Effect of electron-hole asymmetry on optical conductivity in 8-Pmmn borophene
Sonu Verma Alestin Mawrie Tarun Kanti Ghosh
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        We present a detail theoretical study of the Drude weight and 
        optical conductivity of 8-Pmmn borophene having tilted anisotropic 
        Dirac cones. 
        We provide exact analytical expressions of xx and yy components of 
        the Drude weight as well as maximum optical conductivity. We also obtain 
        exact analytical expressions of 
        the minimum energy (ϵ_1) required to trigger the optical transitions 
        and energy (ϵ_2) needed to attain maximum optical conductivity. 
        We find that the Drude weight and optical conductivity are highly anisotropic as a 
        consequence of the tilted Dirac cone. The tilted parameter can be extracted by 
        knowing ϵ_1 and  ϵ_2 from optical measurements. The maximum 
        values of the components of the optical conductivity do not depend on the carrier 
        density and the tilted parameter. The product of the maximum values of the anisotropic 
        conductivities has the universal value (e^2/4ħ)^2.
        The tilted anisotropic Dirac cones in 8-Pmmn borophene can be realized by 
        the optical conductivity measurement.
        
        Effect of electron-hole asymmetry on optical conductivity in 8-Pmmn borophene
            Sonu Verma, Alestin Mawrie and Tarun Kanti Ghosh
        
        Department of Physics, Indian Institute of Technology-Kanpur,
        Kanpur-208 016, India
            
        =========================================================================================================================================
        
        
        
        § INTRODUCTION
        
        Graphene is the first atomically thin two-dimensional (2D) material 
        having isotropic Dirac cones realized in a laboratory 
        <cit.>. 
        Since then, there have been numerous attempts to synthesize 
        more and more new 2D materials having Dirac cones.
        Several quasi-2D materials possessing Dirac cones such as 
        silicene<cit.>, germanene<cit.>,
        and MoS_2<cit.> have been synthesized experimentally and are being studied
        theoretically.
        
        
        Recently, there has been intense research interest in
        synthesis of 2D crystalline boron structures, referred to as borophene.
        Several attempts have been made to synthesize a stable structure of
        borophene but only three different quasi-2D structures of borophene have 
        been synthesized <cit.>. Various numerical experiments have 
        predicted a large number of borophene structures with various geometries 
        and symmetries <cit.>. The orthorhombic 8-Pmmn borophene is 
        one of the energetically stable structures, having ground state energy lower 
        than that of the α-sheet structures and its analogues.
        The Pmmn boron structures have two non-equivalent sub-lattices.
        The coupling and buckling between two sub-lattices and vacancy give rise 
        to the energetic stability as well as tilted anisotropic Dirac cones <cit.>.
        The coupling between different sub-lattices enhances the strength of the
        boron-boron bonds and hence gives rise to structural stability.
        The finite thickness is required for energetic stability of 2D
        boron allotropes. 
        The orthorhombic 8-Pmmn borophene possesses tilted anisotropic Dirac cones and
        is a zero-gap semiconductor. It can be thought of as topologically 
        equivalent to the distorted graphene.
        
        In the last couple of years, there have been several theoretical 
        studies on 8-Pmmn borophene.
        Very recently, electronic properties of 8-Pmmn borophene have been studied using the 
        first-principle calculations and have shown the Dirac cones arising from the p_z 
        orbitals of one of the two inequivalent sub-lattices <cit.>.
        Zabolotskiy and Lozovik proposed a tight-binding Hamiltonian 
        for 8-Pmmn borophene and obtained a low-energy effective Hamiltonian in
        the continuum limit <cit.>. The effective Hamiltonian successfully described
        all the main features of the quasi-particle spectrum as predicted in  ab initio 
        calculation.
        A similar Hamiltonian has been considered in Ref. <cit.> for studying 
        quinoid-type graphene due to mechanical deformation and 
        organic compound α-(BEDT-TTF)_2I_3.
        
        
        The anisotropic plasmon dispersion of borophene is predicted in Ref. <cit.>. 
        The magnetotransport coefficients have been investigated very recently <cit.>.
         
        
        The frequency-dependent optical conductivity is associated with the transitions 
        from a filled band to an empty band, whereas the zero-frequency Drude weight is 
        due to the intra-band transitions. The real part of the complex optical
        conductivity is connected with the absorption of the incident photon energy.
        Its measurement is an important tool for 
        extracting the shape and nature of the energy bands. 
        There are several theoretical and experimental studies on the optical conductivity
        in various monolayer quantum materials such as graphene 
        <cit.>, 
        silicene <cit.>, 
        MoS_2<cit.>, and 
        surface states of topological insulators <cit.>.
        
        
        
        In this paper, we study zero-frequency Drude weight and frequency-dependent optical 
        conductivity of 8-Pmmn borophene. 
        We find that the Drude weight and optical conductivity are highly anisotropic due 
        to tilted Dirac cones. 
        
        We obtain an analytical expression of the minimum photon energy 
        required for triggering the optical transitions and of the photon energy at which the 
        conductivities attain maximum value.    
        The maximum value of the optical conductivity along the tilted and perpendicular
        directions are obtained, which are independent of the carrier density and tilting parameter.
        
        
        The spectroscopic measurement of the absorptive part of the optical conductivity can shed
        some light on the anisotropic but tilted Dirac cone.
        
        
        This paper is organized as follows. In Sec. II, we provide basic 
        information of 8-Pmmn borophene in detail. 
        We discuss the Drude weight and absorptive part of the optical conductivity 
        in Sec. III. An alternate derivation of the optical conductivity using 
        Green's function method is provided in the Appendix.
        We provide a summary and conclusions in Sec. IV.
        
        
        
        
        § BASIC INFORMATION
        
        The massless Dirac Hamiltonian associated with the 8-Pmmn borophene 
        in the vicinity of one of the two independent Dirac points is given 
        by <cit.>
            H = v_xσ_x p_x+ v_yσ_yp_y+ v_tσ_0p_y,
        
        where p_μ with μ=x,y are the momentum operators, σ_μ are 
        the 2× 2 Pauli matrices, and σ_0 is the 2 × 2 identity matrix. 
        The velocities are given <cit.> as v_x = 0.86 v_F, v_y= 0.69 v_F, 
        and v_t= 0.32 v_F with v_F = 10^6 ms^-1. The Hamiltonian associated with the
        second Dirac cone has the opposite sign of v_t.
        
        
        The energy dispersion and the corresponding wave functions are given by
        
            E_λ( k) = ħ k  [v_tsinθ_ k + λΔ(θ_ k)]
        
        and
        
            ψ_ k^λ( r) = e^ik·r/√(2)[       1; λ e^i ϕ ],
        
        where λ = ± denotes the conduction and valence bands, respectively,
        θ_ k = tan^-1(k_y/k_x), 
        Δ(θ_ k) = √(v_x^2cos^2θ_ k + 
        v_y^2sin^2θ_ k)
        describes anisotropy of the spectrum and ϕ = tan^-1(v_y k_y/v_x k_x).
        The energy difference between the conduction and valence bands at a given k
        is E_g( k) = E_+( k) - E_-( k)  =2 ħ k Δ(θ_ k).
        Note that the first term in the energy spectrum tilts the Dirac cone and breaks the
        electron-hole symmetry E_λ( k) = - E_-λ( k), 
        even for the isotropic case v_x = v_y. The tilted Dirac cones are depicted in Fig. 1. 
        
        
        
        The Berry connection of 8-Pmmn borophene is given by
        
            A_λ( k) = - v_x v_y/2 Δ^2(θ_ k) θ̂_ k/k,
        
        where θ̂_ k = - sinθ_ k x̂ + cosθ_ k ŷ 
        is the unit polar angle. 
        The corresponding Berry phase is γ_λ = 
        ∮ A_λ( k) · d k= π, exactly the same as in the 
        monolayer graphene case.
        
        
        
        The chirality operator can be defined as
        
            Λ̂ = v_x cosθ_ k σ_x + v_ysinθ_ k σ_y/Δ(θ_ k).
        
        It can be easily checked that the chirality operator Λ̂ commutes with 
        the Hamiltonian H even in the presence of the tilted Dirac cone and 
        the eigenfunctions ψ_ k^λ( r) are also eigenfunctions of the chirality 
        operator with the eigenvalues λ = ± 1, respectively.
        
        
        The role of the Berry phase γ_λ = π and of the
        chiral symmetry preservation  must be reflected in the scattering process. This can be
        easily understood by analyzing the angular scattering probability for the borophene. 
        This is given by the squared moduli of the overlap matrix element between the initial spinor 
        (χ_λ(θ_ k)) and the final spinor (χ_λ(θ_ k^'))
        with |  k  | = | k^' |.
        The angular scattering probability is then 
        
            |f(θ_ k,θ_ k^')|^2     =    
            |⟨χ_λ(θ_ k) 
            |χ_λ(θ_ k^')⟩ |^2 
                =    1/2[1 + v_x^2cosθ_ kcosθ_ k^' +
            v_y^2sinθ_ ksinθ_ k^'/Δ(θ_ k)Δ(θ_ k^')] .
        
        It is to be noted that the wave functions do not depend on the tilt parameter v_t.
        Hence the Berry connection and |f(θ_ k,θ_ k^')|^2 are
        also independent of the tilt parameter.
        It also shows that |f(θ_ k,θ_ k^')|^2 is independent of the bands
        and vanishes exactly when 
        θ_ k^' - θ_ k = π. It implies that the 
        backscattering is completely absent, similar to the graphene case.
        The absence of backscattering survives even for tilted anisotropic energy spectrum.
        This is due to the conservation of the chirality and/or the ∓π Berry phase.  
        
        
        
        
        
        
        
        
        
        The density of states is given by
        
            D(E)     =     g_s g_v ∫d^2k/(2 π)^2δ(E - E_λ( k)) 
                =     N_0|E|/π^2ħ^2v_F^2,
        
        where the spin degeneracy g_s =2 and the “valley degeneracy" g_v =2<cit.>. 
        Also, the constant N_0 is given by
        
            N_0 = 
            ∫_0^2π v_F^2    d θ_ k/[v_t sinθ_ k + 
            λΔ(θ_ k)]^2 = 15.2263.
         
        For a given carrier density n_c, the Fermi energy is 
        E_F =  ħṽ_F√(π n_c) with ṽ_F = √(2π/N_0) v_F
        and the associated anisotropic Fermi wave vectors are  
        obtained as 
        
            k_F^λ (θ_ k)     =    E_F /ħ |v_t sinθ_ k + λΔ(θ_ k)|.
        
        
        The components of the velocity operator along the x- and
        y-directions are 
        v̂_x = v_x σ_x and v̂_y = v_t σ_0 + v_y σ_y.
        The expectation values of these operators are given by
        ⟨v̂_x⟩_λ = [ v_x^2/Δ(θ_ k)] cosθ_ k 
        and ⟨v̂_y⟩_λ = v_t + [v_y^2/Δ(θ_ k)] sinθ_ k, 
        respectively.
        
        
        
        
        § OPTICAL CONDUCTIVITY
        
        We consider n-doped 8-Pmmn borophene subjected to zero-momentum 
        electric field E∼μ̂ E_0 e^i ω t with
        oscillation frequency ω (μ̂ = x̂, ŷ).
        The complex charge optical conductivity tensor is given by 
        Σ_μν(ω) = δ_μνσ_D(ω) + 
        σ_μν(ω),
        where μ,ν=x,y, σ_D(ω) = σ_d/(1- i ωτ) is 
        the dynamic Drude conductivity due to the intra-band transitions, with 
        σ_d being the static Drude conductivity and 
        σ_μν(ω) being the complex optical conductivity due to
        transitions between valence and conduction bands.
        It should be mentioned here that Re σ_D and Re σ_μν
        correspond to the absorption of the photon energy.
        
         Drude weight: 
        The Drude weight at vanishingly low-temperature is given by <cit.>
            D_μν^ = g_s g_v e^2/4π∫ d^2 k  ⟨v̂_μ⟩_⟨v̂_ν⟩_δ(E( k)-E_F^ ),
        
        
        On further simplification, we obtain
        
            D_μν^ = e^2/ħE_F/πħδ_μν(δ_μ x N_1^ + δ_ν y N_2^),
        
        where N_1 = 4.686 and N_2 = 2.673. 
        
        
        
        
        
        
        
        
        
        
        
        In this case, the Drude weight is anisotropic, unlike the monolayer graphene case 
        where the Drude weight D_w^G = (v_F e^2/ħ) √(π n_c) is isotropic.
        
         Optical Conductivity: Within the linear response theory, 
        the Kubo formula for the optical conductivity tensor σ_μν(ω) 
        is given by
        
            σ_μν(ω)     =    1/ħ(ω+i η)∫_0^∞dt 
            e^i(ω+iη)t⟨ [ĵ_μ(t), ĵ_ν(0) ] ⟩,
        
        where 
        ⟨ [ĵ_μ(t), ĵ_ν(0) ]⟩ = 
        ∑_m,n[f(E_n)-f(E_m)] e^i(E_n-E_m)t/ħj_μ^nmj_ν^mn, 
        ĵ_μ = e v̂_μ is the charge current density with μ = x,y, 
        f(E) is the Fermi-Dirac 
        distribution function and η→ 0^+. 
        Here E_n and E_m are the discrete energy levels of the system. 
        So changing the sum into integration over momentum space, the real part of 
        the charge optical conductivity is given by
        
        
            Re σ_μν(ω)    =    e^2/4πω∫ d^2k [f(E_-(k)) - f(E_+(k))] 
                   v_μ^-+( k)  v_ν^+-( k) 
            δ(E_g( k) - ħω).
        
        The final expression of the real part of the optical conductivity tensor is given by
        
            Re  σ_μν(ω)  
             =  e^2/4 πħ∫_0^2π  dθ_ kv_x^2 v_y^2/Δ^4(θ_ k ) 
            [ f(E_-) - f(E_+) ]  
            [(δ_μ xsin^2θ_ k + δ_ν ycos^2θ_ k)  
            δ_μν-(1-δ_μν)sinθ_ kcosθ_ k ],
        
        where 
        E_±(k_ω(θ_ k),θ_ k) ≡ E_± with
        k_ω(θ_ k) = ω/[2 Δ(θ_ k)].
        
        First of all, we find that the real part of the off-diagonal optical conductivity 
        Re σ_xy(ω) vanishes exactly.
        For monolayer graphene (v_t=0 and v_x=v_y=v_F), Eq. (<ref>) gives featureless 
        isotropic optical conductivity which has a step-like shape with a step height 
        σ_0 = e^2/4ħ at ħω = 2 E_F^0 = 2 ħ v_F √(π n_c).  
        Whereas tilted Dirac cones in borophene provide a distinct anisotropic optical
        conductivity which can be seen in the subsequent discussion.
        
        We analyze the real part of the optical conductivity by solving Eq. (<ref>) numerically
        for electron density n_c = 1.0 × 10^16 m^-2 at T=0. 
        The plots of Re σ_xx(ω) and Re σ_yy(ω) 
        as a function of photon energy ħω are shown in the lower panel of Fig. 2.
        It exhibits anisotropic nature of the optical conductivity. 
        We plot ϵ_-(θ_ k) = 2 ħ k_F^-(θ_ k) Δ (θ_ k)
        in the top panel of Fig. 2. The shaded region in the top panel contributes to the 
        optical conductivity.  
        The optical transition from the valence band to the conduction band takes place when
        the photon energy satisfies the inequality ħω≥ϵ_-(θ_ k).
        The optical transition begins at ħω = 0.113 eV,
        which corresponds to 
        ϵ_1 = ϵ_-(π/2) =  2E_F v_y/v_y + v_t < 2 E_F. 
        Moreover, the optical conductivities attain a maximum value when 
        ħω = 0.359 eV, which corresponds to
        ϵ_2 = ϵ_-(3π/2) = 2E_Fv_y/v_y - v_t > 2E_F.
        Note that the two energy scales ϵ_1 and ϵ_2 depend on the
        carrier density, tilted parameter v_t and the velocity v_y along the tilted direction.
        
        
        
        We have checked numerically that 
        Re σ_xx(ω) =   Re σ_yy(ω) when 
        ħω≃ 2 E_F.
        By knowing the energies ϵ_1 and ϵ_2 from an experimental measurement,
        one can extract the tilted parameter v_t using the relation 
        
            v_t  = v_y E_F [ 1/ϵ_1 - 1/ϵ_2].
        
        Analyzing the lower panel of Fig. 2, the maximum attainable absorptive part of the 
        conductivity along the tilted direction is σ_yy^ max =  σ_0 (v_y/v_x) < σ_0 
        and its orthogonal axis is σ_xx^ max =  σ_0 (v_x/v_y) > σ_0.
        It is interesting to note that σ_xx^ max and σ_yy^ max 
        do not depend on the carrier density as well as the tilted parameter v_t.
        Moreover, σ_xx^ max > σ_yy^ max and the product of these two conductivities 
        σ_xx^ max·σ_yy^ max = (e^2/4ħ)^2 is universal.
        
        
        To confirm the results of the optical conductivity, we 
        analyze the joint density of states which is given by
        
            D(ω) = ∫_0^2πdC [f(E_-(k_ω,θ_ k)) - f(E_+(k_ω,θ_ k)) ]/4 π^2 |∂_k E_g( k)|_E_g=ħω,
        
        where C is the line element along the contour.
        In the middle panel of Fig. 2, we show the joint density of states versus the photon
        energy ħω.
        One can easily see that that the van Hove singular points 
        are at θ_ k = θ_s = π/2,3π/2.
        The region of zero optical conductivity is nicely captured by the joint density
        of states.  
        
        
        
        
        
        
        
        
        The absorptive part of the optical conductivity arises due to the transitions 
        from the valence band to the conduction band for a given momentum as demonstrated 
        in Fig. 3. 
        The green (dashed) and red (solid) arrows indicate the allowed and forbidden transitions,
        respectively. 
        One can easily see from this sketch that there are no allowed transitions 
        for the photon energy ħω < ϵ_1 as a result of the Pauli blocking. 
        One can also notice that the transitions are allowed even for ħω > ϵ_2. 
        
        
        
        
        § SUMMARY AND CONCLUSIONS
        
        We have presented detailed theoretical studies of the Drude weight and optical conductivity 
        of the 8-Pmmn borophene. 
        The exact analytical expressions of the Drude weight, components of the optical conductivity and
        the onset energy needed for initiating the optical transitions are provided.
        We also obtain an analytical expression for the photon energy required to attain
        maximum optical conductivity.
        We find that the Drude weight and the absorptive parts of the optical conductivity 
        are strongly anisotropic as a result of the tilted Dirac cones.
        The tilted parameter v_t and the velocity components
        (v_x,v_y) can be extracted from experimental measurements. We have shown that
        the product of the maximum values of the anisotropic conductivities 
        is always universal.
        
        
        
        § ACKNOWLEDGEMENTS
        
        We would like to thank Arijit Kundu and SK Firoz Islam for useful discussion.
        
        
        
        § ALTERNATIVE DERIVATION OF THE OPTICAL CONDUCTIVITY
        
        We provide an alternative derivation of the optical 
        conductivity using Green's function method..
        The Kubo formula for the optical conductivity 
        is given by 
        
            σ_μν(ω)=ie^2/ω1/(2π)^2∫ d^2k 
               × T∑_n Tr⟨v̂_μĜ( k,ω_n) 
            v̂_νĜ( k,ω_n+ω_l) ⟩_iω_n 
            →ω + iδ.
        
        Here, T is the temperature and n and l are integers, where ω_l=(2l+1) π T 
        and ω_n = 2n π T are the fermionic and bosonic Matsubara frequencies, respectively.
        
        The Green's function of the Hamiltonian in Eq. (<ref>) is given by
        
            Ĝ( k,ω)     =    ∑_λ[ σ_0 - λ/Δ(θ_ k)(v_xcosθ_ k σ_x + v_y sinθ_ k σ_y) ] 
               ×    G_λ( k,ω),
        
        where
        
        G_λ( k,ω) = [iħω + μ - E_λ( k)]^-1.
        
        
        
        Using this Green's function, the following trace is obtained as
        
            Tr⟨v̂_xĜ( k,ω_n)v̂_x 
            Ĝ( k, ω_n + ω_l)⟩ = ∑_λ,λ^'[v_x^2/2(1-λλ^')+λλ^'v_x^4cos^2θ_ k/Δ^2(θ_ k)]G_λ( k,ω_l)
            G_λ^'( k,ω_l+ω_n).
        
        
        
        Using the well-known identity
        
            T ∑_s [1/iħω_n+μ-E_λ·1/iħ(ω_l+ω_n)+μ-E_λ^']= 
            f(E_λ)-f(E_λ^')/iħω_n-E_λ^'+E_λ,   if λ≠λ^'
            
                0,                 otherwise.
        
        One can further simply the above equation as
        
            T ∑_n Tr⟨v̂_x Ĝ( k,ω_n) v̂_x 
            Ĝ( k,ω_l+ω_n) ⟩ = 
            v_x^2v_y^2sin^2θ_ k/Δ^2(θ_ k)[f[E_-( k)] - f[E_+( k)]/iħω_n-E_+( k) + E_-( k)+ 
            f[E_+( k)]-f[E_-( k)]/iħω_n-E_-( k)+E_+( k)].
        
        
        It can be seen that the second term turns out to be zero as 
        a result of the conservation of energy.
        Using the result of Eq. (<ref>) into Eq. (<ref>), we have
        
            σ_xx(ω)=i e^2/(2π)^2ω∫_0^∞∫_0^2π  dk dθ_ kv_x^2v_y^2   k sin^2θ_ k/Δ^2(θ_ k)f[E_-( k)]-f[E_+( k)]/iħω_n-E_+( k)+E_-( k)|_iω_n→ω+iδ.
        
        The real part of the optical conductivity is given by
        
            Re σ_xx(ω) =e^2/4π∫ dk dθ_ kv_x^2v_y^2  k sin^2θ_ k/Δ^2(θ_ k)[ f(E_-( k)) - f(E_+( k))] δ(ħω - 2ħ kΔ(θ_ k)).
        
        The above Eq. (<ref>) can be further simplified to 
        
            Re σ_xx(ω) =e^2/16π∫_0^2π dθ_ kv_x^2v_y^2sin^2θ_ k/Δ^4(θ_ k)[ f(E_-(k_ω(θ_ k))) - f(E_+ (k_ω(θ_ k)))]
        
        where k_ω(θ_ k) = ω/2Δ(θ_ k). 
        
        
        
        
        
        
        
        Similarly, the yy and yx components of the optical conductivity can be obtained as
        
            Re σ_yy(ω)     =      - e^2/16π∫_0^2π 
            dθ_ kv_x^2v_y^2cos^2θ_ k/Δ^4(θ_ k)[ f(E_+ (k_ω(θ_ k))) - f(E_- (k_ω(θ_ k)))], 
            
            
            
            
             Re σ_yx(ω)     =    e^2/16π∫_0^2π 
            dθ_ kv_x^2v_y^2sinθ_ kcosθ_ k/Δ^4(θ_ k)[f(E_+ (k_ω(θ_ k))) - f(E_- (k_ω(θ_ k))) ] .
        
        
        Equations (<ref>), (<ref>) and (<ref>) can be written in a 
        compact form as given in Eq. (<ref>).
        
        
        
        
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        Phys. Rev. B  87, 035438  (2013).
        
        silicene-op-con2
        C. J. Tabert and E. J. Nicol,
        Phys. Rev. B  87, 235426  (2013).
        
        mos2-op-con
        Z. Li and J. P. Carbotte,
        Phys. Rev. B  86, 205425 (2012).
        
        mos2-op-con1
        I. Milosevic, B. Nikolic, E. Dobardzic, M. Damnjanovic, I. Popov,
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        Phys. Rev. B  76, 234414 (2007). 
        
        ti-op-con
        P. D. Pietro, F. M. Vitucci, D. Nicoletti, L. Baldassarre, 
        P. Calvani, R. Cava, Y. S. Hor, U. Schade, and S. Lupi,
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        ti-op-con1
        Z. Li and J. P. Carbotte,
        Phys. Rev. B  87, 155416 (2013).
        
        ti-op-con2
        X. Xiao and W. Wen,
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        lozovik1
        Y. E. Lozovik (private communication, September 2016).
        
        
        mermin
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