for OPA and TPA processes. Sec-

a r X i v : 1 2 0 9 . 6 5 9 7 v 2

[ c o n d - m a t . m e s - h a l l ] 1 5 N o v 2 0 1 3

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tion IV presents the discussion. Section V concludes thispaper.

II. MODEL AND THEORY

The low-energy physics of a TI thin lm is character-ized by two copies of the topological surface states on thetop and bottom surfaces. In the simplest TIs such as theBi2Se3 family, each surface state has a single Dirac cone.The surface state wavefunction is localized on the surfaceand decays exponentially away from the surface with acharacteristic “penetration depth” ξ . For the Bi 2Se3 fam-ily of materials ξ ∼1 nm. For ultra-thin lms with thick-ness comparable with ξ , the overlap between the surfacestate wavefunctions from the two surfaces of the lm be-come non-negligible and hybridization between them hasto be taken into account. In a thin lm TI on a substrate,the chemical potentials of the top and bottom surfacesare inequivalent and the Dirac points are generically atdifferent energies. Considering the inter-surface coupling

and the chemical potential difference one can write downthe following low energy effective model of the thin lmTI which matches well with experiment, 22,23

H0 = vτ z⊗(σx ky −σy kx ) + ∆ h

2 τ x⊗1 + ∆ ib τ z⊗1, (2)

k

E(k)

α1β 1

γ

(a) (b)

TI thin-lm

E1

E2

E3

E4

β 2

α 2

α 3

α 4

δ

(c)

D 21 D 43 D 23D 41

FIG. 1. (color online) (a) Coupling of Dirac cones on op-posite surfaces of a thin-lm TI. (b) Surface band structurewith divergences in JDOS for one- ( α 1 ) and two-photon ( δ )transitions. The doping level of the system is in the gap. Thebands and their labels are in the same color. The possibleone-photon optical transitions are indicated by arrows withan index of α 1 , α 2 , α 3 , α 4 and two-photon optical transitionsby β 1 , β 2 , γ , δ . (c) Optical selection rules for direct interbandtransition. Dji denotes the optical transition from band E i toE j . The ring denotes the constant energy contour for verticaltransition, and arrow indicates the polarization of the opticaleld.

where v is the Dirac velocity, and σi (i = 1, 2, 3) and τ j( j = 1, 2, 3) are Pauli matrices acting on spin space andopposite surfaces, respectively. Time-reversal invariancefollows from [Θ, H0] = 0, where Θ = 1 ⊗iσyK and K iscomplex conjugation. ∆ h is the hybridization betweenthe two surface states. ∆ ib is the inversion symmetrybreaking, which can be substantially modied throughelectrical gating. 22 Here for simplicity we neglect the

higher-order terms in k, and we will discuss the effectof higher-order terms at the end of the paper. The sur-face band dispersion is

E (k ) = ∓ ( v |k |∓∆ ib )2 + (∆ h / 2)2 , (3)

The energy gap is E edge = ∆ h at the wavevector |k | =∆ ib / v. In the following, we consider the doping level of the system is always in the gap. 24 E 1 and E 3 bands areoccupied, while E 2 and E 4 bands are unoccupied. Thusthe low energy optical absorption by the surface statescan occur with photon energy ranging from E edge to gapof the bulk bands.

The direct electron-photon interaction is the dipole in-terband optical transitions, which is determined by min-imal coupling, i.e. , by replacing k by k −eA / c in themodel Eq. ( 2), which leads to the interaction Hamilto-nian

H1 = −ec

vτ z⊗(σx Ay −σy Ax ) . (4)

Here A = Ae is the optical vector potential with ampli-tude A and polarization e . The amplitude A is relatedto the light irradiance by I = √ ωω2A2 / 2πc, and ωis the dielectric constant of the material. Here we haveneglected the small wave vector of light. Taking into ac-count the momentum conservation k for the initial

|ψc

and nal |ψv states, only vertical excitation processescontribute to absorption.

The optical selection rules of the interband transitionare obtained by the polarization operator

Dji (k ) ≡ ψj (k )|H1 |ψi (k ) , (5)

here Dji denotes the transition from state |ψi to |ψj .The explicit form of the wavefuctions |ψi (i = 1 , 2, 3, 4)are

|ψ1,2(k ) =

1

N 1,2

( v | k |− ∆ ib )∓√ ( v | k |− ∆ ib ) 2 +∆ 2h / 4

∆ h / 2 ie− iθ k

( v | k |− ∆ ib )∓√ ( v | k |− ∆ ib ) 2 +∆ 2h / 4

∆ h / 2

ie− iθ k

1

|ψ3,4(k ) = 1N 3,4

( v | k | +∆ ib )± √ ( v | k | +∆ ib ) 2 +∆ 2h / 4

∆ h / 2 ie− iθ k

−( v | k | +∆ ib )± √ ( v | k | +∆ ib ) 2 +∆ 2

h / 4∆ h / 2

−ie− iθ k

1

with N 1,2 and N 3,4 are normalization factor, and θk isthe azimuth angle between k and kx -axis. Thus we have

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the transition for the surface bands D21 ∝ k̂ , D43 ∝−k̂ ,

D41 ∝ n̂ × k̂ and D23 ∝ −̂n × k̂ , where n̂ is a unitvector normal to the momentum k . Such selections rulesis shown in Fig. 1(c). We can see clearly that the opticalselection rules in a thin lm TI is very different fromthat in a conventional direct-gap semiconductor such asGaAs, where the spin wavefunction remains unchanged.This unique optical transition selection rule is due to the

spin momentum locking of surface states.The energy spectrum of the effective model ( 2) isshown in Fig. 1(b). For nite chemical potential offset∆ ib , the coupling between the two surface states leadsto avoided crossing of the energy dispersion at nitewavevectors, and the valence band maxima and conduc-tion band minima coincide on a one-dimensional (1D)ring at |k | = ∆ ib / v in momentum space. This featureis essential for optical properties of the system, since thedensity of states of both conduction and valence bandsdiverge at the same wavevectors, enabling a divergence inthe probability of the optical transition process markedby α1 in Fig. 1(b). Other important optical transitions

beside α 1 are also shown in Fig. 1(b).

III. RESULTS

A. One-photon absorption

The OPA coefficient is

α(ω) = ωW 1

I , (6)

where W 1 is the transition probability rate for OPA perunit area

W 1 = 2π k f = i | ψf |H1 |ψi |2 δ (E fi (k ) − ω) , (7)

where E f i (k ) ≡E f (k ) −E i (k ). Fig. 2(a) shows the OPAspectrum when the doping level is in the gap. Thus al-

FIG. 2. (color online) (a) Contour plot of OPA spectrafor thin lm TIs (logarithmic scale). ∆ h / ∆ ib can be tunedby lm thickness and electrical gating. α(ω) is in units of πα/ √ ω . (b) Line cut for the 4-QLs Bi 2 Se3 thin lm with∆ h = 70 meV and ∆ ib = 53 meV. The edge of the interbandtransition E edge is indicated by an arrow; important featuresare labeled α 1 −α 4 .

lowed optical transitions are E 1 → E 2 , E 3 → E 4 , E 1 →E 4 , E 3 → E 2 . These optical processes α i (i = 1 , 2, 3, 4)contribute to different features at different frequencies asmarked in Fig. 2(a). In the following we will discuss thecontribution of these processes in more details.

1 . When the optical frequency ω < ∆ h , no real OPAwill occur, for the energy conservation of the optical tran-sition cannot be satised.

2 . As the frequency is larger at ω = ∆ h , the opticaltransition α1 can occur, and it will lead to the band edgesingularity in OPA spectrum at the gap energy E edge asshown in Fig. 2. This singularity is directly related tothe JDOS divergence of the surface states in E 1 and E 2bands. Explicitly, the summation over the delta func-tion δ (E 21 (k ) − ω) in Eq. ( 7) can be converted into anintegration over energy by a JDOS,

N (ω) = 14π2 dS k

|∇k [E 2(k ) −E 1(k )]|, (8)

where S k

is the constant energy surface dened byE 21 (k ) = ω. When ω ≤ 4∆ 2ib + ∆ 2

h , the JDOS forE 1 and E 2 bands becomes

N (ω) = 12π

ω

2ω2 −∆ 2h

. (9)

It becomes singular when ω = ∆ h at nite wavevec-tor. This square-root divergent Van Hove singularityof the JDOS at the band edge is characteristic of one-dimensional behavior. 25 For large photon frequencies, theE 1 →E 2 contribution to α (ω) is proportional to ω− 3 .

3 . When the optical frequency is increased at ω =

4∆ 2ib + ∆ 2h , besides the E 1 → E 2 transition, othertransitions E 1 → E 4 , E 3 → E 4 and E 3 → E 2 start tooccur at k = 0, as is labeled by α2 in Fig. 1(b). Theseprocesses will lead to a step discontinuity in the OPAspectra. In particular, OPA from E 3 → E 4 is exactlyzero at ω = 4∆ 2

ib + ∆ 2h and has ω− 2 dependence when

ω/ ∆ ib 1.4 . For frequency ω 4∆ 2

ib + ∆ 2h , the transitions

α 3 (E 1 → E 4) and α4 (E 3 → E 2) will occur far awayfrom the avoided crossing wavevector ∆ ib / v. In thislimit the inter-surface coupling can be neglected, andthe transition occurs within each surface. It has beenstudied in the graphene context 26 –28 that such a tran-sition in a 2D Dirac fermion system leads to a univer-sal frequency-independent contribution πα/ 2√ ω to OPAwith α ≡e2 / c the ne-structure constant. This contri-bution dominates the absorption probability in the highfrequency limit as shown in Fig. 2(b).

In short, as the frequency increases, the OPA spectrumof a TI thin lm rst has a square-root singularity atband edge ω = ∆ h , and then has a step discontinuityat ω = 4∆ 2

ib + ∆ 2h , and approaches πα/ 2√ ω in the

high frequency limit.

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FIG. 3. (color online) (a) Contour plot of TPA spectra of transitions from valence bands to E 4 for thin lm TIs (logarithmicscale). ∆ h / ∆ ib can be tuned by lm thickness and electrical gating; Γ / ∆ ib = 0 .05. β is in units of (4 π 2 / ω ∆ 4

ib )(ve2 /c )2 . (b)Line cut for ∆ h / ∆ ib = 1 .32 (4-QLs Bi 2 Se3 ) and ∆ h / ∆ ib = 1 .6. ∆ h / ∆ ib = 1 .32 has two resonances, labeled β 1 and β 2 . (c)The maximum of β vs. ∆ h / ∆ ib . It should be noticed that the resonance peak at ω = ∆ h / 2 is not shown in this gure as itoriginates from the E 1 →E 2 transition (see Eq. (11) and the text).

B. Two-photon absorption

For the surface state of a bulk TI, the TPA coefficientis obtained by including all possible intermediate statesin the surface bands, which leads to

β thick = 2π2

ωω4 3ve2

c

2

. (10)

There is no resonant feature or Van Hove singularity.In a thin lm new resonant features will appear due

to the inter-surface coupling. There are four allowedtransitions, E 1 → E 4 → E 2 and E 3 → E 4 → E 2 ,

E 1 → E 2 → E 4 (β 1 , β 2) and E 3 → E 2 → E 4 (γ ). Weconsider the case that the doping level is in the gap, sothe latter two with nearly resonant condition dominatethe optical process, as shown in Fig. 1(b). Both transi-tions are included in the calculation of the TPA coeffi-cient if ω ≥ ∆ 2

ib + (∆ h / 2)2 , as both of them satisfyenergy conservation. In Fig. 3 we show numerical resultsfor TPA coefficients and corresponding optical processes.When ∆ h / 2 < ω < ∆ 2

ib + (∆ h / 2)2 , the optical tran-sitions from valence bands to conduction band E 4 is for-bidden by energy conservation, so that β = 0. As thephoton frequency becomes larger, β (ω) has two resonancefrequencies, corresponding to transitions β 1 and β 2 , withthe resonance condition E 4

−E 2 = E 2

−E 1 = ω indi-

cated by Eq. ( 1). These features represent large tunableabsorptive nonlinearities, making thin lm TIs promisingTPA materials for applications. The double resonance isat k1,2 = (5 µ ± 9∆ 2

ib −4∆ 2h )/ 4 v, and it disappears

when ∆ h / ∆ ib reaches a critical value ∆ h / ∆ ib ≥ 1.5 asshown in Fig. 3(a). In practice, the energy ω in Eq. (1)needs to be replaced by ω + iΓ in order to take into ac-count the effect of carrier damping. Here, Γ is assumedto be a constant and inversely proportional to the de-phasing time τ . In our calculation we set Γ / ∆ ib = 0.05.

Fig. 3(b) shows the TPA spectrum for representative val-ues of ∆ h / ∆ ib . β (ω) shows as double resonance featurefor ∆ h / ∆ ib = 1 .32, which corresponds to the experimen-tal values observed for 4 quintuple layers (QLs) Bi 2Se3lm.22,23 The strongest resonance feature occurs at β 1since the optical transition matrix elements at β 1 arelarger than that at β 2 . In contrast, there is no strongresonance for ∆ h / ∆ ib = 1 .60. The process γ does not sat-isfy the resonant condition for any photon frequency, soit has little contribution to the TPA coefficient. Fig. 3(c)shows the maximum of β (at β 1) versus the parameter∆ h / ∆ ib . In particular, although the resonant conditionis satised at ω = ∆ ib when ∆ h = 0, the associated tran-sition from E 1

→ E 2 vanishes. The strongest β occurs

at ∆ h / ∆ ib ≈0.5.The double resonance feature of the TPA coefficient

from transition E 1 → E 4 is due to the Rashba-typesplitting, compared to the single resonance in bilayergraphene. 29 The Rashba splitting also gives rise to thedivergent JDOS at the band edge, which has alreadybeen shown in the OPA coefficient. Obviously, there isno resonant intermediate states in the TPA process fromE 1 → E 2 , however, with the divergent JDOS and nitetransition matrix elements at |k | = ∆ ib , the TPA con-tributed by the process δ around the gap is

E 1

→E 2 : β (ω)

∝

1

ω3 (2 ω)2 −∆ 2h

. (11)

It has a singular feature centered around ω = ∆ h / 2 dueto the Van Hove edge singularity. It shows ω− 3 depen-dence in the resonant region ω ∼∆ h / 2 and ω− 9 depen-dence in the off-resonant regions of ω ∆ h , while TPAof gapless surface states in TIs has a ω− 4 dependence forall photon frequency.

In short, at a given value of ∆ h , the TPA spec-trum of a TI thin lm has a singularity at band edge

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ω = ∆ h / 2. In addition, for small ∆ h there are twoother resonance peaks appearing in the frequency range

ω > ∆ 2ib + (∆ h / 2)2 which merge into one peak and

disappear at a certain ∆ h .

IV. DISCUSSION

Taking into account of both OPA and TPA, the changein the intensity of the light as it passes through the sam-ple is given by

∆ I = −α I −β I 2 . (12)

The total absorption coefficient is given by α total =α + β I . The nonlinearity is characterized by the ra-tio β I /α which depends on the intensity I . Since forthe TI lm we have shown that β has resonance fea-tures occurring at frequencies different from that of α,the ratio β/α can be greatly enhanced, enabling the re-alization of strong nonlinearity at low intensity of light.For the 4-QL Bi 2Se3 , the parameters are estimated by∆ ib = 53 meV. 30,31 In our calculation we have takenτ ∼ 2.3 ps, which leads to Γ = 2 .9 meV. For the reso-nant frequency of β 1.25 < ω/ ∆ ib < 1.8, the conditionβ I /α ∼ 1 can be satised for an (optical) electric eldstrength of 10 5 V/m. Such eld strengths potentiallycould be realized at low incident optical powers usingphotonic resonators with high ratio of quality-factor tomode-volume. 32

The interaction of surface states with phonons and im-purities will cause electron relaxation. In fact, the de-phasing time of the topological surface states observedin experiments.may be longer than the τ we take. Fromthe femtosecond time- and angle-resolved photoemis-sion spectroscopy, 30 a long-lived population of a metallicDirac surface state ( > 10 ps) in Bi2Se3 has been found.Such long dephasing time is directly related to the spintexture of surface states. The electron-phonon interac-tion in TI is also weak because the small Fermi sur-face limits the number of phonon modes coupled withelectrons. According to Ref. 31, the broadening isΓ < 1 meV at T < 50 K.

In general, the electron hole pair at two surfaces un-der external bias will have Coulomb interaction. In themean eld approximation, the interaction here would pre-fer the exciton condensate when the chemical potentialis smaller than ∆ h .33 This is exactly the case considered

here that the doping level is always in the gap. (It shouldbe pointed out that there are many interesting effectswhen the chemical potential is in the conduction band,which need further investigation. 34 ) Such exciton con-

densate will enhance the hybridization of the two surfacestates, and thus the resonance frequency of the system. 22

There are higher order terms such as hexagonal warp-ing term proportional to k3 in the surface state dispersionrelation. 35 With such terms the gap due to avoid crossingof surface states is no longer uniform around the cross-ing wavevectors, and the JDOS becomes nite. However,the warping parameter is very small in Bi 2Se3 when the

crossing energy is lower than 0.22 eV (dened respect tothe Dirac point), so that the JDOS enhancement givenby our low energy effective theory remains valid. 36

It worths mentioning that the gap of TI thin lms isalways less than the bulk gap 0.3 eV, which is in theTHz frequency range. However, Raman processes maylead to transition between E 1 and E 2,4 in the opticalfrequency range where the intermediate states are high-energy bulk states, 37,38 such Raman process should begreatly enhanced due to the divergent JDOS and couldbe more conveniently accessed in experiments and appli-cations.

V. CONCLUSION

In conclusion, the TI thin lm has interesting lin-ear and non-linear optical properties. The unique bandstructure of the coupled topological surface states givesrise to a divergent edge singularity in one-photon absorp-tion process and a signicant enhancement in two-photonabsorption process. The tunable one-photon and two-photon absorption in this system may nd applicationsin spintronics, such as optical generation of spin currentand charge current. 39 Such unique linear and nonlinearoptical properties make ultra-thin lms of TIs promis-ing material building blocks for tunable high-efficiencynanophotonic devices.

ACKNOWLEDGMENTS

We are grateful to Y. Cui, H. Y. Hwang, R. B. Liuand S. C. Zhang for insightful discussions. This work issupported by the Defense Advanced Research ProjectsAgency Microsystems Technology Office, MesoDynamicArchitecture Program (MESO) through contract Nos.N66001-11-1-4105 (J. W. and X. L. Q.) and N66001-11-

1-4106 (H. M.), and by the US Department of Energy,Office of Basic Energy Sciences, Division of MaterialsSciences and Engineering, under contract No. DE-AC02-76SF00515.

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