Polarization-dependent photocurrents in polar stacks of van der Waals solids
PPolarization-dependent photocurrents in polar stacks of van der Waals solids
Y. B. Lyanda-Geller,
1, 2
Songci Li, and A. V. Andreev Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907 USA Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907 USA Department of Physics, University of Washington, Seattle, WA 98195 USA (Dated: September 7, 2015)Monolayers of semiconducting van der Waals solids, such as transition metal dichalcogenides(TMDs), acquire significant electric polarization normal to the layers when placed on a substrateor in a heterogeneous stack. This causes linear coupling of electrons to electric fields normal to thelayers. Irradiation at oblique incidence at frequencies above the gap causes interband transitionsdue to coupling to both normal and in-plane ac electric fields. The interference between the twoprocesses leads to sizable in-plane photocurrents and valley currents. The direction and magnitude ofcurrents is controlled by light polarization and is determined by its helical or nonhelical components.The helicity-dependent ballistic current arises due to asymmetric photogeneration. The non-helicalcurrent has a ballistic contribution (dominant in sufficiently clean samples) caused by asymmetricscattering of photoexcited carriers, and a side-jump contribution. Magneto-induced photocurrent isdue to the Lorentz force or due to intrinsic magnetic moment related to Berry curvature.
PACS numbers: 73.63.-b, 78.67.- n, 73.50.Pz, 72.15.Gd
Introduction.
Since the discovery of graphene [1] awhole class of novel two dimensional (2D) materials,called van der Waals solids (vdWs), has been identi-fied [2, 3]. In these materials 2D monolayers with strongin-plane bonding are coupled by weak van der Waals in-teractions. Few-monolayer thick structures of vdW mate-rials have electronic and optical properties that can differdrastically from those of the bulk phases [4]. vdW ma-terials exhibit phenomena associated with valley degreesof freedom, such as valley Hall currents [5] and valley-selective carrier photoexcitation by circularly polarizedlight [6], related to topological properties of the bands,such as Berry curvature and valley-dependent magneticmoment [7]. Stacking of monolayers of different vdWsolids enables fabrication of novel artificial structureswith interesting electronic properties [2, 3].In their natural form most vdW materials are nonpo-lar. When monolayers of different vdW materials arestacked in a heterostructure or placed on a substrate,an electric dipole moment perpendicular to the layersarises. This allows for photogalvanic effects (PGE): elec-tric currents due to illumination by light in the absenceof external electric field. In particular, a photocurrentarises between the top and bottom contacts of a het-erojunction fabricated from monolayers of different vdWsolids [8]. This current does not depend on the polariza-tion of light, is caused by spatial separation of photoex-cited electrons and holes in a junction, and belongs toa class of effects in which the direction of the photocur-rent is governed by spatial inhomogeneities of the sam-ple or its illumination. Another class of effects, in whichthe direction of photocurrent or photovoltage is deter-mined by the polarization of light [9, 10], occurs evenin uniformly illuminated spatially homogeneous solids.Recently, polarization-sensitive photocurrents were ob- served [11] when the 2D conduction layer formed at theinterface of a WSe stack and the substrate was irradi-ated at frequencies below the band gap.Here we show that coupling of radiation to the elec-tric dipole moment in stacks of undoped semiconductingvdW solids leads to sizable polarization-dependent pho-tocurrents for frequencies above the band gap. Quantuminterference between this coupling and electron couplingto the in-plane electric field component of the radiationresults in carrier photogeneration in the conduction andvalence bands that leads to a valley and net currents.For 2D structures with reflection symmetry brokenby the dipole moment, polarization-dependent PGE cur-rents, to linear order in light intensity, can be expressedin terms of a polar vector d = (0 , , d z ) perpendicular tothe layers by the phenomenological relation j = ξ d × i [ E × E ∗ ] + ζ [ E ∗ ( d · E ) + E ( d · E ∗ )] . (1)Here E is the complex electric field amplitude of amonochromatic light, E ( t ) = (cid:60) ( E e − iωt ), and the (real)phenomenological parameters ξ and ζ describe, respec-tively the circular and linear PGE. In Eq. (1) the elec-tric field of the radiation is assumed spatially uniform andthe photon momentum is neglected. The in-plane pho-tocurrent arises when the sample is illuminated at obliqueincidence, as shown in Fig. 1, and its direction and mag-nitude are determined by the polarization of light.Determination of the physical mechanism of the pho-tocurrent and evaluation of the phenomenological param-eters d , ξ and ζ in Eq. (1) requires a microscopic theory.The photocurrents arise due to: i) asymmetric photo-electron generation (with different generation rate foropposite electron momenta) [12–14], and ii) asymmet-ric kinetics (when light-induced symmetric momentumdistribution leads to the current due to asymmetric scat-tering, due to side jumps, spin relaxation, or evolution a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p FIG. 1. (color online) Schematic representation of thesystem. Irradiation of a semiconducting polar TMD mono-layer by helical light at oblique incidence, θ (cid:54) = 0, generatesa helicity-dependent net photocurrent perpendicular to theplane of incidence yz . For linear polarization, a net currentis generated in the plane of incidence. in magnetic field), [15–21]. These mechanisms describewell experiments detecting polarization-dependent cur-rents in bulk semiconductors, such as Te [22] and GaAs[23–25], and photocurrents in III-V type heterostructures[26]. Both asymmetric photogeneration and kinetics playan important role in the discussion below.When the mean free path of the photoexcited carriersexceeds their de Broglie wavelength, the photocurrent(1) can be expressed in terms of the electron distributionfunction f l ( p , r ) as [17, 27, 28] j = e (cid:88) p ,l [ v l ( p ) + δ v l ( p )] f l ( p ) . (2)The first term here with the group velocity in a band l v l ( p ) = ∂(cid:15) l ( p ) /∂ p describes the ballistic current. Thesecond term is the side jump (shift) current due to thedisplacement R l (cid:48) ,l ( p (cid:48) , p ) of the center of mass of the wavepacket during a transition from state l, p to state l (cid:48) , p (cid:48) (asa result of scattering [29, 30] or photoabsorption [17, 18]).The correction to the velocity is expressed in terms of thetransition probability W l (cid:48) ,l ( p (cid:48) , p ) as δ v l ( p ) = (cid:88) l (cid:48) , p (cid:48) W l (cid:48) ,l ( p (cid:48) , p ) R l (cid:48) ,l ( p (cid:48) , p ) . (3)The magnitude of the side jump is expressed in terms ofthe phase of the transition matrix element T l (cid:48) ,l ( p (cid:48) , p ) as R l (cid:48) ,l ( p (cid:48) , p ) = Ω l (cid:48) ( p (cid:48) ) − Ω l ( p ) − ( ∂ p + ∂ p (cid:48) ) (cid:61) ln T l (cid:48) ,l ( p (cid:48) , p ) , (4)where Ω l ( p ) is the Berry connection in band l . Gaugeinvariance of (4) is obvious: When Ω l ( p ) → Ω l ( p ) − ∂ p χ l ( p ), T l (cid:48) ,l ( p (cid:48) , p ) → T l (cid:48) ,l ( p (cid:48) , p ) e iχ l ( p ) − iχ l (cid:48) ( p (cid:48) ) .In a spatially uniform steady state, and in the ab-sence of static external electric and magnetic fields, thenonequilibrium part δf l ( p ) of the electron distribution function is determined by the balance between the photo-generation due to direct interband transitions J l ( p ) andrelaxation and recombination of photoexcited carriers, (cid:88) l (cid:48) , p (cid:48) w l (cid:48) ,l ( p (cid:48) , p ) [ δf l (cid:48) ( p (cid:48) ) − δf l ( p )] + J l ( p ) = 0 , (5)where w l (cid:48) ,l ( p (cid:48) , p ) is the probability of momentum relax-ation [31]. Below we apply Eqs. (2) and (5) to the studyof polarization-dependent currents (1) in polar stacks ofsemiconducting TMDs, such as MoS and WSe . Asymmetric photogeneration.
Semiconducting TMDat low number of monolayers are direct band semicon-ductors with strong coupling to light and sizable chargecarrier mobility [4, 7, 35]. The inter-layer tunneling isweak and we neglect it. Since in this approximation thetotal in-plane photocurrent is the sum of contributionsof individual layers, we consider the photocurrent in asingle layer of TMD either placed on a substrate or in apolar stack. We assume that photon energy is not toofar from the absorption threshold. In this case only elec-trons with momenta near the K and K (cid:48) points of thehexagonal Brillouin zone absorb light and produce pho-tocurrent, see left panel in Fig. 2. The effective two-bandHamiltonian for such low energy electrons is [5, 6, 36] H = v ( τ z σ x p x + σ y p y ) + ∆ σ z , (6)where the momentum p is measured from the K or K (cid:48) point, v has dimensions of velocity, and ∆ is half thebandgap between the spin-nondegenerate conduction andvalence bands. The Pauli matrices σ i act on the bandpseudospin, and τ z acts on the valley pseudospin.At normal incidence of the radiation, electrons coupleto the in-plane component of the ac electric field. Thecorresponding coupling Hamiltonian is obtained fromEq. (6) by the usual substitution p → p − e A /c , where A is the vector potential and e is the electron charge.This results in valley-selective transitions for circularly-polarized light [6]; Application of an in-plane dc electricfield results in valley current [5]. At oblique incidence,electrons in a polar stack also couple to the normal com-ponent of the ac electric field, E z ( t ). The full couplingof electrons to the (uniform) ac electric field is given by V = − evc ( τ z σ x A x + σ y A y ) + 1 c d z ˙ A z σ z , (7)where the electric field enters through the time derivativeof the vector potential, E = − ˙ A /c , and d z is the differ-ence between the dipole moments of electron states in theconduction and valence bands, which arises as follows. If E bz is a built-in electric field in a polar TMD stack, the to-tal z − component of the electric field is E tz = E bz + E z ( t ).This electric field couples orbitals even in z , that form theconduction and valence bands described by (6), to oddin z higher and lower band states with energies (cid:15) s , with s labeling odd bands. Then the energies of the bottom FIG. 2. (Color online). Left: Direct transitions (red arrow)between electron states in the valence and conduction bandsshown for one of the valleys by black circles. Right: Asym-metric photogeneration rate. of the conduction band and the top of the valence band (cid:15) c ( v ) change: δ(cid:15) c ( v ) = (cid:80) s | ( eE tz z ) c ( v ) s | / ( (cid:15) c ( v ) − (cid:15) s ). Thusa coupling of charge carriers to light linear in electricfield E z ( t ) arises, and the dipole moment difference d z = e (cid:80) s [( E bz z ) cs z sc / ( (cid:15) c − (cid:15) s ) − ( E bz z ) vs z sv / ( (cid:15) v − (cid:15) s )]. Thiscoupling plays a crucial role in generation of polarization-dependent photocurrent in vdW materials. The value of d z can be estimated from the measured [8] dependenceof the band gap on the applied external electric field per-pendicular to the layers, d z = − d ∆ /dE z .Optical transitions between the valence (-) and con-duction (+) band in the K -valley are described by thematrix elements V K − + ( p ) = Ψ K + ( p ) † V Ψ K − ( p ), where thewavefunctions Ψ K ± ( p ) corresponding to energies ± (cid:15) = ± (cid:112) ( vp ) + ∆ are (cid:0) Ψ K ± ( p ) (cid:1) T = (cid:16) ± vp − / (cid:112) (cid:15) ∓ ∆) (cid:15), (cid:112) (cid:15) ∓ ∆ / (cid:15) (cid:17) . (8)Here p ± = p x ± ip y , and the superscript T indicates amatrix transposition. For the K (cid:48) -valley, the wavefunc-tions are obtained by replacing p − in Eq. (8) with − p + .The rate of direct optical transitions, see Fig. 2, in the K ( j = 1) or K (cid:48) ( j = 2) valley, assuming fully occupiedvalence band and empty conduction band, can be deter-mined using the Fermi golden rule, J + ,j = − J − ,j = J j = π (cid:126) | V j − , + | δ ( (cid:126) ω − (cid:15) ), and is given by J j ( p )= 2 π (cid:126) (cid:18) e | E | vω (cid:19) Z ( p ) δ ( (cid:126) ω − (cid:15) ) × (cid:20) − | e z | + (cid:15) (cid:15) − ( − j κ z ∆ (cid:15) + | e z | ( ωd z p ) e (cid:15) − v (cid:15) (cid:2) ( | e x | − | e y | )( p x − p y ) + 2 S xy p x p y (cid:3) + ωe(cid:15) p · (cid:18) ∆ (cid:15) [ κ × d ] + ( − j [ ˆ z × ˆ S d ] (cid:19)(cid:21) . (9)Here ˆ z is the z -axis unit vector, e = E / | E | is the lightpolarization vector, and the pseudovector κ = i e × e ∗ and the tensor ˆ S , S ij = e i e ∗ j + e ∗ i e j , characterize, re-spectively, the helical and the non-helical components oflight polarization, with ( ˆ S d ) T = ( S xz , S yz , S zz ) d z . TheSommerfeld factor [37] Z ( p ) accounts for the Coulomb in-teraction between the photogenerated electron and hole.In the 2D case for a quadratic energy dispersion, Z ( p ) =2 [1 + exp ( − π (cid:126) /pa B )] − [38], where a B = (cid:126) ε/µe isthe exciton Bohr radius, ε is the dielectric constant, p isthe electron or hole momentum, and µ is the reduced ef-fective mass. In our model, p = √ (cid:15) − ∆ /v , µ = ∆ /v .The momentum dependence of the photogeneration inEq. (9) is illustrated in the right panel of Fig. 2. Theasymmetry of photogeneration responsible for the in-plane photocurrent arises from the interference betweencoupling of electrons to the in-plane electric field of lightand the linear Stark coupling to the normal field E z caused by the dipole d . It is described by the two termslinear in the electron momentum in last line of Eq. (9).The different angular harmonics of the nonequilibriumdistribution function relax independently. Therefore forthe photocurrent it is sufficient to consider the first angu-lar harmonic δf (1) l ( p ) of the nonequilibrium distributionfunction, δf (1) l ( p ) = ( A l + τ z B l ) · ˆ p , where A l and B l characterize valley-even and odd asymmetry of momen-tum distribution, respectively, and ˆ p = p / | p | is a unitvector along the electron momentum. The relevant scat-tering probability in Eq. (5) is given by w l (cid:48) ,l ( p (cid:48) , p ) → δ l,l (cid:48) (cid:20) τ l ˆ p · ˆ p (cid:48) + τ z τ skl ˆ z · ˆ p × ˆ p (cid:48) (cid:21) , (10)where 1 /τ l and 1 /τ skl are respectively the transport andskew momentum relaxation rates in band l . Ballistic photocurrent.
The first term in Eq. (2) for thephotocurrent describes charge transfer during ballisticmotion of electrons, and is characterized by the asymmet-ric in momentum part of the distribution function. Thelatter is caused by asymmetric photogeneration or subse-quent asymmetric scattering. The ballistic circular PGEarises directly due to the valley-even asymmetric photo-generation (first term in the last line of Eq. (9)). We findthat the dominant ballistic linear PGE requires a conver-sion, via skew scattering, of the valley-odd photogenera-tion (last term in Eq. (9)) into a valley-even asymmetricmomentum distribution. Skew scattering arises only inthe second Born approximation. As a result, althoughboth transport and skew scattering rates are proportionalto the impurity concentration, the skew scattering rateis smaller in the parameter τ l /τ skl ∼ δ l (cid:28)
1, where δ l is aphase shift of electron scattering off impurities in a band l . The ballistic contribution to linear and circular PGEcoefficients ξ and ζ in Eq. (1) are given by ξ bal = (cid:16) e (cid:126) (cid:17) Z ( p ω ) (cid:2) ( (cid:126) ω ) − (2∆) (cid:3) ∆( (cid:126) ω ) ( τ c + τ v ) , (11a) ζ bal = ξ bal (cid:126) ω ∆ ( τ c + τ v ) (cid:18) τ c τ skc + τ v τ skv (cid:19) . (11b)Here τ c and τ v are the momentum relaxation timesin the conduction and valence bands, and p ω = (cid:126) (cid:112) E exc / ( (cid:126) ω − /a B with E exc = µe / (2 (cid:126) ε ) beingthe exciton binding energy in three dimensions.Taking d z ∼ . A , τ c ∼ τ v ∼ − s (from thereported mobility 200 cm / (V · s) [35]), and the helic-ity κ = 0 .
7, we find the strength of the one mono-layer circular PGE signal ∼ − A / W for ∆ = 0 . (cid:126) ω = 1 .
95 eV. This value exceeds the helicity-dependentspin-galvanic signal in 2D GaAs [26]. The ratio of thenet linear PGE and circular PGE is small as τ /τ sk . Side jump photocurrent.
Since the leading ballistic lin-ear PGE, Eq. (11b), is inversely proportional to the impu-rity concentration, in sufficiently high mobility samples itdominates the side jump current. The side jump current,e.g., due to direct optical transitions to ζ is obtained us-ing Eqs. (3), (4) and the expressions for V K ( K (cid:48) ) − + ( p ). Theresult is ζ dirsj = 8 (cid:16) e (cid:126) (cid:17) Z ( p ω )∆ ( (cid:126) ω ) ω . Other contributions to ζ stem from the asymmetry ofimpurity-assisted photoabsorption or from the side jumpsof photogenerated carriers due to scattering off impuri-ties, and are of the same order of magnitude as ζ dirsj . Valley photocurrent.
In addition to the net current,the asymmetric photogeneration leads to the valley cur-rents equal in magnitude but oppositely directed in the K and K (cid:48) valleys, defined by j balv = e (cid:80) p ,l ( − j v p ,l δf l ( p ).The dominant ballistic contributions to circular and lin-ear valley PGE can be found using Eqs. (5), (9) and (10): j balv = | E | (cid:20) ξ bal (cid:126) ω ∆ (cid:16) ˆ z × ˆ S d (cid:17) + ζ bal ∆ (cid:126) ω ˆ z × [ κ × d ] (cid:21) , (12)where ξ bal and ζ bal are given by Eq. (11). The linearand circular valley PGE are related, respectively, to thenet circular ( ξ bal ) and linear ( ζ bal ) PGE. Therefore at τ l /τ skl ∼ δ l (cid:28) K -valley electrons at theleft boundary of the monolayer with respect to the di-rection of the net linear PGE, and K (cid:48) -valley electronson the right. If intervalley scattering is weak, this accu-mulation can be measured in transport experiments [39].Valley currents can be possibly also captured experimen-tally investigating non-local transport [40] or non-linearphenomena[41]. Magneto-induced photocurrent.
Magnetic field perpen-dicular to the layers, H = H ˆ z , induces a Hall-like current j balH = | E | ˆ z × (cid:104) ξ H κ × d + ζ H ˆ S d (cid:105) . (13)One obvious contribution to (13) arises from the Lorentz-force term ec v l ( p ) × H · ∂δf l ( p ) ∂ p included into the left hand side of the Boltzmann equation (5). The correspondingballistic contributions ξ balH and ζ balH to the coefficients ξ H and ζ H are related to ξ bal and ζ bal in Eq. (11) by ξ balH = ξ bal ω H ( τ c − τ v ) , ζ balH = ζ bal ω H (cid:16) τ c τ skc − τ v τ skv (cid:17) τ c /τ skc + τ v /τ skv , (14)where ω H = 2 eHv / (cid:126) ωc is the cyclotron frequency.A more interesting mechanism of magneto-inducedphotocurrent arises from the opposite magnetic field de-pendence of the band gap in the K and K (cid:48) valleys;∆ → ∆ ± M · H , where M is the orbital magnetic mo-ment in the Bloch state [42] at the K or K (cid:48) points inthe Brillouin zone. The latter is related to the Berrycurvature [42] F jz ( p ) = ∂ p x Ω jy ( p ) − ∂ p y Ω jx ( p ) and in oursystem is given by [5, 6] M jz = e F jz ( p ) (cid:112) ∆ + v p (cid:126) c = ( − j ev ∆2 (cid:126) c (∆ + v p ) . (15)The corresponding contribution to the net ballisticmagneto-induced photocurrent may be expressed as j m = M z H ˆ z × ∂ j balv ∂ ∆ , (16)where j balv is the magnitude of the H = 0 ballistic valleycurrent (12). The magnetic moment contribution (16)is ∼ j balv (cid:126) ω H / ( (cid:126) ω − ∼ j balv ω H τ l /τ sk . Theratio of j m to linear PGE in Eq. (1) at H = 0 is (cid:126) ω H τ sk / [( (cid:126) ω − τ ], which can easily reach ∼ ω H τ , usu-ally defining the Lorentz force effects. The role of (16)is further enhanced by the partial cancellation betweenthe Lorentz force contributions of electrons and holes tolinear and circular PGE in Eq.(14), and the magnetic mo-ment contribution may become the dominant magneto-induced photocurrent in lower mobility samples. Discussion . Besides polar TMD systems, our approachbased on Eqs. (6) and (7) may be used to study lin-ear and circular PGE induced by interband transitionsin polar boron nitride structures. Another interestingsystem is a Bernal stacked graphene bilayer placed ona substrate [43], in which the photocurrents predictedhere can potentially be tuned by gating the system. Wenote that the existence of helicity-dependent current in-duced by an in-plane external magnetic field and Rashba-like spin-orbit effects [44, 45] was recently suggested ingraphene [46]. We expect that photocurrents in polar bi-layer graphene, due to the coupling of light to the orbitaldipole moment d , will be significantly larger.This work was supported by the U. S. Departmentof Energy Office of Science, Basic Energy Sciences un-der awards number de-sc0010544 (YLG) and DE-FG02-07ER46452 (S. L. and A. A.). We are grateful to DavidCobden, Vladimir Falko, Boris Spivak and Xiaodong Xufor useful discussions. [1] A. K. 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