Current induced by a tilted magnetic field in phosphorene under terahertz laser radiation
CCurrent induced by a tilted magnetic field in phosphorene under THz laser radiation
Narjes Kheirabadi
Department of Physics, Sharif University of Technology, Tehran 14588-89694, Iran
We study the cyclotron resonance effect in the first order ac current, magnetic ratchet effect,and second harmonic generation in phosphorene in the presence of a steady tilted magnetic fieldand under THz laser radiation. We show that dependent on the angular frequency of incominglight, there are resonances dependent on the value of ω c + . We have discussed the direction and themagnitude of the deduced currents for different radiation polarizations and we have compared theresults with the situation where the perpendicular magnetic field is zero. I. INTRODUCTION
The discovery of graphene in 2014 has opened a newarea in the field of nanomaterials and their applications[1, 2]. The success of graphene discovery has started apursuit in the research for two–dimensional (2D) materi-als beyond graphene [3]. Consequently, a range of mono-layer 2D materials from graphene to transition metaldichalcogenides (TMDCs) and phosphorene has been fab-ricated [4–7]. Each of these materials has different elec-tronic and optical properties. For example, graphene isa zero bandgap semiconductor, phosphorene has a 2 eVnatural bandgap, and most of the studied TMDCs havebandgaps larger than 1 eV. Hence, based on differentmaterials, different electronic and optoelectronic appli-cations in the field of 2D materials is possible.In this article, the understudy 2D material is phos-phorene. We have selected this material to study itsresponse to THz laser radiation because of some rea-sons. First, the mobility of carriers in phosphorene ishigher than other 2D materials including TMDCs ( ≤ V − s − ) and phosphorene conducts electrons at asimilar rate as that of graphene (650 cm V − s − at roomtemperature). It makes phosphorene suitable for high–frequency electronic applications [8–10]. Second, whilecurrent graphene detector performances are strongly lim-ited by the large dark currents that dominate under anon-zero bias operation, the direct bandgap of phospho-rene makes it a good candidate for applications, includ-ing transparent photovoltaics, photodetection, ultra-fastphotonics, and high–frequency optoelectronics [9, 11–14].Indeed, the I on /I off ratio of 10 has been recently re-ported for phosphorene and makes it suitable for detec-tion of THz frequency light [9, 15]. Third, in phospho-rene, each phosphorus atom is covalently bonded withthree adjacent phosphorus atoms. Therefore, each p or-bital retains a lone pair of electrons. Because of the sp hybridization, phosphorene does not form an atomicallyflat sheet, like graphene. This property causes an intrin-sic in-plane anisotropy that deduces to a specific angle-dependent conductivity [9]. The puckered structure ofphosphorene deduces to a strong anisotropy in electricconducting and it is important to have novel devices withanisotropic properties [14]. While, the photonic, andelectronic properties of graphene, and TMDCs are largelyisotropic and do not exhibit a significant directional de- pendence [7, 10]. Fourth, this inorganic material, phos-phorene, can be easily integrated with other photonicor optoelectronic components of alternative 2D materi-als, like graphene, or with silicon–based technologies [9].Consequently, we have selected phosphorene to study lin-ear, and nonlinear responses of an anisotropic 2D mate-rial under THz laser radiation. The field of nonlinearoptics, where the response is proportional to the higherpowers of the electric field, has become extremely im-portant for ultrafast signal processing. While, linear andnonlinear transport effects in electric field phenomena ingraphene are in the focus of the current research [16–20],the linear and nonlinear current response in phosphoreneare less studied [21]. The aim of this article is the studyof the current response of phosphorene where it is under asteady magnetic field. Overall, our results provide a newunderstanding of the anisotropic nonlinear optical prop-erties of an anisotropic 2D material, which may be usedfor polarized optical applications, optical switching de-vices, and photodetection that relies on the conversion ofabsorbed photons into an electrical signal that are prob-ably the most explored black–phosphorus based photonicdevices [22]. The calculations of this paper are valid for (cid:126) ω ≤ (cid:15) f where ω is the laser radiation angular frequencyand (cid:15) f is Fermi level; semi-classical regime. Hence, theresults of this study are valid for the THz and microwaveradiation types to develop novel electronic devices formicrowave– and THz–optoelectronic.Here, we will show that under a steady tilted magneticfield and THz laser radiation, a first order ac current, asecond order dc current (ratchet effect) and a second or-der ac current (second harmonic generation (SHG)) willbe deduced in phosphorene. The ratchet effect is a non-linear response to a driving light field result in a dc cur-rent. SHG is a second order in electric field effect accord-ingly an ac current with two times the frequency relatedto incoming ac radiation is generated. Additionally, be-cause of the presence of a perpendicular magnetic field,the cyclotron resonance affects the strength of the de-duced currents in phosphorene. Cyclotron resonance ofthe magnetic ratchet effect and SHG has been studied inisotropic materials like 2D electron gas [23] and bilayergraphene [16]. However, this effect has not been studiedin anisotropic materials like phosphorene, which we wantto consider in this article. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b II. BOLTZMANN EQUATION
The tight–binding Hamiltonian of phosphorene underthe effect of an in–plane magnetic field is derived, before[21]. The Hamiltonian of phosphorene in a steady in-plane magnetic field and in the basis of ( A , B , A , B ) T (Fig. 1) is H = U f + f f f + f f ∗ + f ∗ U + δ f ∗ + f ∗ f f ∗ f + f U + δ f (cid:48) + f (cid:48) f ∗ + f ∗ f ∗ f (cid:48) ∗ + f (cid:48) ∗ U . We assume that b = ed B (cid:107) / − e is the electroncharge, d is the interlayer distance, B (cid:107) is the in–planemagnetic field, in–plane momentum is p = ( p x , p y , U and U , δ is the interlayerpotential asymmetry, and a x and a y are the length of theunit cell into the x (armchair) and y (zigzag) directions,respectively (Fig. 1). Hence, we have f = 2 t cos a y ( p y + b x )2 (cid:126) exp (cid:2) i ( p x − b y ) (cid:126) a cos α (cid:3) ,f (cid:48) = 2 t cos a y ( p y − b x )2 (cid:126) exp (cid:2) i ( p x + b y ) (cid:126) a cos α (cid:3) ,f = t exp[ − i (cid:126) p x d (cid:48) sin β ] ,f = 2 t cos a y ( p y + b x )2 (cid:126) × exp (cid:2) − i ( p x − b y ) (cid:126) (2 d (cid:48) sin β + a cos α (cid:3) ,f (cid:48) = 2 t cos a y ( p y − b x )2 (cid:126) × exp (cid:2) − i ( p x + b y ) (cid:126) (2 d (cid:48) sin β + a cos α (cid:3) ,f = 4 t cos[ p x (cid:126) ( d (cid:48) sin β + a cos α p y (cid:126) a sin α ,f = t exp[ i p x (cid:126) ( a x − d (cid:48) sin β )] . In above equations, the intralayer coupling is t , inter-layer couplings are t , t and t , d (cid:48) is the distance between B and A atoms, α is (cid:92) A B A = (cid:92) B A B = 96 . ◦ , β is (cid:92) A B A − ◦ = 108 . ◦ − ◦ = 18 . ◦ [24, 25], the up-per layer is located at d/
2, and the lower layer is locatedat − d/ E (cid:107) ( t ) = E (cid:107) exp( − iωt ) + E ∗(cid:107) exp( iωt ) , where E (cid:107) = ( E x , E y ), and E ∗(cid:107) = ( E ∗ x , E ∗ y ). Out of an equilibrium condition caused by the laserradiation, the electron distribution function is dependenton the momentum p and time t ; f ( p , t ). If we considerthat the under study system is spatially homogeneous, f ( p , t ) satisfy the following Boltzmann kinetic equation − e (cid:18) E (cid:107) + V g × B ⊥ (cid:19) · (cid:79) p f ( p , t ) + ∂f ( p , t ) ∂t = S { f } , (1)where the group velocity is V g = V g,x cos φ ˆi + V g,y sin φ ˆj , φ is the polar angle of momentum, B ⊥ is the perpen-dicular magnetic field and S { f } is the collision integral.Additionally, V g,x = p/m xx and V g,y = p/m yy , where p = | p | , m xx and m yy are effective mass along the x and y directions, respectively [26]. To solve the Boltz-mann equation, Eq. 1, we consider that the distributionfunction is the following series f ( p , t ) = (cid:88) n,m f nm e imφ − inωt , where f nm coefficients are functions of the total energyof an electron, (cid:15) , and m and n are integers. Then, wemultiply the Boltzmann equation in exp( − ijφ + ilωt ), j and l are integers, and integrate over a period of 2 π of angle φ and a period of time; t . Therefore, coupledequations between different f nm coefficients are achievedin the following form γ l,j f lj = α j − f l − j − + (cid:101) α j − f l +1 j − + β j +2 f lj +2 + (cid:101) β j − f lj − + η j +1 f l − j +1 + (cid:101) η j +1 f l +1 j +1 + δS lj . (2)In Eq. 2, γ l,j = τ − | j | , p − ilω + ijω c + , where τ − | j | , p isthe relaxation time of the jth angular harmonic of theelectron distribution function, ω c + = eB ⊥ / (2 m + ) is thecyclotron frequency including the cyclotron resonance ef-fect, and 1 /m + = 1 /m xx +1 /m yy . Also, linear in electricfield α , β , and η operators are α j = e ( E x − iE y )2 (cid:18) − jp + ∂∂p (cid:19) , ˜ α j = e ( E ∗ x − iE ∗ y )2 (cid:18) − jp + ∂∂p (cid:19) ,β j = 12 ω c − (cid:0) ip ∂∂p + j (cid:1) , (cid:101) β j = 12 ω c − (cid:0) − ip ∂∂p + j (cid:1) ,η j = e ( E x + iE y )2 (cid:18) jp + ∂∂p (cid:19) , ˜ η j = e ( E ∗ x + iE ∗ y )2 (cid:18) jp + ∂∂p (cid:19) . Here, we have ω c − = eB ⊥ / (2 m − ) and 1 /m − = 1 /m xx − /m yy . In Eq. 2, δS lj is the correction to the scatteringcaused by an in–plane magnetic field.FIG. 1: (Top left) the structure of phosphorene and related coupling parameters t i ( i = 1 , , ..., x is armchairand y is zigzag direction. (Top right) the unit cell of phosphorene. The side view of four atoms in the unit cell ofphosphorene. A and B atoms are located on the lower layer, and A and B are located on the upper layer. Theintralayer distance between atoms in one unit cell is a , and d (cid:48) is the distance between B and A atoms in differentlayers. Parameters U , U , δ indicate different on–site energies. (Bottom) The side view of phosphorene includingcoupling parameters, atomic distances and the (cid:92) A B A angle [24].For an anisotropic 2D electron gas like phosphorene,where ξ is the unit matrix of the electric field, the relax-ation time of the jth angular harmonic, τ − | j | , p , is [26, 27] τ − | j | , p ( ξ, p )= 2 π (cid:126) (cid:88) p (cid:48) |(cid:104) p (cid:48) | δH | p (cid:105)| δ ( (cid:15) p − (cid:15) p (cid:48) ) × (cid:26) − [ ξ. V g ( p (cid:48) )] τ | j | , p (cid:48) [ ξ. V g ( p )] τ | j | , p (cid:27) . (3)Note that for isotropic materials, we have ω c − = 0 and ω c + = eBV g /p and τ | j | , p ( ξ, p ) = τ | j | . So, Eq. 2 and 3 arevalid for isotropic materials, as well [16, 23, 28, 29].For static impurities, we can show that [16, 17, 21] δH = N imp (cid:88) j =1 ˆ Y u ( r − R j ) , (4)where N imp is the number of impurities, u ( r − R j ) de-scribes the spatial dependence of the impurity potentialand ˆ Y is a dimensionless matrix describing the additionaldegree of freedom related to the structure within the unitcell. We neglect interference between different impuri-ties, we use the Fourier transform of the impurity poten-tial and perform a harmonic expansion of the impuritypotential; so, we have | ˜ u ( p (cid:48) − p ) | = (cid:88) m (cid:48) ν m (cid:48) e im (cid:48) ( φ (cid:48) − φ ) . In addition, we assume that electrons are trapped in ahuge box with the length L and under a periodic poten-tial. Hence, the current density is J = − gL (cid:88) p e V g f ( p , t ) , where g is the spin degeneracy factor ( g = 2). In thermalequilibrium, and where there is not any applied electricfield, only the f harmonic that is given by the Fermi–Dirac distribution function is nonzero. However, when anac electric field is applied to the system, other harmonicsarise and this gives the possibility of having a linear (firstorder) or nonlinear (higher orders) in electric field cur-rents. In this study, we consider all nonzero first orderor second order in electric field currents. Hence, basedon coupled equations (Eq. 2), f lj harmonics in terms ofthe equilibrium distribution function ( f ) should be cal-culated. In addition, we assume that the system is a de-generate electron gas at low temperature condition; so,we have ∂f /∂(cid:15) ≈ − δ ( (cid:15) − (cid:15) f ). III. PHOSPHORENE
To find f lj harmonics in Eq. 2, we should find the cor-rection to the scattering caused by the in–plane magneticfield, δS lj . We have δS lj = L (cid:90) ∞ (cid:20) (cid:88) m f lm Γ( (cid:15) ) d(cid:15) × (cid:90) π δW p (cid:48) p (cid:18) e imφ (cid:48) − ijφ − e imφ − ijφ (cid:19) dφ π dφ (cid:48) π (cid:21) , (5)where Γ( (cid:15) ) is the electronic density of states per spin andper unit area and δW p (cid:48) p is the change of the scatteringrate caused by the in–plane magnetic field. The scat-tering rate of an electron passes through a phosphorenein a steady magnetic field to the linear order in B (cid:107) andmomentum is calculated before [21]. Accordingly, for thecase of the asymmetric disorder where the lower layer( ζ = 1) or the upper layer ( ζ = −
1) symmetry is brokenby the disorder ( z → − z asymmetry), ˆ Y in Eq. 4 is equalto (cid:0) ˆ I + ζ ˆ σ z ⊗ ˆ I (cid:1) /
2, where ˆ I is the 2 × σ z is a Pauli matrix. Consequently, δW p (cid:48) p has thefollowing general form δW p (cid:48) p ( ζ, U , U , δ )= 2 π (cid:126) n imp L | ˜ u ( p (cid:48) − p ) | δ ( (cid:15) p (cid:48) − (cid:15) p ) × (cid:26) (cid:126) C ( ζ, U , U , δ ) b y p (cos φ + cos φ (cid:48) )+ 1 (cid:126) C ( ζ, U , U , δ ) b x p (sin φ + sin φ (cid:48) ) (cid:27) . (6)In Eq. 6, n imp = N imp /L , C coefficients are dependenton on–site energies and disorder types. Consequently,according to Eq. 5, we can show that δS l = 0 ,δS l = Λ (cid:0) C B y + iC B x (cid:1) f l ,δS l − = Λ (cid:0) C B y − iC B x (cid:1) f l − ,δS l = Λ (cid:0) C B y − iC B x (cid:1) f l ,δS l − = Λ (cid:0) C B y + iC B x (cid:1) f l − , where Λ = edπn imp ΩΓ( (cid:15) ) p/ (cid:126) and Ω = − ( ν − ν ). IV. FIRST ORDER AC CURRENT
The first order ac current is the result of f and f − harmonics and the complex conjugate of those terms; f nm = (cid:0) f − n − m (cid:1) ∗ . We can show that the first order ac cur-rent is J = 2 Re { σ E e − iωt } , where σ is the conductivity tensor. The conductivitytensor components are σ ii = σ i σ (cid:48) ii and σ ij = σ i σ (cid:48) ij ,where i and j indicate the x or y directions and σ i = ge (cid:15) ) C ph pV g,i , (7) σ (cid:48) xx = σ (cid:48) yy = (1 − iωτ , p ) τ , p (1 − iωτ , p ) + ( ω c + τ , p ) , (8) σ (cid:48) xy = − ( ω c + + ω c − + iω c − / τ , p (1 − iωτ , p ) + ( ω c + τ , p ) (9) σ (cid:48) yx = ( ω c + − ω c − − iω c − / τ , p (1 − iωτ , p ) + ( ω c + τ , p ) (10)Here, we have ω c + + ω c − = eB ⊥ /m xx and ω c + − ω c − = eB ⊥ /m yy . These calculations are valid for a degenerateelectron gas, (cid:15) f (cid:29) k B T , all parameters are evaluated onthe Fermi surface and above results are in agreement withthe results related to isotropic materials [16]. In Eq. 7,we have ∂∂p = C ph p ∂∂(cid:15) ,C ph = s (cid:126) (cid:20) γ E g + (cid:0) η v/c + ν v/c (cid:1)(cid:21) , where s is the band index and it is +1 for the conductionband and − E g is the direct energygap, E g = 0 .
912 eV, γ = 0 .
480 eVnm, η v = 0 .
038 eVnm , ν v = 0 .
030 eVnm , η c = 0 .
008 eVnm , and ν c = 0 . are from Ref. [30].According to Eqs. 8 to 10, there are resonances at ω c + = ± ω and at these frequencies the current is muchlarger than at ω c + = 0. V. RATCHET EFFECT
The ratchet effect is a nonlinear response to a drivinglight field where f and f − harmonics result in a dccurrent. The ratchet current is calculated based on thefollowing equation J = − geL (cid:88) p V g (cid:0) f e iφ + f − e − iφ (cid:1) . For Θ = | E x | − | E y | , Θ = E x E ∗ y + E ∗ x E y , and Θ = i (cid:0) E x E ∗ y − E ∗ x E y (cid:1) , we can show that the current is J x = | E | (cid:0) B (cid:48) y Re [ M ,x ] + B (cid:48) x Im [ M ,x ] (cid:1) +Θ (cid:0) B (cid:48) y Re [ M ,x ] − B (cid:48) x Im [ M ,x ] (cid:1) +Θ (cid:0) B (cid:48) y Im [ M ,x ] + B (cid:48) x Re [ M ,x ] (cid:1) +Θ (cid:0) B (cid:48) y Re [ M ,x ] + B (cid:48) x Im [ M ,x ] (cid:1) ,J y = | E | (cid:0) − B (cid:48) y Im [ M ,y ] + B (cid:48) x Re [ M ,y ] (cid:1) − Θ (cid:0) B (cid:48) y Im [ M ,y ] + B (cid:48) x Re [ M ,y ] (cid:1) +Θ (cid:0) B (cid:48) y Re [ M ,y ] − B (cid:48) x Im [ M ,y ] (cid:1) +Θ (cid:0) − B (cid:48) y Im [ M ,y ] + B (cid:48) x Re [ M ,y ] (cid:1) , where B (cid:48) x = C B x and B (cid:48) y = C B y . In addition, M coefficients are M ,i = − ge C ph p (cid:0) γ − , + 1 γ , (cid:1) × (cid:26) V g,i Λ γ , γ , p Γ( (cid:15) ) + C ph (Γ( (cid:15) ) V g,i Λ γ , γ , p (cid:1) (cid:48) (cid:27) ,M ,i = ge C ph p Λ (cid:0) γ − , γ − , + 1 γ , γ , (cid:1) × (cid:26) V g,i γ , p Γ( (cid:15) ) − C ph (Γ( (cid:15) ) V g,i γ , p (cid:1) (cid:48) (cid:27) ,M ,i = ge C ph p Λ (cid:0) γ , γ , − γ − , γ − , (cid:1) × (cid:26) V g,i γ , p Γ( (cid:15) ) − C ph (Γ( (cid:15) ) V g,i γ , p (cid:1) (cid:48) (cid:27) , where ( . . . ) (cid:48) ≡ ∂ ( . . . ) /∂(cid:15) and all parameters are evaluatedon the Fermi surface. Additionally, for ω c + = 0, wherethe perpendicular magnetic field is zero, the above resultsare in complete agreement with previous results [21].Moreover, M ,i is response to the linearly polarizedlight, M ,i is response to the unpolarized light and M ,i isresponse to the circularly polarized light. For linearly po-larized light, we can assume that E ∗ x = E x = ( E /
2) cos θ and E ∗ y = E y = ( E /
2) sin θ , where θ is the polarizationangle. We also assume that B (cid:48)(cid:107) = (cid:0) B (cid:48) x + B (cid:48) y (cid:1) / and ϕ (cid:48) = arctan (cid:0) B (cid:48) y /B (cid:48) x (cid:1) . For linearly polarized light, wecan show that the current in the x and y directions are J x = E B (cid:48)(cid:107) (cid:26) | M ,x | sin( ϕ (cid:48) + χ ,x )+ | M ,x | cos(2 θ + ϕ (cid:48) − χ ,x + π (cid:27) , (11) J y = E B (cid:48)(cid:107) (cid:26) | M ,y | cos( ϕ (cid:48) + χ ,y )+ | M ,y | sin(2 θ + ϕ (cid:48) − χ ,y + π (cid:27) , (12)where χ ,i = arg ( M ,i ) and χ ,i = arg ( M ,i ). For un-polarized light, M ,i related terms are equal to zero,but M ,i related terms survive. In addition, M ,i re-lated current has a notable resonance for ω c + = ± ω andthe strength of the ratchet effect is highest for ω c + = 0(Fig. 2). Current induced by unpolarized light, M ,i hasresonances for ω = ± ω c + that deduces to a dc currentthat is much larger in comparison to where ω c + = 0; B ⊥ = 0. The other resonance is related to ω = ± ω c + .However, in this case, the deduced dc current is smallerthan where ω c + = 0.For circularly polarized light, we can show that E ∗ x = E x = E / E ∗ y = − E y = − iµE /
2, where µ = 1( − M coef-ficients considering a momentum-independent scatteringtime τ = τ = τ and ωτ = 5.this radiation type, where χ ,i = arg ( M ,i ), we can showthat the current density is J x = E B (cid:48)(cid:107) (cid:26) | M ,x | sin( ϕ (cid:48) + χ ,x )+ µ | M ,x | sin( ϕ (cid:48) + χ ,x ) (cid:27) , (13) J y = E B (cid:48)(cid:107) (cid:26) | M ,y | cos( ϕ (cid:48) + χ ,y )+ µ | M ,y | cos( ϕ (cid:48) + χ ,y ) (cid:27) . (14)The above equations show that the response to circularlypolarized light is dependent on the radiation helicity. Forthe M ,i , the major resonance effects are related to ω = ± ω c + and there is also a resonance effect for ω = ± ω c + and both cause a current that is stronger than the currentwhere B ⊥ = 0 (Fig. 2). According to Eqs. 11 to 14, thedirection of the ratchet current for circularly and linearlypolarized lights are dependent on the ϕ (cid:48) and χ phases. VI. SHG EFFECT
Assuming Θ = E x − E y , and Θ = 2 E x E y and Θ = E x + E y , we can show that the general current form fora degenerate electron gas is J x = 2 Re (cid:26)(cid:2) Θ (cid:0) N ,x B (cid:48) y + N ,x B (cid:48) x (cid:1) +Θ (cid:0) − N ,x B (cid:48) y + N ,x B (cid:48) x (cid:1) +Θ (cid:0) N ,x B (cid:48) y + N ,x B (cid:48) x (cid:1)(cid:3) e − iωt (cid:27) , (15) J y = 2 Re (cid:26)(cid:2) Θ (cid:0) N ,y B (cid:48) y − N ,y B (cid:48) x (cid:1) + Θ (cid:0) N ,y B (cid:48) y + N ,y B (cid:48) x (cid:1) +Θ (cid:0) − N ,y B (cid:48) y + N ,y B (cid:48) x (cid:1)(cid:3) e − iωt (cid:27) , (16) N coefficients in Eqs. 15 and 16 are N ,i = − ge (cid:2) V g,i C ph ΛΓ( (cid:15) ) (cid:0) γ , γ , γ , + 1 γ , − γ , − γ , − (cid:1) + C ph pγ , (cid:0) Γ( (cid:15) ) V g,i Λ γ , γ , p (cid:1) (cid:48) + C ph pγ , − (cid:0) Γ( (cid:15) ) V g,i Λ γ , − γ , − p (cid:1) (cid:48) (cid:3) ,N ,i = − ge i (cid:2) V g,i C ph ΛΓ( (cid:15) ) (cid:0) γ , γ , γ , − γ , − γ , − γ , − (cid:1) + C ph pγ , (cid:0) Γ( (cid:15) ) V g,i Λ γ , γ , p (cid:1) (cid:48) − C ph pγ , − (cid:0) Γ( (cid:15) ) V g,i Λ γ , − γ , − p (cid:1) (cid:48) (cid:3) ,N ,i = ge (cid:2) V g,i C ph ΛΓ( (cid:15) ) (cid:0) γ , γ , γ , + 1 γ , − γ , − γ , − (cid:1) − C ph p Λ γ , γ , (cid:0) Γ( (cid:15) ) V g,i γ , p (cid:1) (cid:48) − C ph p Λ γ , − γ , − (cid:0) Γ( (cid:15) ) V g,i γ , − p (cid:1) (cid:48) (cid:3) ,N ,i = − ige (cid:2) V g,i C ph ΛΓ( (cid:15) ) × (cid:0) γ , γ , γ , − γ , − γ , − γ , − (cid:1) − C ph p Λ γ , γ , (cid:0) Γ( (cid:15) ) V g,i γ , p (cid:1) (cid:48) + C ph p Λ γ , − γ , − (cid:0) Γ( (cid:15) ) V g,i γ , − p (cid:1) (cid:48) (cid:3) . Here, all parameters are evaluated on the Fermi surface.Note that, for zero perpendicular magnetic field, ω c + = 0, N ,i and N ,i vanish. Also, assuming ψ ,i = arg ( N ,i ), ψ ,i = arg ( N ,i ) and ψ ,i = arg ( N ,i ), we can show thatthe current density deduced by a linearly polarized lightis J x = E B (cid:48)(cid:107) (cid:26) | N ,x | sin(2 θ + ϕ (cid:48) ) cos(2 ωt − ψ ,x )+ | N ,x | cos(2 θ + ϕ (cid:48) ) cos(2 ωt − ψ ,x )+ | N ,x | sin ϕ (cid:48) cos(2 ωt − ψ ,x )+ | N ,x | cos ϕ (cid:48) cos(2 ωt − ψ ,x ) (cid:27) , (17) J y = E B (cid:48)(cid:107) (cid:26) − | N ,y | cos(2 θ + ϕ (cid:48) ) cos(2 ωt − ψ ,y )+ | N ,y | sin(2 θ + ϕ (cid:48) ) cos(2 ωt − ψ ,y )+ | N ,y | cos ϕ (cid:48) cos(2 ωt − ψ ,y ) − | N ,y | sin ϕ (cid:48) cos(2 ωt − ψ ,y ) (cid:27) . (18)Consequently, ϕ (cid:48) affects the deduced current directionand ψ phases determine the time lag between the incom-ing radiation and the deduced current. For unpolarizedlight, only N ,i and N ,i related currents survive.For circularly polarized light, the current density is J x = E B (cid:48)(cid:107) (cid:26) | N ,x | sin(2 ωt − ψ ,x + µϕ (cid:48) )+ | N ,x | cos(2 ωt − ψ ,x + µϕ (cid:48) ) (cid:27) ,J y = E B (cid:48)(cid:107) (cid:26) − | N ,y | cos(2 ωt − ψ ,y + µϕ (cid:48) )+ µ | N ,y | sin(2 ωt − ψ ,y + µϕ (cid:48) ) (cid:27) . Accordingly, ϕ (cid:48) and ψ phases determine the time lagbetween the incoming radiation and the deduced cur-rent. For N ,i and N ,i , the major resonance hap-pens if ω = ± ω c + . And, there is another resonance at ω = ± ω c + /
2. The deduce dc current in these two casesis stronger than where ω c + = 0. For N ,i and N ,i , themajor resonance effect is related to ω = ± ω c + and then ω = ± ω c + . There is also a resonance at ω = ± ω c + / ω c + = 0; there isnot any perpendicular applied magnetic field. VII. DISCUSSION
To discuss about the strength of the effect, first weshould determine C coefficients in Eq. 6. To estimatethese prefactors in phosphorene, we have substituted thevalues of coupling parameters, t i ( i = 1 , , ..., a x , a y , α , β and d (cid:48) in the de-rived change of scattering rate (Eq. 6 and Fig. 1). Con-sequently, we can show that C coefficients are dependenton the disorder type that is on the top or bottom layerand on-site energies. Further details may be found inRef. [21]. Accordingly, it has been shown that to havenonzero C coefficients, z → − z symmetry and inter layersymmetry should be broken; U − U (cid:54) = 0. For instance,it has been shown that for the valence band, where thedisorder is on the lower layer, δ = 20 meV, U = 0 eV and U = 40 meV, C = 0 .
034 ˚A and C = − . × − ˚A . For the valence band, we also have m xx = 0 . m , m yy = 1 . m ( m is electron free mass) [26]. We alsoassume that ν is independent of energy and it is equalto what has been calculated for bilayer graphene [17], B x = B y = 7 T, and inter layer distance is d = 2 .
13 ˚A[24, 25]. For n imp = 10 m − , for impurity distance 0nm, according to Ref. [26], we can assume that for allharmonics and in the x and y directions, the relaxationtime is 0 . µ A / cm; similar to what has been observed for the caseof monolayer graphene [18].Furthermore, we consider the effect of the change ofinter layer asymmetry, U − U , on ϕ (cid:48) . For the valenceband, when the lower layer is disordered, ζ = 1, U = 0eV and assuming δ = 20 meV, when B x = B y and U changes from 0 to 40 meV, the ϕ (cid:48) is − π/ − accuracy. On the other hand, if we consider that for aconstant inter layer asymmetry, B (cid:107) rotates in the planeof phosphorene, where δ = 20 meV, U = 40 meV andthe lower layer is disordered, because of the magnitude ofthe C /C , ϕ (cid:48) has three amounts dependent on the direc-tion of the magnetic field in the plane of phosphorene. Ifthe applied magnetic field is parallel or antiparallel to thex direction, armchair edge, the ϕ (cid:48) is equal to zero. Unlessthat is − π/ B x B y > π/ B x B y < ϕ (cid:48) is a rectangular step function. We can also havethe same results for the conduction band [21]. This hap-pens for anisotropic phosphorene because the magnitudeof C and C that are coefficients of the magnetic fieldin the y and x directions, respectively, are different inthe scattering rate; there is an in–plane anisotropy. Forisotropic materials, B x and B y appears with the samemagnitude of coefficients in the scattering rate [16].Additionally, a semi-Faraday effect has been predictedin SHG current in isotropic materials [16]. It means thatfor an incoming in–plane polarized light, by the changeof the direction of the in–plane magnetic field, the de-duced SHG current polarization direction rotates. Inanisotropic phosphorene, for linear polarized light, ac-cording to Eqs. 17 and 18, SHG current is dependent on ϕ (cid:48) , that is dependent on the direction of the magneticfield rather than its magnitude and it could be ± π/ ω c + = eB ⊥ / (2 m + ) for a perpen-dicular magnetic field, B ⊥ = 1 T, for electron carriers( m xx = 0 . m , m yy = 0 . m ) is 7 . × rads − . Forthe cyclotron resonance condition ω = ω c + , this ω c + cor-responds to a linear frequency of light f ≈ .
11 THz. Onthe other hand, in the introduction section, it is men-tioned that the results of this paper is valid for (cid:126) ω ≤ (cid:15) f .For f ≈ .
11 THz, (cid:126) ω is 0 . (cid:15) f is (cid:126) πn/m d where n is the electron orhole density and m d = √ m xx m yy [26]. So, for carrierdensity 10 m − [21], Fermi energy is 7 meV is around20 times larger than the THz radiation energy to havea resonance current. Hence, the considered semiclassicalregime is correctly used in this paper. VIII. CONCLUSION
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