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Excitonic Terahertz Absorption in Semiconductors with Effective-Mass

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Excitonic Terahertz Absorption in Semiconductors with Effective-Mass Excitonic terahertz absorption in semiconductors with effective-mass anisotropies P. Springer,∗ S. W. Koch, and M. Kira Department of Physics and Material Sciences Center, Philipps-Universit¨atMarburg, Renthof 5, 35032 Marburg, Germany (Dated: September 24, 2018) A microscopic approach is developed to compute the excitonic properties and the corresponding terahertz response for semiconductors characterized by anisotropic effective masses. The approach is illustrated for the example of germanium where it is shown that the anisotropic electron mass in the L-valley leads to two distinct terahertz absorption resonances separated by 0:8 meV. I. INTRODUCTION in two clearly separated exciton resonances in the THz absorption spectrum. Terahertz (THz) spectroscopy has been broadly ap- plied, e.g., to investigate transient photoconductivity [1], inter-molecular vibrations [2], high-harmonic genera- II. THEORY WITH MASS ANISOTROPY tion [3], and the transition energies between excitonic eigenstates in quantum many-body systems [4{6]. For To identify the main consequences of mass anisotropy direct-gap semiconductors with isotropic effective-mass in the THz absorption spectra, we use Ge as a prototype configurations, the excitonic THz excitation dynamics system and a 2-band model to describe the energy dis- and the resulting spectra have been studied extensively persion. Germanium is an indirect semiconductor whose both theoretically [7{10] and experimentally [11{14]. conduction (c) and valence (v) bands are centered around In comparison to direct-gap systems, the correspond- the L and Γ points, respectively, separated by the wave ing investigations in indirect semiconductors such as sili- vector k0, as indicated in the inset of Fig. 1. For excita- con (Si) and germanium (Ge) are more elaborate because tions close to the band gap Eg, it is sufficient to describe optical excitations are accompanied by strong dephasing the electronic energies via [21], due to intervalley scattering [15] and the indirect exci- 2 2 tons typically involve states characterized by strongly v h ~ k Ek = −Ek = − ; (1) anisotropic masses [16]. Experimentally, excitonic fea- 2mh 2 2 tures have been observed in Ge [17] and Si [18] and THz c e X ~ [(k − k0) · ej] studies have been reported recently [19, 20]. Ek = Ek = Eg + ; (2) 2me;j Whereas most of the isotropic exciton properties can j be determined analytically [7, 8], even the linear eigen- for the holes (h) and the electrons (e), respectively. The value problem must be solved numerically for anisotropic indices j = fx; y; zg denote the Cartesian components conditions. These subtleties complicate the microscopic and k0 is aligned with the ez direction, as shown in Fig. 1. analysis of the linear and nonlinear optical experiments, Although k in general defines a group of energy minima, and in particular also of the THz absorption measure- 0 we first evaluate the theory for one single k0 and gener- ments. alize the results for multiple k in Sec. IV B. In Ge, the To deal with this problem, we develop in this paper a 0 group of k0 points to the eight L centers. These lie in microscopic approach and an ensuing numerical scheme the center of the hexagonal planes of the truncated octa- to efficiently evaluate the excitonic properties in sys- hedron which defines the first Brillouin zone. tems with anisotropic effective masses. To illustrate the In Ge, the valence band is isotropic with mass mh = scheme, we analyze Ge and show that the THz absorp- 0:33 m while the conduction band masses are m = tion exhibits distinct resonances related to the L-valley 0 e;x me;y ≡ m? = 0:0815 m0 and me;z ≡ mk = 1:59 m0 at electron-mass anisotropy. the L valley [22]. Figure 1 illustrates the directions of The paper is organized as follows: In Sec. II, we extend the anisotropic effective mass tensor as an ellipsoid with the generalized Wannier equation to systems with mass the xy-plane mass given by m and with m in the z- anisotropy and discuss the system Hamiltonian and basic ? k arXiv:1602.02972v1 [cond-mat.mtrl-sci] 9 Feb 2016 direction. THz absorption equations. In Sec. III, we present an efficient numerical scheme to obtain the radial solutions of the Wannier equation. We analyze the modifications A. System Hamiltonian of the selection rules and the THz absorption spectra for different polarizations in Sec. IV. For the example The many-body Hamiltonian is H^ = H^ + V^ + H^ of Ge, we then show that the mass anisotropy results 0 THz where the non-interacting electrons are described by [23] ^ X h c y v y i H0 = Eka^c;ka^c;k + Eka^v;ka^v;k ; (3) ∗ [email protected] k 2 ! ! K exciton states φλ and their binding energies Eλ have to be computed from the generalized Wannier equation [23, 27] R ~ R e h X R 0 Eλφλ (k) = Ekφλ (k) − 1 − fk − fk Vjk−k0jφλ (k ) ; k0 (7) Energy e(h) Momentum where fk is the electron (hole) distribution. Further Coulomb correlation effects, such as excitation induced dephasing [28, 29], could be included via complex scat- tering matrices [30], but are omitted here for simplicity. Non-vanishing carrier distributions renormalize the electron{hole pair energy 2 2 ~ X ~ [k · ej] X e h Ek = − Vjk−k0j fk0 + fk0 ; (8) 2µj j k0 −1 −1 after we have introduced a reduced mass µj = me;j + Figure 1. Schematic illustration of the system. In the xy- m−1. Since two of the three µ are identical in Ge, the plane (dark grey), the effective electron mass is denoted by h j energy dispersion (8) simplifies to m? (white arrow), in z-direction it is mk (red arrow). The THz field (red) is either polarized parallel or perpendicular 2k2 ~2k2 (blue arrows) to the z-axis which is aligned with k0. The ~ ~ ? k X e h Ek = + − Vjk−k0j fk0 + fk0 ; (9) inset schematically depicts the band structure of Ge. 2µ? 2µ k k0 −1 −1 −1 y with µ?(k) = m?(k) + mh and the momentum k = with Fermionic creation (annihilation) operatorsa ^λ,k (k?; kk), both being decomposed into directions perpen- (^aλ,k) for conduction (λ = c) and valence band (λ = v), dicular (?) and parallel (k) to k0, as shown in Fig. 1. respectively. The Coulomb-interaction is given by [24] ~ In general, Ek is anisotropic for µ? 6= µk. For non- vanishing carrier distributions, the Wannier equation ^ 1 X X y y V = Vqa^λ,ka^ν;k0 a^ν;k0+qa^λ,k−q ; (4) defines a non-Hermitian eigenvalue problem with left- 2 0 L R λ,ν k;k ;q and right-handed solutions φλ and φλ , respectively. As shown in Ref. [31], these solutions are connected via containing the usual Coulomb matrix element Vq. For L R e h φλ(k) = φλ (k)=(1−fk −fk ). Due to the mass anisotropy weak THz fields, the light-matter coupling follows λ and the fk dependence, Eq. (7) cannot be solved ana- from [25] lytically. In Sec. III, we therefore present a method to numerically determine the anisotropic exciton wave func- ^ X λ y R HTHz = − jk · ATHza^λ,ka^λ,k ; (5) tions φλ . λ,k Once the exciton wave functions are known, we can di- rectly evaluate the THz absorption via the susceptibility with the current-matrix elements in the effective-mass ν ν ν ν ? approximation, X Sλ(!)nλ − [Sλ(−!)nλ] χ(!) = 2 ; (10) 0! ( ! + iγJ ) λ,ν ~ jej k X (k − k0) · ej jh = − ~ ; je = −|ej e ; (6) k m k ~ m j h j e;j derived in Ref. [23]. The susceptibility defines the linear absorption α(!) = !=crIm[χ(!)] yielding the THz Elliott and a THz field ATHz(t) ≡ A(t)eA. Due to the mass formula, where cr is the speed of light within the medium. e anisotropy in jk, the THz interaction is sensitive to the Equation (10) also contains a decay constant γJ for the polarization eA of the applied field. THz current, as well as a THz response function β ν ν X (Eβ − Eλ)Jλ Jβ Sλ(!) = ; (11) B. Anisotropic Excitons Eβ − Eλ − ! − iγ β ~ ν To compute the THz probe absorption spectrum, we where γ is the dephasing constant, and nλ assign exciton have to specify the initial many-body state of the semi- correlations. The transition-matrix element conductor. Here, we consider a situation where the L- ν X L ? R point electrons and the Γ -point holes have formed bound Jλ = φλ(k) j(k)φν (k) (12) electron{hole pairs, i.e. excitons [26]. The corresponding k 3 R 1.0 First of all, we expand φλ into spherical harmonics 0, 0 { } 2, 0 1 l R X X m { } φλ (k) = Rλ,l;m(k)Yl (θ; ') : (13) )(norm.) 0.5 l=0 m=−l k ( Inserting Eq. (13) into Eq. (7) and projecting spherical ,l,m (a) harmonics yields an eigenvalue problem for the radial GS 0.0 part alone R 2 2 0.0 0.4 0.8 1.2 ~ k (1) EλRλ,l;m(k) = l;mRλ,l;m(k) Wave vector ka0 2µz 1 3.4 Z 0 0 2 0 e h − dk (k ) Vk;k0 fk0 + fk0 Rλ,l;m(k) 3.3 0 1 (meV) Z 1 3.2 e h 0 0 2 l 0 − 1 − fk − fk dk (k ) Vk;k0 Rλ,l;m(k ) ! (b) 0 GS 3.1 " E 2 2 ~ k X (2) + l+ξ;mRλ,l+2ξ;m(k) 2 4 6 8 2µz ξ=±1 Number of included l states (3) + l;m+ξRλ,l;m+2ξ(k) Figure 2.
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