A fully quantum mechanical model for electron transport in quantum well infrared photodetectors is

presented, based on a self-consistent solution of the coupled rate equations. The important macroscopic

parameters like current density, responsivity and capture probability can be estimated directly from this

first principles calculation. The applicability of the model was tested by comparison with experimental

measurements from a GaAs/AlGaAs device, and good agreement was found. The model is general and can

be applied to any other material system or QWIP design

Harrison, P.

Ikonic, Z.

Indjin, D.

Jovanovic, V.D.

English

White Rose Research Online

This is a repository copy of Physical model of quantum-well infrared photodetectors.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/1116/Article:Jovanovic, V.D., Harrison, P., Ikonic, Z. et al. (1 more author) (2004) Physical model of quantum-well infrared photodetectors. Journal of Applied Physics, 96 (10). pp. 269-272. ISSN 1089-7550 https://doi.org/10.1063/1.1756691eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request.      White Rose Consortium ePrints Repository http://eprints.whiterose.ac.uk/  This is an author produced version of a paper published in Journal of Applied Physics. White Rose Repository URL for this paper: http://eprints.whiterose.ac.uk/archive/00001116/    Published paper Jovanovic, V.D. and Harrison, P. and Ikonic, Z. and Indjin, D. (2004) Physical model of quantum-well infrared photodetectors. Journal of Applied Physics, 96 (10). pp. 269-272.  Repository paper Jovanovic, V.D. and Harrison, P. and Ikonic, Z. and Indjin, D. (2004) Physical model of quantum-well infrared photodetectors. Author manuscript available at: http://eprints.whiterose.ac.uk/archive/00001116/   White Rose Consortium ePrints Repository eprints@whiterose.ac.uk  A physical model of quantum well infrared photodetectors (QWIPs)V. D. Jovanovíc∗, P. Harrison, Z. Ikoníc, D. IndjinSchool of Electronic and Electrical Engineering,University of Leeds, Leeds LS2 9JT, United KingdomAbstractA fully quantum mechanical model for electron transport in quantum well infrared photodetectors ispresented, based on a self-consistent solution of the coupled rate equations. The important macroscopicparameters like current density, responsivity and capture probability can be estimated directly from thisfirst principles calculation. The applicability of the model was tested by comparison with experimentalmeasurements from a GaAs/AlGaAs device, and good agreement was found. The model is general and canbe applied to any other material system or QWIP design.∗ Electronic mail: eenvj@leeds.ac.uk1-2000 -1500 -1000 -500 0 50000.10.20.30.40.50.6reference planecentral QWecontinuumgrowth directionFIG. 1: Schematic diagram of the conduction band profile and electron scattering transport in QWIPs.In the past decade quantum well infrared photodetectors (QWIPs) have reached a technologi-cal maturity as devices offering excellent performance in the mid- (3−5µm) and long-wavelength(8−14µm) infrared spectral range [1]. A large number of papers have been published, coveringdifferent aspects of QWIP design, modelling and characterization, delivering tunable, broadbandand multicolor operation [2–8]. Moreover, large, highly uniform QWIP focal plane arrays havebeen reported with a wide range of possible applications [9, 10]. However, despite the extensiveamount of experimental and theoretical efforts, not much work has been done on a microscopicquantum description of the processes that govern both vertical and parallel electron transport inperiodic quantum structures, involving bound-bound and bound-continuum intersubband transi-tions [6, 11–13]. In order to ensure further improvement of the QWIP technology, primarily byusing novel structures and material systems, a thorough understanding of fundamental physicalprocesess in QWIPs, as well as a first principles simulation tool are neccesary.In this paper, two main topics are addressed: (i) development of a fully quantum mechanicalmodel for the simulation of electron transport in QWIPs yielding physical observables like darkcurrent density, responsivity, and capture probability, (ii) verification of the model by comparisonwith experimental data for GaAs/AlGaAs QWIPs detecting around 8.5µm.Consider a multiple quantum well (MQW) structure with a large number of periods in an ex-ternally applied electric field. The energy spectrum is formally continuous, but to a very goodapproximation can be considered as consisting of quasi-discrete states (resonances). Based on thewavefunction localization properties, these states can be associated to different single QWs (i.e.periods) of the MQW, so that each period has an indentical set ofN states in the energy rangeof interest. Electron scattering occurs between states within the same period, and between states2associated to different periods, the latter clearly becoming less effective for more distant periodsbecause of reduced wavefunction overlap. Assuming an identical electron distribution in each pe-riod, one may consider some ’central‘ period and take itsP nearest neighbours on either side, andwrite the scattering rate equations in the steady-state :dnidt= 0 =N∑j=1, j 6=in jWj,i −niN∑j=1, j 6=iWi, j+P∑k=1N∑j=1, j 6=i{n j[Wj,i+kN +Wj+kN,i]−ni[Wi+kN, j +Wi, j+kN]}+ξ×C(Φ,n1,n2, ..) (1)wherei + kN is the ith state of thekth neighbouring period, Wi, j is the total scattering rate fromstatei into state j, ni is the electron concentration of theith state andξ is equal to 1 for lightand 0 for dark conditions. The first two sums in Eq.1 are due to intra-period, and the third dueto inter-period scattering, while the termC(.) describes the non-scattering contribution to elec-tron transition rates. This is the rate at which electrons transfer between pairs of states due tointersubband absorption of incident radiation. After solving for electron densitiesni , macroscopicparameters of the system like current density, capture probability, responsivity, and drift velocitycan be estimated.The current density can be calculated by subtracting the current density component due to elec-trons scattering into the next periods of the MQW structure from the component due to electronsscattering back, as is usually done in quantum cascade laser simulations [14]. If we put a referenceplane somewhere in the right barrier of the central QW, the current density flowing throught thatcross section can be written as (Fig. 1):J =P∑k=1N∑i=1N∑j=1k ·ni(Wi, j+kN−Wi+kN, j)(2)This expression defines both the dark and light current dependence on the value of the param-eterξ. The factork in the summation, effective for non-nearest-neighbour scattering, comes fromscatterings from any QW left of the centre well into any QW right of it, or vice versa (i.e. skip-ping the central well, but going through the reference plane). This current density component ispresented with the dashed arrow on Fig. 1. With the current density known, the responsivity canbe expressed asR=J(ξ = 1)−J(ξ = 0)(hc/λ)Φ(3)3whereλ is the detection wavelength andΦ the total optical flux.Another important parameter is the capture probability in standard drift-diffusion simulationsusually taken as an empirical parameter representing quantum mechanical behavior of the QWs.Here we define the capture probability as the ratio of the electron current density component due toscattering from continuum states into quasi-discrete states of the QWs and the current componentin continuum [12]. If the number of quasi-discrete states within the central QW isNb then thecapture probability can be expressed aspc =P∑k=1JkJ=P∑k=1N∑i=NbNb∑j=1niWi, jJ(4)A very important issue is the choice of wavefunctions assigned to a certain period, as thecontinuum states are not confined within a defined area and a wrong choice can lead to duplicatedstates and false estimate of the carrier dynamics. We define states belonging to the consideredperiod as those having a better overlap integral with the ground state of that period than withthe ground states of the neighbouring periods. For this purpose we define the overlap integral asOi j =R|ψi(z)|2|ψ j(z)|2dz.In order to reduce the number of scattering rate processes necessary to calculate the electrondistribution and the corresponding current density (note that the number of total scattering rateprocesses is equal toN2(2P+1)−N), we introduce the ‘tight-binding’ approximation assumingthat only a few closest neighbors interact, and setP = 2. The choice of quantum scattering mech-anisms depends on the material and doping density, as well as the detection wavelength. Thek-space scattering rate averaging assumed Fermi-Dirac distribution within each subband with aunique electron temperature.The optical perturbation constant, in the steady state conditions, is modelled assuming lineardependence on optical flux as:C(Φ,ni ,nf ) = Ai f (ni ,nf )×Φ (5)whereAi f is fractional optical apsorption on thei → f transition calculated as in Ref. [15]. Ingeneral,Ai f is a function of subband sheet electron densities. However, it is valid to assume thatthe electron concentration in the ground state is much larger than in the continuum states (notethat this is justified under low illumination conditions, a reasonable assumption for most QWIPstructures), which setsAi f as a constant in the self-consistent process. Furthermore, the diffractiongrating influences the optical constant to some extent, and can in principle be included in themodel, but this is out of the scope of this work.45 10 15 20 25 30Number of wavefunctions10-1610-15Current density [a.u.]2V1V0.5V0.67VFIG. 2: Current density vs. number of wavefunctions for four different values of bias (i.e. 0.5, 0.67, 1, and2V)The energies and wavefunctions were calculated by solving the envelope function Schr¨ dingeequation, within the effective mass approximation, implementing the finite difference solver. Thisapproach intrinsically enforces the ’hard wall’ boundary conditions, which in turn defines thecontinuum discretation.We should note that the presented model is general and can be applied to any QW structure,with an arbitrary number of discrete states, and delivers a direct estimate of the physical observ-ables, as well as input parameters used in other phenomenological simulations (capture proba-bility, drift velocity). Moreover, the simulation is independent of the materials used, and can beemployed for evaluation of QWIP characteristics, for materials where empirical estimates as wellas experimental data are not available.Next we present numerical results obtained from our model, for the usual GaAs/AlGaAs QWIPstructure design for detection at∼ 8.5µm (see Ref. [16]), as well as a comparison with experi-mental data. The modelled QWIP consists of 40 periods, with well and barrier widths of 40 and342 Å, respectively and an Al content in the barrier layer of 26.3%. The wells are assumed tobe doped with an effective sheet carrier density of 1.8×1011cm−2 and the temperature was set to77K. Electron -longitudinal optical (LO) phonon scattering was taken as the main scattering mech-anism and calculated as in Ref. [17]. Electron-electron and impurity scattering was considered tobe negligible as continuum states are hardly populated and the doping density is low.The a priori task was to verify the first assumption of our model, that the continuum can be, forthis purpose, accurately presented by a finite set of discrete wavefunctions. In order to prove that,we have calculated the current density as a function of the number of wavefunctions used in our50 0.5 1 1.5 2 2.5 3 3.5 4Bias [V]10-610-510-410-3J dark [A/cm2 ]T = 77 K0 0.5 1 1.5 2 2.5 3 3.5Bias [V]020406080100p c [%]calculated pcexponential fitFIG. 3: Calculated (solid line and circles) and experimental (dashed line and squares) dark current den-sity/voltage characteristics at 77 K. Inset: Calculated capture probability vs. bias (circles) and the appropri-ate exponential fit (dashed line).basis for a few values of the external bias (Fig. 2). It can be seen that the current density convergeswith an increasing number of states. It is expected that the necessary number of wavefunctionsincreases with the increase of the bias, as the energy range between the two ground states of theadjacent periods increases. It is interesing to note that the convergence corresponds to an averagespacing between adjacent continuum states of a few meV.The next step was to compare the calculated dark current density with experimental results. InFig. 3 both the calculated and the experimental current/voltage characteristics are presented. Agood agreement is achieved for the whole range of examined voltages with an average discrepancyof a factor of two. This may be partly due to leakage on contacts, or due to scattering mechanismsnot taken into account in the simulation, and is within expectation. In the inset of Fig. 3 thecalculated capture probability is given as a function of the applied bias. It exhibits steep decay atlower biases (from 0.3 to 0.5V) and saturation at higher voltages (> 1V). The simulated depen-dence generally follows the exponential decay (usually assumed in the literature; see Ref.[16] and[18]), except in the mid-range of voltages between, 0.5 and 1V, where the exponential dependenceoverestimates the capture probability.Fig. 4 shows the responsivity as a function of the applied voltage for detection at 8.5µm.The absorption linewidth was assumed to be 50meV for the bound-continuum transition (∆λ/λ ∼33%). The responsivity curve exhibits nonlinear behavior at lower biases, reaching a peak valueof around 0.3A/W. Furthermore, a decrease at higher biases is observed, which is the consequenceof the absorption line shifting to higher wavelengths with an increase of the applied bias. As the60.5 1 1.5 2 2.5 3 3.5Bias [V]00.10.20.3R [A/W]T = 77 K4 6 8 10 12 14λ [µm]00.10.2R [A/W]λ = 8.5µm @ 1VFIG. 4: Calculated responsivity vs. applied bias at 77K for the fixed wavelength of 8.5µm. Inset: wavelengthdependence of the responsivity for 1V biasexperimental data for the calculated QWIP structure do not exist we have compared our resultswith available data for a fairly similar structure (see Fig.10 in Ref. [16]) and again found goodagreement, as well as with both the high and low bias behaviour. The wavelength dependenceof the responsivity at 1V bias is given in the inset. The responsivity approximately follows theLorentzian shape profile as expected, with peak at∼ 8.5µm.In conclusion we presented a detailed, fully microscopic model for electron transport in QWIPs.The model is based on fully self-consistent rate equations approach within the ‘tight-binding’approximation, applied by including the interaction with two nearest neighbors. Macroscopicparameters like current density, responsivity and capture probability are obtainable without needfor empirical parameters. In order to examine the accuracy of the simulation we have applied itto a GaAs/AlGaAs QWIP, and found very good agreement with the experiment for dark current,capture probability, as well as both the bias and spectral dependence of the responsivity. Thepresented model can be applied to any QWIP device, and can be very useful for examining novelstructures and materials.The authors would like to acknowledge discussions with Richard Soref. V. D. J. would like tothank the School of Electronic and Electrical Engineering for funding.[1] A. Rogalski, J. Appl. Phys.93, 4355 (2003).[2] L. Jiang, S. S. Li, M. Z. Tidrow, W. R. Dyer, W. K. Liu, J. M. Fastenau, T. R. Yurasits, Appl. Phys.Lett.79, 2982 (2001).7[3] M. Z. Tidrow, X. Jiang, S. S. Li, K. Bacher, Appl. Phys. Lett.74, 1335 (1999).[4] H. C. Liu, R. Dudek, A. Shen, E. Dupont, C. Y. Song, Z. R. Wasilewski, M. Buchanan, Appl. Phys.Lett. 79, 4237 (2001).[5] V. Ryzhii, M. Ryzhii, H. C. Liu, J. Appl. Phys.92, 207 (2002).[6] M. Ryzhii, V. Ryzhii, M. Willander, J. Appl. Phys.84, 3403 (1998).[7] J. L. Pan, C. G. Fonstad, IEEE J. Quant. Electron.35, 1673 (1999).[8] M. Ershov, H. C. Liu, J. Appl. Phys.86, 6580 (1999).[9] S. D. Gunapala, S. V. Bandara, J. K. Liu, E. M. Luong, N. Stetson, C. A. Shott, J. J. Bock, S. B. Rafol,J. M. Mumolo, M. J. McKelvey, IEEE Trans. Electron. Dev.47, 326 (2000).[10] S. D. Gunapala, S. V. Bandara, A. Singh, J. K. Liu, S. B. Rafol, E. M. Luong, J. M. Mumolo, N. Q.Tran, D. Z.-Y. Ting, J. D. Vincent, C. A. Shott, J. Long, P. D. LeVan, IEEE Trans. Electron. Dev.47,963 (2000).[11] N. E. I. Etteh, P. Harrison, J. Appl. Phys.92, 248 (2002).[12] M. A. Gadir, P. Harrison, R. A. Soref, Appl. Phys. Lett.81, 4272 (2002).[13] O. O. Cellek, C. Besikci, Semicond. Sci. Technol.19, 183 (2004).[14] D. Indjin, P. Harrison, R. W. Kelsall, Z. Ikonić, Appl. Phys. Lett.82, 1347 (2003).[15] V. Jovanovíc, D. Indjin, Z. Ikoníc, V. Milanović, J. Radovanović, Solid State Commun.121, 619(2002).[16] L. Thibaudeau, P. Bois, J. Y. Duboz, J. Appl. Phys.79, 446 (1996).[17] P. Harrison,Quantum Wells, Wires and Dots: Theoretical and Computational Physics(John Wiley &Sons Ltd, Chichester,1999)[18] B. F. Levine, J. Appl. Phys.74, R1 (1993).8

Physical model of quantum-well infrared photodetectors

V. D. Jovanović

P. Harrison

Z. Ikonić

D. Indjin

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Physical model of quantum-well infrared photodetectors

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