Optoelectronic response calculations in the framework of k \cdot p coupled to Non-equilibrium Green’s functions for 1D systems in the ballistic limit

Optoelectronic response calculations in the framework of k p coupled to Non-equilibrium Green’s functions for 1D systems in the ballistic limit

Abstract

We present theory of the carrier-optical interaction in 1D systems based on the nonequilibrium Green’s function formalism in the 4x4 k p model. As a representative parameters we chose the GaAs. Although theory is presented in 4x4kp many subbands, results and discussion section is based on the simplified model such as 2x2 kp model (two transverse modes). Even though 2x2 kp model is simple enough it shows many phenomena that have not been seen before. We focus mainly on the ballistic extraction of photogenerated free carriers at the radiative limit which is described by the self-energy term derived in dipole approximation and solved in self-consistent manner with Keldysh quantum kinetic equations. Any relaxation or non-radiative recombination mechanisms as well as excitonic features are neglected. Effect of non-locality of electron-photon self energy term is considered and discussed. Spontaneous emission is also considered and shown to be small in short devices under medium bias conditions. Electron and hole spatial current oscillations are seen and discussed. It is shown that neglecting off-diagonal correlation in the band index not only produces quantitatively wrong results but it also alters the qualitative picture. All simulations are done in the full-rank approximation , with all spatial and band correlation effects are kept intact. This allows us to study not only quantitative effects but also qualitative behaviour.

pacs:
85.60.Gz, 85.35.Ds, 73.50.Pz, 85.35.Be, 85.30.De

I Indroduction

The past several years has seen a growing interest in nanowires (NWs) such as SiNWsJ. Ramanujam (2011); Verma, Buin, and Anantram (2009), GeNWsCao et al. (2009), and GaAsNWs because of their excellent optoelectronic properties Agarwal and Lieber (2006). As few examples, recent experimental work Thunich et al. (2009) on the photocurrent response of freely suspended single 140 nm GaAsNWs has shown current as high as 0.45 nA for a titanium:sapphire laser light intensity of 100 . Experimental work on the effect of strain on GaAsNWs, approximately 80 nm in diameterZardo et al. (2012), and theoretical work on much smaller diameter SiNWsShiri et al. (2012), have also shown a direct-to-indirect bandgap transition, which can potentially be used for laser applications. At the same time, it has also been found that surface passivation of the GaAsNW with AlGaAs increases the photoluminescence (PL) lifetime, and minority carrier diffusion lengths, significantlyDemichel et al. (2010); Chang et al. (2012). The bandgap in GeNWs is also found to be dependent upon the type of surface passivation as well as strain Sk et al. (2013), which has a consequence on the optoelectronic response of the NW. Concomitantly, GaAs p-i-n NW structures have also shown excellent solar power harvesting capabilityColombo et al. (2009). The above results (as well as several references contained therein) highlight the significance of obtaining a detailed understanding of the photo response of NWs and 1D devices. As these nanostructures are being used for image photo detectors, calculations of the photo current response become important. Of particular importance is the behavior of smaller diameter NWs, in keeping with the trend towards smaller feature sizes. Over the years several theoretical work have been reported to understand the optoelectronic response of NWsWitzigmann et al. (2009); Fedoseyev, Turowski, and Wartak (2007). Just to name a few advanced works in the field of NEGF coupled to photonic field which recently appeared are the works of Aerberhard et al.Aeberhard and Morf (2008); Aeberhard (2011a) , SteigerSteiger (2009) and Henrickson et al.Henrickson (2002) and Stewart et.al Stewart and Léonard (2004) which use either Tight Binding(TB) or bulk 2D k p modeling. The limitation of the TB is the system size, whereas the limitation of the bulk 2D k p system is applicability to 2D systems such as quantum wells, superlattices. In this work we discuss an approach suitable for modeling the photo current response of sub-10 nm diameter NWs. The basis of our work is the band structure calculation by utilizing a 1D 4x4 k p model, with transport calculations utilizing non-equilibrium Green’s function (NEGF) formalism. For small structures, semiclassical simulations, such as Monte Carlo, are reasonably accurate, but they may not capture the details of charge distribution in its entirety, particularly in the problem being addressed. On the other hand, NEGF based quantum mechanical approach may provide a more accurate estimation especially in the phase-coherent regime. Moreover, NEGF allows to incorporate phase-breaking(not considered here) processes vis self-energies. To our knowledge this is the first work which couples 4x4 k p and NEGF to compute the photo response of the 1D nanostructures. We have used 4x4 k p (applicable to direct band gap materials) to keep things simple, although conceptually there is no restriction and the model can be easily transferred to the indirect band-gap materials such as Si, Ge by using a larger dimensional k p such as 15x15, 24x24, 30x30 k p models for sub-10 nm 1D devices. This approach takes into account correlations between different band indices as well as spatial correlation allowing us to study the effect of non-locality of electron-photon self-energy. We believe that the proposed method provides a good compromise between computational speed and modeling complexity. The paper is divided into different sections. Section II focuses on the theory, particularly band structure calculation, electron-photon interaction, transport formalism, as well as mode-space approach and physical observables. Device setup and numerical parameters are discussed in Section III. Section IV comprises of results and discussion, and conclusions are drawn in Section V.

Ii Theory

ii.1 Hamiltonian

Starting point of the work was the calculation of the band structure using the original Kane Kane (1982) 4x4 k p scheme and using GaAs as a representative material. For the computation of the photoresponse (discussed below), we use a modified 2x2 scheme (or two subband model). Originally k p was done for the direct bandgap materials, although usage of it has been extended to indirect bandgap materials (Si and Ge)Boujdaria et al. (2002); El kurdi et al. (2003); Ridene et al. (2001); Boujdaria and Zitouni (2004); Zitouni, Boujdaria, and Bouchriha (2005); Cardona and Pollak (1966), and one can adapt the present method to originally indirect bulk materials.

Hamiltonian is given in basis of cell-periodic zone centered (k=0) Bloch functions Bahder (1990)

(II.1)

where , is the effect of remote bandsKane (1982) , and

(II.2)
(II.3)
(II.4)

with being modified Luttinger parameters are related to the original Luttinger parameters by

(II.5)
(II.6)
(II.7)

where ()Foreman (1997) are specific material parameters. Renormalisation is required so as to subtract effects of conduction band in the original 3x3 k p model Pidgeon and Brown (1966). Making transformation from -space representation to real space representation one has to use momentum operators which are given by

(II.8)

Since sine waves naturally satisfy infinite barrier boundary conditions, they are chosen as basis functions in the transverse direction. This corresponds to the situation of free-standing p-i-n junction. Along the transport direction (-axis) we adopt the following basis functions

(II.9)

where is the Heaviside function, and is the interlayer spacing. Electronic wavefunction in the aforementioned basis is written as

(II.10)

where , and with denoting and denoting . The total Hamiltonian Shin (2009) in basis

(II.11)

where  is the block matrix of the size and given by

(II.12)

with

(II.13)
(II.14)
(II.15)
(II.16)

where is the Kronecker delta and if , otherwise 0 and

(II.17)

is orthogonal transformation to of Hartree  potential (which is obtained self-consistently solving NEGF-Poisson equation), with being the 4x4 identity matrix. Similarly, the inter-layer coupling matrix can be written in similar manner

(II.18)

, where

(II.19)
(II.20)

One should mention that form rectangular grid. Further simplification such as Hamiltonian size reduction in k p basis by taking only vectors inside the circleShin (2009)can be done to minimize memory usage and computational power . Moreover, one can get further matrix size reduction by employing the mode-space approach. Mode-space is crucial for the recursive algorithm in NEGF implementation and charge distribution construction. It was shownPourfath, Baumgartner, and Kosina (2008) that in case of electron-photon interaction one cannot easily use recursive approach since self-energies are highly non-local and in this case one has to take more off-diagonal blocks. In other words, more correlations between electron Green’s functions have to be kept when dealing with electron photon interaction.

ii.2 Electron-photon interaction. Monochromatic excitation.

The electron-photon interaction part of Hamiltonian reads as

(II.21)

where the photon field is quantized and is given by

(II.22)
(II.23)

where are the photon creation and annihilation operators, respectively, - is the polarization vector, - is the photon wavevector and - is the photon energy. Sum is over all photon wavectors and energies. where is the absorbing volume. The incident photon flux is related to photon occupation number via

(II.24)

where - is the intensity of the EM field and - is the speed of light. Equation (II.21) in the second quantized form can be written as

(II.25)

with - being electron creation operator in the state symmetry , transverse subband , and position and - being electron destruction operator in the state of symmetry , transverse subband , and position

Carrying out explicitly matrix element of (II.25) in dipole approximation with wire dimensions much smaller than a wavelength ( and taking into account only inter-subband excitations(CB-VB), i.e. considering only CB-VB transitions, we arrive at

(II.26)

where

(II.27a)
(II.27b)

Total Matrix () becomes

(II.28)

where,

(II.29)
(II.30)
(II.31)
(II.32)

where x,y,z stands for either x,y or z EM field polarization, l.c. stands for the linear polarization which is linear combination of the x,y and z axis.

ii.3 NEGF and Self-Energies

Green’s functions are assumed to be in steady state with electron Green’s function being at zero temperature (although temperature comes via Fermi levels) and photon Green’s functions being unperturbed by electronic elementary excitations. Within Keldysh formalism the Dyson’s equations of motion for the electronic Green’s functions in matrix notation are given by

(II.33a)
(II.33b)
(II.33c)
(II.33d)

where is the boundary self-energy, which incorporates effect of semi-infinite contact(coupling to contacts). Contacts are are assumed to be with equilibrium with right and left leads respectively and are perfect absorbersLake and Datta (1992). is the electron-photon self-energy describing electron-photon interaction, where

(II.34)

where are the block matrices of size that are related to surface Green’s functions via

(II.35a)
(II.35b)

where,

(II.36a)
(II.36b)

are surface Green’s functions corresponding to left and right lead, respectively. Equations on the are matrix quadratic equations. There are many ways of calculating the solution to (II.36). Simplest solution is just straightforward iteration, although this is very slowly converging process. Therefore, we have adopted the improved version of Anderson mixing Thompson, Rasmussen, and Lookman (2004) which is also simple in implementation. Lesser(in-scattering) boundary self energy in case of equilibrated contacts is given by

(II.37a)
(II.37b)

where are the Fermi levels at the left and right lead respectively, and is the level broadening.

Light-matter interaction leads to electron-hole pair generation and electron-hole recombination by absorbing/emitting a photon. This process is inelastic, and in general is phase-breaking. In order to incorporate this interaction into NEGF formalism in the first order Born-approximation(one-photon processes) one has to utilize Wick’s theorem and Langreth contour rules as it was done in several worksLake and Pandey (2007); Steiger (2009) and in the original Henrickson’sHenrickson (2002) papers.

Most self-energies of this form, including electron-photon,(fermion-boson interaction in the limit one elementary exciation) are current conservingMahan (1987). In order to achieve current conservation one has to utilize self consistency among Green’s functions and self-energies - in other words use self-consistent Born approximation(SCBA) or one can use current conserving schemes using Non-self consistent Born Approximation described in LakeR. K. Lake and Jovanovic (1997) et.al. A detailed derivation of the self-consistent Born approximation approach is given in the work of Jiang et al.Jiang, Wang, and Li (2011). Lesser and greater parts, are given by

(II.38a)
(II.38b)
(II.38c)
(II.38d)

where are the self energies associated with photon absorption, stimulated emission and spontaneous emission, respectively. The derivation is very similar to the work of Jiang et al.Jiang and Wang (2011) One should note that spontaneous emission term is integrated over broad energy range in CB and VB energy regions and is only dependent on joint density of states and occupation numbers at energies which differs by photon energy. are the minimal and maximal photon energies dictated by material and device parameters. is the full electron-photon interaction Hamiltonian in the basis . Strictly speaking, one has to be careful considering since originally it couples only bulk CB-VB bands. In other words, if one wants to consider inter-subband excitations such as CB-CB or VB-VB (either within CB or VB manifolds), the has to be modified accordingly to include intraband coupling in the original bulk model since one 3D band gives raise to many 1D subbands. In case of short-channel devices under certain biases the spontaneous term is assumed to be smallAeberhard and Morf (2008) and as will be shown later can be neglected. Real part of the retarded self energy is neglected since it leads just to energy renormalizationAeberhard and Morf (2008), and only imaginary part of the is important and given by ( notation is omitted throughout for simplicity)

(II.39)

ii.4 Mode space and Physical quantities

In case of mode-spaceShin (2009) transformation one defines mode in the following manner

(II.40)

which satisfies 2D-sliced Schrodinger equation at slice

(II.41)

Original eigenfunction of (II.10) is given in terms of modes as

(II.42)

In order to have self-consistent NEGF with Poisson one has to compute 3D electron density in the real space representation. An incompleteLuisier, Schenk, and Fichtner (2006) calculation consists of writing 3D electron density in real space J.Wang, Polizzi, and Lundstrom (2004) neglecting the mode correlation effects as

(II.43)

where , superscripts stand for the real-space and mode-space representations respectively. stands for the diagonal matrix element of mode at block . are unitary transformation matricesShin (2009) defined as block diagonal matrices built from respectively, where

(II.44)

is the size of and

(II.45)

is the size of

Current flowing between layers ,and can be written as

(II.46)

Similar approach has been applied in the study of thermal expansion of single-wall carbon nanotubes and grapheme sheetsJiang, Wang, and Li (2009)

Iii Numerical Details

The device under study is a p-i-n structure and is depicted in Fig. 1. The device is 42 nm long, with a square cross-section of 10nm x 10nm. The doping on both the and ends is assumed to be . Furthermore, length of the and region was set nm and inter-layer spacing nm . Current conserving grid was chosenSteiger (2009) as with total number of energy grid points with being conduction and valence band cut-off energies chosen accordingly to the region of interest. defines by how many energy points separated and . varied between 800 and 2000 points to make sure convergence is achieved in energy space. was set to 140 points. The potential profile is assumed to be uniform in the cross-sectional area. 1D potential profiles and Fermi-levels were obtained by nextnano simulator Birner et al. (2007) with the parameters being =1.42eV, =0.067, =0.082 where parameters are bandgap, effective conduction mass, effective valence mass(light hole) respectively. Although, strictly speaking there is no physical justification for this, but it does not affect the physical picture except consideration of the boundary effects in which we are not interested at the moment. 1D Potential profiles and Quasi-Fermi levels were fed into optical NEGF simulator based on 2 subband model, which is written as

(III.1)
(III.2)

In order to avoid spurious solutions in - space we took cross-sectional area such that condition on the envelope function is satisfied so, that plane-wave expansion lies in the first Brillouin-zoneRideau et al. (2009); Foreman (1995); Burt (1992); Yang and Chang (2005)

(III.3)

with . In addition we set with optimizing the parameters Foreman (1997, 2007) Veprek, Steiger, and Witzigmann (2007) such that bulk effective masses of hole and electrons are reproduced. Moreover, the original Hamiltonian can be modified to avoid spurious solutions Kolokolov, Li, and Ning (2003). Going from - space representation to real-space representation with finite differences being the basis one has another source of spurious solutionsCartoixà, Ting, and McGill (2003); Eissfeller and Vogl (2011). Such solutions can be avoided by using certain finite element basisEissfeller and Vogl (2011). However in general, there is no common remedy for this type of problemEissfeller and Vogl (2011). In particular, to avoid this type of problem, one either chooses inter-layer spacing accordingly to the parameters Cartoixà, Ting, and McGill (2003), or as we did, fix the and vary the parameters to reproduce the bulk effective masses of conduction band and light-hole bands (we have assumed that charge carrier effective masses of 10nm x 10nm are bulk values). Parameters after fitting are . SCBA computations are aborted once convergence is achieved by monitoring the norm of the total photocurrent