Self-consistent quantum-kinetic theory for interplay between pulsed-laser excitation and nonlinear carrier transport in a quantum-wire array

# Self-consistent quantum-kinetic theory for interplay between pulsed-laser excitation and nonlinear carrier transport in a quantum-wire array

Jeremy R. Gulley Department of Physics, Kennesaw State University, Kennesaw, Georgia 30144, USA    Danhong Huang Air Force Research Laboratory, Space Vehicles Directorate, Kirtland Air Force Base, New Mexico 87117, USA
February 28, 2019
###### Abstract

We propose a self-consistent many-body theory for coupling the ultrafast dipole-transition and carrier-plasma dynamics in a linear array of quantum wires with the scattering and absorption of ultrashort laser pulses. The quantum-wire non-thermal carrier occupations are further driven by an applied DC electric field along the wires in the presence of resistive forces from intrinsic phonon and Coulomb scattering of photo-excited carriers. The same strong DC field greatly modifies the non-equilibrium properties of the induced electron-hole plasma coupled to the propagating light pulse, while the induced longitudinal polarization fields of each wire significantly alters the nonlocal optical response from neighboring wires. Here, we clarify several fundamental physics issues in this laser-coupled quantum wire system, including laser influence on local transient photo-currents, photoluminescence spectra, and the effect of nonlinear transport in a micro-scale system on laser pulse propagation. Meanwhile, we also anticipate some applications from this work, such as specifying the best combination of pulse sequence through a quantum-wire array to generate a desired THz spectrum and applying ultra-fast optical modulations to nonlinear carrier transport in nanowires.

## I Introduction

Several studies on strong light-matter interactions in semiconductors Henneberger et al. (1992); Lindberg and Koch (1988); Buschlingern et al. (2015) were reported in the past three decades. However, most of these studies involved some non-self-consistent phenomenological models with model-parameter inputs from experimental observations. For example, a very early work on spectral-hole burning in the gain spectrum in Ref. [Henneberger et al., 1992] assumed a constant pumping electric field by fully neglecting the field dynamics but including the electron dynamics instead under the energy-relaxation approximation. Later, such a simplified study was improved by including full quantum kinetics for collisions between pairs of electrons in Ref. [Lindberg and Koch, 1988] within the second-order Born approximation. However, a spatially-uniform electric field was still adopted in their model and the dynamics of this pumping electric field was not taken into account. Only very recently, a self-consistent calculation based on coupled Maxwell-Bloch equations was carried outin Ref. [Buschlingern et al., 2015] for a multi-subband quantum-wire system. However, some phenomenological parameters were introduced for optical-coherence dephasing, spontaneous-emission rate and energy-relaxation rate. Although the field dynamics was solved using Maxwell’s equations in Ref. [Buschlingern et al., 2015], only the propagating transverse electromagnetic field was studied while the localized longitudinal electromagnetic field was excluded. It is interesting to point out that none of this early work on strong light-matter interactions has ever considered the drifting effects of electrons under a bias voltage in a self-consistent way.

For the first time, we have established a unified quantum-kinetic model for both optical transitions and nonlinear transport of electrons within a single frame. This unified quantum-kinetic theory is further coupled self-consistently to Maxwell’s equations for the field propagation so as to study strong interactions between an ultrafast light pulse and driven electrons within a linear array of quantum wires beyond the perturbation approach. In our theory, the optical excitations of quantum-wire electrons by both propagating transverse and localized longitudinal electric fields are considered. Meanwhile, the back action of optical polarizations Iurov et al. (2017a), resulting from induced dynamical dipole moments and plasma waves due to electron-density fluctuations, on transverse and longitudinal electric fields is also included. Moreover, the semiconductor Bloch equations Lindberg and Koch (1988) (SBEs) are generalized to account for possible crystal momentum altering (non-vertical) transitions of electrons under a spatially nonuniform optical field, as well as the drifting of electrons under a net driving force including electron momentum dissipation.

It is well known that photons do not interact directly with themselves. Instead, they interact indirectly through exciting electrons in nonlinear materials. Although the strong interaction of photons in a laser pulse with electrons in quantum wires is extremely short in time and confined only within a micro-scale, photons still acquire “fingerprints” from the configuration space of excited electron-hole pairs. These pairs can be detected through either a delayed light pulse in the same direction or another light beam in different directions as a stored photon quantum memory Lvovsky et al. (2009) from the first light pulse. Within the perturbation regime, the nonlinear optical response of incident light can be studied, such as the Kerr effect and sum-frequency generation. Shen (1984) On the other hand, for optical reading, writing, and memory these ultrafast processes cannot be fully described by perturbation theories since they involve an ultrafast and strong interaction between a light pulse and the material.

From the physics perspective, in this work we want to focus on three fundamental issues for a pulsed-laser irradiated quantum wire system. They are: variations in local transient photo-current and photoluminescence spectra by an incident laser pulse, changes in propagation of laser pulses by nonlinear photo-carrier transport in a micro-scale quantum-wire array, and optical reading of photon quantum memory (or electronic-excitation configurations) stored in a micro-scale quantum-wire array by a laser pulse. From the technology perspective, however, we look for a specification of the best combination of pulse sequence through a quantum-wire array to generate a desired terahertz spectrum and a realization of ultra-fast optical control of nonlinear carrier transport in wires by laser pulses.

The rest of this paper is organized as follows. In Sec. II, we first establish a self-consistent formalism for propagation of laser pulses and generation of local optical-polarization fields by photo-excited electron-hole pairs in quantum wires. After this, we develop in Sec. III another self-consistent theory for pulsed-laser excitation of electron-hole pairs and nonlinear transport of photo-excited carriers under a DC electric field. Meanwhile, we also derive dynamical equations in Sec. IV for describing back actions of electrons in quantum wires on interacting laser photons. In Sec. V, we present a discussion of numerical results for transient properties of photo-excited carriers and laser pulses as well as for light-wire interaction dynamics. Finally, conclusions are given in Sec. VI with some remarks.

## Ii Pulse Propagation

The pulse propagation is governed by Maxwell’s equations which we solve using a Psuedo-Spectral Time Domain (PSTD) method Taflove and Hagness (2000). Under this scheme, derivatives in real position () space are evaluated in the Fourier wavevector () space, in which Maxwell’s equations take the form

 i\boldmathq⋅~\boldmathD(% \boldmathq,t) =~ρqw(\boldmathq,t) , (1a) i\boldmathq⋅~\boldmathB(% \boldmathq,t) =0 , (1b) i\boldmathq×~\boldmathE(% \boldmathq,t) =−∂∂t~\boldmathB(% \boldmathq,t) , (1c) i\boldmathq×~\boldmathH(% \boldmathq,t) =∂∂t~\boldmathD(% \boldmathq,t) . (1d)

Here, and represent the electric and magnetic fields, and are the auxiliary electric and magnetic fields, and is the charge-density distribution in the quantum wires embedded within a dielectric host. The two-dimensional (2D) Fourier transforms with respect to spatial positions are defined by

 ~f(\boldmathq) =∫d2\boldmathre−iq⋅rf(\boldmathr) , (2a) f(\boldmathr) =1(2π)2∫d2\boldmathqeiq⋅r~f(\boldmathq) . (2b)

In this work for non-magnetic materials, we neglect magnetic effects on the propagation and on the quantum wires so that the auxiliary magnetic field with as the vacuum permeability. We further divide the fields into transverse and longitudinal contributions with respect to , defined by and , respectively. By definition and from Eqs. (1a) and (1b) then, the longitudinal components of the auxiliary fields are given at all times by

 ~\boldmathD∥(\boldmathq,t) =\boldmath^eq[~ρqw(\boldmathq,t)iq] , (3a) ~\boldmathH∥(\boldmathq,t) =0 , (3b)

is a unit vector specifying the direction, includes the longitudinal polarization fields, , of quantum wires, and the longitudinal-optical conductivity, , is determined from the equation: Forstmann and Gerhardts (1986) .

We further recast the electric field in terms of the electric displacement , the polarization fields of the host material, , and the quantum wires, . The dispersion in the host material will be important for ultrashort pulses. We therefore use a frequency () dependent dielectric function for the host, , where is a static and uniform background constant and is the polarizability of the th local Lorentz oscillator for bound electrons such that with as the vacuum permittivity, where . By solving a time-domain auxiliary differential equation for each th oscillator, Taflove and Hagness (2000) we get .

Therefore, the time-evolution of the transverse auxiliary fields for different light pulses can be obtained from Eqs. (1c) and (1d):

 ∂~\boldmathD⊥(\boldmath% q,t)∂t =i\boldmathq×~\boldmathH⊥(\boldmathq,t) , (4a) ∂~\boldmathH⊥(\boldmath% q,t)∂t =−iϵ0c2\boldmathq×~% \boldmathE⊥(\boldmathq,t) , (4b)

where is the vacuum speed of light, includes the transverse polarization fields, , of quantum wires, and the transverse-optical conductivity can be determined from Forstmann and Gerhardts (1986) . At all times, the longitudinal () and transverse () components of are evaluated through Jackson (1975)

 ~\boldmathE⊥,∥(\boldmathq,t)=~\boldmathD⊥,∥(\boldmathq,t)−∑i~\boldmathP⊥,∥i(\boldmathq,t)+~\boldmathP⊥,∥qw(\boldmathq,t)ϵ0ϵb . (5)

Note that in Eq. (5) has often been omitted. Instead, it enters directly into Eq. (1d) as a term , mainly flowing along the quantum-wire direction in a 2D field system.

We orient all wires along the direction, and split into vectors parallel and perpendicular to . Note that the directions of and are not related to longitudinal and transverse contributions of an electromagnetic field. The quantum-wire source terms in Maxwell’s equations are the sum of the contributions from each quantum wire and are expressed as Iurov et al. (2017b)

 ~ρqw(\boldmathq,t) =∑j~ρ1Dj(q∥,t)e−iq⊥⋅R⊥j−q2⊥/4α2 , (6a) ~\boldmathP{∥,⊥}qw(% \boldmathq,t) =∑σ=x,y~Pσqw(% \boldmathq,t)~\boldmathGσ{∥,⊥}(\boldmathq)=∑je−iq⊥⋅R⊥j−q2⊥/4α2∑σ=x,y~Pσj(q∥,t)~\boldmathGσ{∥,⊥}(\boldmathq) , (6b)

where label two of three independent dipole directions in a two-dimensional propagating system for electrons within a quantum wire, the centered transverse position of the th quantum wire in real space is denoted by , and the width of each wire is . The total electric field is a complex field and . In addition, we would like to emphasize that the quasi-one-dimensional (quasi-1D) quantum wire is still treated as a bulk semiconductor material for optical transitions of electrons. The polarization field should point to the direction of 2D dipole moments. For centrosymmetric GaAs cubic crystal with isotropic band structures at -point, the unit vector in the dipole direction is found to be with as two coordinate unit vectors. The 1D field sources, and , in Eqs. (6a) and (6b) are calculated from the solutions to the SBEs in the 1D momentum space of the wire as described below. Moreover, in Eq. (6b) represent the two vector projection functions for longitudinal () and transverse () directions of the polarization field, respectively. Specifically, we can write them down as Huang et al. (2006)

 ~\boldmathGx∥(\boldmathq) =(\boldmath^eq⋅\boldmath^ex)\boldmath^eq=q⊥q2⊥+q2∥(q⊥\boldmath^ex+q∥% \boldmath^ey) , (7a) ~\boldmathGy∥(\boldmathq) =(\boldmath^eq⋅\boldmath^ey)\boldmath^eq=q∥q2⊥+q2∥(q⊥\boldmath^ex+q∥\boldmath^ey) , (7b) ~\boldmathGx⊥(\boldmathq) =−(\boldmath^eq×\boldmath^eq×\boldmath^ex)=q∥q2⊥+q2∥(q∥\boldmath^ex−q⊥% \boldmath^ey) , (7c) ~\boldmathGy⊥(\boldmathq) =−(\boldmath^eq×\boldmath^eq×\boldmath^ey)=−q⊥q2⊥+q2∥(q∥\boldmath^ex−q⊥%\boldmath$^e$y) , (7d)

where for our chosen .

## Iii Laser-Semiconductor Plasma Interaction

For photo-excited spin-degenerate electrons and holes in the th quantum wire, the quantum-kinetic semiconductor Bloch equations are given by Haug and Koch (2009); Kuklinski and Mukamel (1991); Buschlingern et al. (2015)

 dnej,k(t)dt= (8a) dnhj,k′(t)dt= 2ℏ∑kIm{\boldmathpj,k,k′(t)⋅\boldmathΩj,k′,k(t)}+∂nhj,k′(t)∂t∣∣ ∣∣rel , (8b) iℏd\boldmathpj,k,k′(t)dt =[εek+εhk′+εG+Δεej,k+Δεhj,k′−iℏΔehj,k,k′(t)]% \boldmathpj,k,k′(t)−[1−nek(t)−nhk′(t)]ℏ\boldmathΩj,k,k′(t) (8c) +iℏ∑q≠0Λej,k,q(t)% \boldmathpj,k+q,k′(t)+iℏ∑q′≠0Λhj,k′,q′(t)\boldmathpj,k,k′+q′(t) ,

where are potentially two equations with respect to that are formally combined into one vector equation (8c), correspond to the dipole directions, the spin degeneracy of carriers is included, is the bandgap of a host semiconductor including size-quantization effects of quantum wires, the retarded interwire electromagnetic coupling has been included in Eqs. (5), (6b) and in Eqs. (25a) and (25b). In Eqs. (8a)-(8c), and are the electron (e) and hole (h) occupation numbers at momenta , and , respectively, and , represent their transition momenta. The quantum coherence between electron and hole states coupled to the electric field is , is the renormalized Rabi frequency, and indicate their kinetic energies, and and are the Coulomb renormalization Huang and Manasreh (1996a) of the kinetic energies of electrons and holes. Moreover, is the diagonal dephasing rate Lindberg and Koch (1988) (quasi-particle lifetime), while and are the off-diagonal dephasing rates Lindberg and Koch (1988) (pair-scattering) for electrons and holes (see Appendix D for details).

In deriving the above equations, the electron and hole wave functions in a quantum wire are assumed to be , where represents the length of a quantum wire, are the ground-state wavefunctions of electrons and holes in two transverse directions, , are the electron and hole effective masses, are the level separations between the ground and the first excited state of electrons and holes due to finite-size quantization, and in Eqs. (6a)-(6b) is given by , and the local position vector just as earlier. The dipole-coupling matrix element is calculated as for the isotropic interband dipole moment at the -point Huang and Cardimona (2001) and is the free-electron mass. If the quantum-kinetic occupations in Eqs. (8a) and (8b) are replaced by their thermal-equilibrium Fermi functions and the Rabi frequencies in Eq. (8c) are also replaced by for an incident electric field, we arrive at the optical linear-response theory from Eq. (8c) after neglecting all dephasing terms.

In Eq. (8c), and are the kinetic energies of electrons and holes. Their correction terms, and , are given by: Huang and Manasreh (1996a)

 Δεej,k =2∑qnej,q(t)Veek,q;q,k−∑q≠knej,q(t)Veek,q;k,q−2∑q′nhj,q′(t)Vehk,q′;q′,k , (9a) Δεhj,k′ =2∑q′nhj,q′(t)Vhhk′,q′;q′,k′−∑q′≠k′nhj,q′(t)Vhhk′,q′;k′,q′−2∑qnej,q(t)Vehq,k′;k′,q , (9b)

which also account for the excitonic interaction energy. The Coulomb-interaction matrix elements, , and , introduced in Eqs. (9a), (9b), (25a) and (25b) are explicitly given in Appendix B.

In the presence of many photo-excited carriers, i.e., for the total numbers of electrons and holes , the Coulomb interaction will be screened by a dielectric function in the Thomas-Fermi limit Huang and Manasreh (1996b), e.g., , and . Using the high-density random-phase approximation (RPA) at low temperatures, is calculated as Gumbs and Huang (2011)

 ϵ1D(q∥,t)=1−limω→02βm∗eπℏ2q∥ln{ω2−[Ω−e(q∥,t)]2ω2−[Ω+e(q∥,t)]2}K0(q∥Re)−limω→02βm∗hπℏ2q∥ln{ω2−[Ω−h(q∥,t)]2ω2−[Ω+h(q∥,t)]2}K0(q∥Rh) , (10)

where is the absolute value of the electron wave number, with as the average dielectric constant of the quantum wire, is the modified Bessel function of the third kind, , are the Fermi wavelengths, , is the thickness of a quantum wire, and are the linear densities of photo-excited carriers.

The additional relaxation terms in Eqs. (8a) and (8b) are given by Huang et al. (2004)

 ∂nej,k(t)∂t∣∣ ∣∣rel =∂nej,k(t)∂t∣∣ ∣∣scat−Rj,sp(k,t)nej,k(t)nhj,k(t)+Fej(t)ℏ∂nej,k(t)∂k , (11a) ∂nhj,k′(t)∂t∣∣ ∣∣rel =∂nhj,k′(t)∂t∣∣ ∣∣scat−Rj,sp(k′,t)nej,k′(t)nhj,k′(t)−Fhj(t)ℏ∂nhj,k′(t)∂k′ . (11b)

Here, on the right-hand side, the first term describes non-radiative energy relaxation through Coulomb and phonon scattering, the second term corresponds to spontaneous recombinations of e-h pairs, and the last term represents carrier drifting in the presence of an applied DC electric field.

The Boltzmann-type scattering terms for non-radiative energy relaxation in Eqs. (11a) and (11b) are given by Huang and Gumbs (2009)

 ∂nej,k(t)∂t∣∣ ∣∣scat= We,(in)j,k(t)[1−nej,k(t)]−We,(out)j,k(t)nej,k(t) , (12a) ∂nhj,k′(t)∂t∣∣ ∣∣scat= Wh,(in)j,k′(t)[1−nhj,k′(t)]−Wh,(out)j,k′(t)nhj,k′(t) , (12b)

where the explicit expressions for scattering-in, , and scattering-out, , rates for electrons and holes are presented in Appendix C.

For hot photo-excited carriers in non-thermal occupations, the time-dependent spontaneous-emission rate, , introduced in Eqs. (11a) and (11b) for each quantum wire is calculated as Huang and Lyo (1999)

 Rj,sp(k,t) =3d2cvϵ0√ϵr∫∞0dω′{ℏω′ρ0(ω′)L(ℏω′−εG−εek−εhk−εj,c(k,t),ℏγeh) ×M(ℏω′−εG−εj,c(k=0,t),ℏγeh)} , (13)

where is the Lorentzian line-shape function, is the broadened step function, with as the lifetime of photo-excited non-interacting electrons (holes), and is the density-of-states for spontaneously-emitted photons in vacuum. Moreover, the Coulomb renormalization of the transition energy in the th quantum wire is found to be

 εj,c(k,t) =∑qnej,q(t)(Veek,q;q,k−Veek,q;k,q)+∑q′nhj,q′(t)(Vhhk,q′;q′,k−Vhhk,q′;k,q′) −∑q≠knej,q(t)Vehq,k;k,q−∑q′≠knhj,q′(t)Vehk,q′;q′,k−Vehk,k;k,k , (14)

where the first two terms are associated with the Hartree-Fock energies Huang and Manasreh (1996b) for electrons and holes, while the remaining terms are related to the excitonic interaction energy.

The net driving forces, and , introduced in Eqs.(11a) and  (11b) for electrons and holes, including the resistive ones from the optical-phonon scattering of photo-excited carriers, can be calculated from Huang et al. (2005)

 Fej(t) =−eEdc−2∑k,qℏq{Θemj,e(k,q,t)−Θabsj,e(k,q,t)} , (15a) Fhj(t) =+eEdc−2∑k′,q′ℏq′{Θemj,h(k′,q′,t)−Θabsj,h(k′,q′,t)} , (15b)

where is the applied DC electric field. In Eqs. (15a) and (15b), the emission (em) and absorption (abs) rates for longitudinal-optical phonons in intrinsic and defect-free quantum wires are given by Huang et al. (2005)

 Θemj,e(k,q,t)= 4πℏ∣∣Vepk,k−q∣∣2nej,k(t)[1−nej,k−q(t)][N0(Ωph)+1] ×L(εek−q−εek+ℏΩph−ℏqvej(t),γe)θ(ℏΩph−ℏqvej(t)) , (16a) Θabsj,e(k,q,t)= 4πℏ∣∣Vepk,k−q∣∣2nej,k−q(t)[1−nej,k(t)]N0(Ωph) ×L(εek−εek−q−ℏΩph+ℏqvej(t),γe)θ(ℏΩph−ℏqvej(t)) , (16b)
 Θemj,h(k′,q′,t)= 4πℏ∣∣Vhpk′,k′−q′∣∣2nhj,k′(t)[1−nhj,k′−q′(t)][N0(Ωph)+1] ×L(εhk′−q′−εhk′+ℏΩph−ℏq′vhj(t),γh)θ(ℏΩph−ℏq′vhj(t)) , (17a) Θabsj,h(k′,q′,t)= 4πℏ∣∣Vhpk′,k′−q′∣∣2nhj,k′−q′(t)[1−nhj,k′(t)]N0(Ωph) ×L(εhk′−εhk′−q′−ℏΩph+ℏq′vhj(t),γh)θ(ℏ