C-W and I-V characteristics of quantum well varactorsa)

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1 CW and V harateristis of quantum well varatorsa) J. P. Sun, R. K. Mains, W. L. Chen, J. R. East, and G.. Haddad Center for HighFrequeny Miroeletronis, Solid State Eletronis Laboratory, Department of Eletrial Engineering and Computer Siene, The University of Mihigan, Ann Arbor, Mihigan 4819 (Reeived 5 Marh 1992; aepted for publiation 21 May 1992) A theoretial model for quantum well varators is presented. The model is used to alulate the devie CV and V harateristis and very good agreement has been found between the alulated and measured results. Based on the model, a triple barrier double well varator has been designed and fabriated. A very high apaitane ratio within a very small bias range is ahieved, as designed. Details of the design alulations and experimental results are presented.. NTRODUCTON Quantum well varators 2 are very attrative andidates for frequeny multiplier appliations with potentially superior performane ompared to Shottky barrier varators at millimeter and submillimeter wavelengths. The potential advantage of quantum well varators lies in their symmetrial, highly nonlinear apaitane harateristis over a small bias range. The symmetrial devie CV harateristi eliminates the seond harmoni omponent, resulting in more effiient harmoni onversion for frequeny tripler appliations. Several novel varator strutures have been proposed and investigated over the last few years, inluding quantum barrier varators (QBV) or single barrier varators (SBV), double barrier quantum well varators (QWV>,3 barrierintrinsin (BN) strutures, and barriernn+ (BNN) strutures. However, reent CV measurements of quantum well varators have shown that there are disrepanies between a simple model based on the depletion approximation and the experimental data,2 3 and the devies so far fabriated show less nonlinearity than theoretially expeted, with the apaitane ratio (C ax/cmin) typially being 23 within a bias voltage of l2 V. Some of these results are shown in Fig. 1. Moreover, the simple model annot predit the quantummehanial urrent density of the devie whih should be kept suffiiently low in order to minimize the iruit loss. n addition, it is diffiult to optimize the devie CV harateristis in terms of the doping profile using the simple model. n this artile, we present a theoretial model for quantum well varators based on a selfonsistent alulation of the harge and potential distributions in the devie. The eletron onentrations in the ontat regions are alulated using a ThomasFermi equilibrium approximation, while the onentration in the quantum barrier/well region is determined using a quantum alulation. The model an be used for the design of multiple quantum well varators in terms of their CV and V harateristis and for opti mization in terms of devie struture and doping profile. A triplebarrier, doublewell (TBDW) varator has )A part of the results has been presented at 1991 nternational Semiondutor Devie Researh Symposium, Charlottesville, VA, Deember 46, Proeedings of the 1991 SDRS, University of Virginia, Charlottesville, VA, Deember 4, 1991, pp been designed and fabriated based on the model. n Se., the formulation of the devie model is outlined. The devie design and fabriation are desribed in Se.. Setion V presents and disusses a omparison between the alulated and measured results, and the artile is summarized in Se. V.. METHOD OF CALCULATON A. Selfonsistent harge and potential distributions For an aurate evaluation of the quantum mehanial urrent density and the dependene of the apaitane on the applied voltage of a quantum devie struture suh as the quantum well varator in this work, a selfonsistent solution to Poisson s equation and the Shrodinger equation is desired. However, a full selfonsistent model based on the above equations has a oneptual diffiulty in that the sattering mehanisms must be invoked to aount for the band bending or the formation of an aumulation layer near the barrier.5*6 However, if the urrent density is alulated by sending eigenstate eletrons from the ontats, the pure state Shrodinger equation is not adequate to formulate the sattering proesses. Therefore, quantum kineti formalisms suh as the Wigner funtion approah may have to be employed, whih usually demand extensive omputer resoures and are not onsidered to be very suitable for routine analysis and design purposes. The devie model of the quantum well varator presented here is based on a selfonsistent alulation of the potential and harge distributions in the devie, similar to a model for the resonant tunneling diode,7 in whih the approximations and detailed disretized formulations have been desribed. To find the potential and harge distributions, the ontat regions are treated using the ThomasFermi equilibrium model, i.e., arrier onentrations are obtained assuming onstant Fermi levels within eah ontat, determined by the equilibrium FermiDira statistis. Within the barrier/ well region, the eletroni harge is obtained from an eigenvalue solution of the timeindependent Shriidinger equation. When biased, the Fermi levels in the ontat regions are separated by the applied bias voltage and onneted by a linearly varying Fermi level in the barrier/well region. The onentration obtained inside the barrier/well region is then selfonsistently oupled to the Thomas Fermi model through mathing onditions. 234 J. Appl. Phys. 72 (6), 15 September /92/182347$ Amerian nstitute of Physis 234

2 VOLTAGE (V) VOLTAGE (V Note that the Fermi levels are assumed to be onstant, respetively, in the left and right ontat regions and that the ondution band edge varies with distane in the ontat regions aording to: E,(x) =E,( o ) qv(x). The Fermi integral in Eq. (2) an be expanded by a rapidly onverging power series in terms of the ratio r =n,(x>/n,, based on an extension of the JoyeDixon approximation: * VOLTAGE (V) with the oeffiients Ak determined using a leastsquares fit tehnique. Up to N= 12 terms of the series are retained to yield the orret Fermi energy value aurate to within 1e4 ev at 8 K for the doping level in the ontat regions. We now have (3) 3 VOLTAGE (V) EF&(x) = k,t Equation (2) is then written as log(f)+.&(f)i Aog(F)+ ila,(f)i (4) FG. 1. CVharateristis for varators of various strutures from Ref. The devie alulations an be divided into three parts, i.e., a ThomasFermi alulation and two quantum alulations, desribed as follows: For a varator diode struture involving quantum barrier/well regions between two ontat regions (f. Fig. 3), the eletron eletrostati potential and harge distributions of the devie are related by Poisson s equation azv(x) ax2 = ; [N&l n,(x), and the eletron onentration outside the barrier/well region an be expressed, within the ThomasFermi approximation, in terms of loal Fermi levels, 2 m x1 & n,(x) = * N(E)f(E)dE=N, s 7S n 1 +exq (2) where N(E) is the density of states in the ontat regions, and f(e)=1 /[ l+ew(f)] is the FermiDira distribution funtion, N==2[mfkBT/(2&)]3 2 is the effetive density of states in the ondution band, and EE,(x) &~&) X=kBT v= kbt EFEd w > 4W) bt + kbt where V(x) will have to be onsistent with the solution of Eq. ( 1). This leads to a pair of oupled nonlinear equations with two unknowns, i.e., the eletroni harge and potential at every spatial point in the ontat regions. The alulation to solve Eqs. ( 1) and (2) simultaneously is referred to as the ThomasFermi alulation, whih is arried out using the NewtonRaphson iterative method with a sparse matrix tehnique. Note that the Thomas Fermi alulation is per se a semilassial treatment and the above proedure is used to determine n,(x) only outside the barrier/well region. For harge inside the quantum wells as well as the quantum mehanial urrent through the devie, quantum alulations must be performed. Therefore, two quantum alulations are invoked; the lirst is an eigenvalue alulation for the harge in the quantum barrier/well region, and the seond is a traveling wave alulation for the urrent density through the devie. The eigenvalue alulation is performed using an eigenvalue solver to solve the Shrodinger equation with the assumption that the barrier/well region onstitutes a losed system. We have performed transmission alulations for multiple quantum well strutures with open boundary onditions; for the energies not very lose to the top of the barriers, the eigenenergies obtained for the losed system math losely with the transmission peaks and therefore represent a good approximation for the energy levels in the quantum wells. The traveling wave alulation will be disussed in Se. B. For both of the quantum alulations, the timeindependent Shriidinger (effetive mass) equation is used: (5) 2341 J. Appl. Phys., Vol. 72, No. 6, 15 September 1992 Sun et al. 2341

3 #a 1 W(x) ~ 2 dx m*(x) ax +4(x)$(x) ==EW), (6) where &E,(x) is the ondution band edge inluding the heterojuntion potential offset and m*(x) is the spatially variable effetive mass, both of whih an be readily speified in a disretized sheme. The harge in the quantum wells is given by %vw=g2 m*kbt C ln[ 1 +exp( E iy))] i X 1 hg%x) 12, (7) where i is the subband index in the quantum wells, and the Fermi level EF is taken to be linearly varying in the barrier/well region from the left ontat to the right ontat. The differene of the Fermi levels at the ontats is determined by the applied bias voltage. The ThomasFermi alulation arried out previously and the quantum harge alulation are now oupled using mathing onditions. These onditions express the fat that at the left and right boundaries of the barrier/well region, the disontinuity of eletri field aross the region is related to the integral of the harge inside, and that the potential disontinuity is related to this harge and the boundary eletri fields. n addition, the potential in the quantum well(s) region between the ontats is modified aording to the alulated spae harge there. When onvergene is ahieved to a presribed tolerane, the traveling wave alulation is performed to alulate the urrent density. B. Current density n order to onvert the threedimensional eletron distribution for a onedimensional alulation, the eletron wave funtion must be multiplied by a weighting funtion that effetively integrates over transverse wave vetors. The weighting funtion is given by where f( El+ E,) is the FermiDira distribution funtion, El=fi2g/2m* is the longitudinal energy of the eletron with wavevetor kl, E,=#kT/2m* is the transverse energy of the eletron with k, and e=g+ e for x being the growth diretion. The urrent density through the devie is alulated from the timeindependent Shrijdinger equation results aording to the expression: Tot= 4 W(k)m 1 +4(x> ( 1iYx) ~*(x>t, (9) k 1 where a superposition over suitably normalized quantum states $k is performed. C. Capaitane alulation The apaitane of the quantum well varators an be represented by the devie small signal apaitane (gq dipok (1) K2 where the harge Qdipole is the dipole harge integrated over. the depletion side of the devie. We note that for strutures with a uniform doping outside the barrier/well region, only one harge dipole exists, and integration over either side of the dipole yields the orret apaitane value. However, for a nonuniform doping protile suh as the one for TBDW varators, multiple harge dipoles exist and are must be used to define the integration domain. in the devie struture (shown in Fig. 3), the thin, highlydoped Nf regions next to the undoped spaers will be referred to as A regions for onveniene. f we draw a Gaussian surfae enlosing the left ontat region, and let it advane toward the barrier/well, the net harge enlosed by the surfae at zero bias will be zero at the N layer, positive at the A region, and zero again at the right of the spaer. For the present struture, the harge dipole for the apaitane evaluation is onsidered to be divided by the barrierspaer interfae at the right hand side. When the left ontat is biased as the athode, the harge integration is then taken to inlude all the points at the righthand side of the barrier/well region, abbreviated as RHS. The dipole harge is then given by: Qdipole=,, q(x)dxy (11) where q(x) is the net harge density at x, obtained from the selfonsistent alulation. The physial signifiane of this integration will be illustrated when the alulated harge and potential distributions for various bias voltages are displayed in Se. V DEVlCE DESiGN AND FABRCATON A. Devie design The above proedure was used to model the CV and V harateristis of various types of quantum well varators. The alulated CV urves for two varator strutures (a SBV and a superlattie varator) are shown in Fig. 2. For a SBV with the struture of GaAs/AlAs(2 A)/GaAs, the alulated apaitane ratio (Cmax/Cmin) is about 2 [Fig. 2(a)], omparing favorably with the experimental CV urve of the SBV in Fig. 1. Note also that the SBV diodes usually suffer from rather large leakage urrent density due to tunneling from the roaas to XalAs valleys. For a superlattie varator (SLV), whih has a struture of GaAs/SL/GaAs, with the SL onsisting of 1 periods of AlAs(2 A)/GaAs(2 A), the leakage urrent density is signifiantly supressed due to the effet of the Wannier Stark ladder. However, sine the hange of depletion width under bias is insignifiant ompared with the SL width for this partiular struture, the alulated apaitane ratio is less than 3 [Fig. 2(b)], in agreement with our initial experimental data J. Appl. Phys., Vol. 72, No. 6, 15 September 1992 Sun et al. 2342

4 CV Curves for a QB Varator TripleBarrier DoubleWell Varator (GuAs/AAs/G~As ngaas/lnalas system 3 K (a).. N+ ngaas 4 A (a) Voltage (V) CV Curves for a SL Varator (GaAs/AAs/GaAs K T = ;: x 8.75 (b) FG. 2. Calulated CVharateristis this model. Voltage (V) for (a) SBV and (b) SLV using n order to improve the varator performane, the following design ideas were implemented: ( 1) the single barrier should be split into multiple barriers and quantum wells to suppress the X valley urrent or the urrent tunneling through the resonant energy levels in the ase of double barrier quantum well strutures; (2) the maximum and minimum apaitanes should be separately on N+ ngaas 4 A (b) FG. 3. (a) TBDW struture showing the ondution band edge, and (b) MBEgrown layers used for TBDW varator fabriation. trolled, so that more design freedom an be obtained; (3) the bias voltage at whih the transition ours between the maximum and minimum apaitanes an be ontrolled by the sheet harge density and the thikness of the A layers; (4) the number of barriers/wells an be optimized in terms of the requirement of apaitane ratio, series resistane, and aeptable leakage urrent density. The TBDW varator we designed and fabriated has a lattiemathed ngaas/nalas struture, with the layers shown in Fig. 3 (b). The struture is designed suh that, at zero bias, the A regions are not fully depleted and the devie apaitane is approximately the apaitane of the TBDW (CO). Under a small bias voltage, however, as one A region is fully depleted, the devie apaitane is redued to the apaitane (C, > between an aumulation layer on one side of the barrier/well region and the ontat on the other side. Sine C, an be designed to be muh smaller than CO, very high apaitane ratio an be obtained. Also, 2343 J. Appl. Phys., Vol. 72, No. 6, 15 September 1992 Sun et al. 2343

5 .8.8~ 1 O.6.6 _ D..2. g z (a),.....; ; Distane (mirons) () Distane (mirons).3 5. t z lb) Distane (mirons) > ). a2 a : E m. z z f G () T 1.OOl O.75.5 _ &.25 :.5 :.75,". 1.oo Distane (m i~rons) FG. 4. Charge and potential distributions of the TBDW varator under (a), (b).5, ().2, and (d).75 V bias voltages. the bias voltage at whih the sharp drop in apaitane ours an be designed by applying Gauss s law to the harge in the A regions. These features an be seen from the alulated energy and harge distributions shown in Se. V. B. Devie fabriation The no.,,gau,as and n.~2a1.48as layers for the devie struture were grown at a substrate temperature of 52 C using a Riber MBE reator. A 5 A ngaas buffer was grown on a (1) z + np substrate to redue defets and silion was used for the ntype dopant. The devie fabriation was arried out using a standard diode proess. An image reversal proess was used to ahieve the liftoff of the frontside Ni/Ge/Au/Ti/Au ontat metals, and wet seletive ething was used to form diode mesas. The diameter of the diodes ranges from 9 to 1 pm. The CV and 1v urves of the TBDW varator were measured in a ryogeni ooling stage at 8 K. The measured data were taken on the 9 pm devies. V. RESULTS AND DSCUSSON A. Charge and potential distributions The eletroni harge and potential (&) distributions at various bias voltages are shown in Fig. 4, where the bias voltage applied aross the diode an also be found from the potential differene between the ontats. For this struture, under zero bias, the harge and potential distributions 2344 J. Appl. Phys., Vol. 72, No. 6, 15 September 1992 are symmetrial. The zerobias apaitane (Co) is then due to the eletroni harge at the spaer regions. For the struture under onsideration, the spae harge inside the barrier/well region is negligible, C is essentially the parallel plate apaitane determined by the thikness of the barrier/well region and an be designed to be large enough to suppress the leakage urrent. However, for strutures, involving deep quantum wells, the bound state harge inside the quantum wells an ontribute signifiantly to the apaitane evaluation. Note that the eletroni harge in the A regions is lower than the loal doping level beause the thikness of the A layer is not suffiient to allow the harge to reah its equilibrium value. When a.5 V bias is applied between the left and right ontats, eletrons are injeted from the left ontat, resulting in an eletron aumulation at the LHS; the eletrons at the RHS are extrated into the right ontat, resulting in a surplus of positive harge at the RHS. This is shown in Fig. 4(b) where the eletroni harge density is rapidly dereasing and an eletri field is established aross the barrier/well region. When the bias voltage is inreased to.2 V, the eletroni harge at the RHS in the spaer and A layers is essentially depleted [Fig. 4()]. The aumulated eletroni harge at the LHS must now be balaned by the positive harge at the NN+ region at the right ontat. The apaitane of the devie (C, ) at this bias voltage is now due to the aumulated eletroni harge at the LHS and the positive harge (ionized donors) at the right on Sun eta/. 2344

6 CV Curves for TBDW Varator (lngaas/lnaas V Curves for TBDW Varator (lngaas/inaias 8 K 8 K!i 1.5 u ir 8 s *O Voltage Q (a) Voltage (V) FG. 5. Calulated (dotted line) and measured (solid line with dots) CV harateristis of TBDW varator. V Curves for TBDW Varator (ngaas/inaas tat being separated by the thik N layer. The devie apaitane is thus approahing its minimum value ( Cmin). As the bias voltage is further inreased [Fig. 4(d)], more positive harge will be needed at the right ontat, but the depletion width at the RHS will only inrease slightly sine the right ontat is heavily doped. The devie apaitane is therefore approximately onstant at its minimum value. B. CV and V harateristi of the TBDW devies The alulated and measured CV and V harateristis at 8 K are shown in Figs. 5 and 6, respetively. Comparing the alulated and measured apaitanes, very good agreement between the theoretial model and experimental data is ahieved with Cmax/Cmin typially greater than 5 within a small bias range of ho.2 V, as designed. The measured CV urve, however, has nonideal symmetry and a flat apaitane region for small bias. The nonideal symmetry is probably due to the nonideal doping profile near the A regions, while the flat region may indiate that the total amount of harge in the A layers is more than required. The onset of the onstant apaitane region in the measured urve is just visible with the slope hanging at about.2 V, the leakage urrent beyond that point is too large to allow a reliable apaitane measurement. The results for the urrent density in Fig. 6 also show very good agreement between the alulated and measured urves. However, the urrent density alulation inludes a number of simplifiations in addition to the effetive mass approximation. First, the leakage urrent. through the pos 1 1 1l i? EL? 1Z l3e +< 3 EN; 1o3 L3 1o4 1 o5 1 o6 lb) Voltage (V) FG. 6. Calulated (dotted line) and measured (solid line with dots) V harateristis of TBDW varator. (a) urrent density in linear sale, (b) urrent density in logarithmi sale. sible impurity states in the barriers as well as the leakage through possiblesurfae states have been ignored. Seond, our model does not inlude sattering mehanisms and therefore annot model possible urrent transport through the X valleys12 3 and other satteringrelated proesses. We expet that our urrent alulation will deviate more 2345 J. Appl. Phys., Vol. 72, No. 6, 15 September 1992 Sun et al. 2345

7 from the experimental values for strutures with a thik (greater than 2 A14) barrier or several more barriers. Further work is desired to improve the urrent density alulation for suh strutures. V. CONCLUSON A theoretial model was presented for modeling the CV and V harateristis of quantum well varators. The alulated results ompared favorably with the experimental data. The model was shown to be very useful for design and optimization of quantum well varators. A TBDW varator was suessfully designed and fabriated that exhibits superior apaitane ratio (typially 5 or greater) within a very small bias range ( f.2 V) with an aeptable urrent density level. The Wannier Stark ladder is effetively utilized in the TBDW varator to suppress the leakage urrent, while the Adoped layers ontrol the transition from maximum to minimum apaitane. Sine the maximum and the minimum apaitane of the devie an be ontrolled separately, design optimization of multiple quantum well varators an be readily performed for speifi appliations. On the other hand, further improvement of the urrent alulations is desired to inlude effets suh as the rx transport. ACKNOWLEDGMENTS The authors wish to thank Professor P. Bhattaharya and Y. C. Chen for providing the MBE materials used in this study. This work was supported by the Army Researh Offie under the UR program Contrat No. DAAL E. Kollberg and A. Rydberg, Eletron. Lett. 25, 1696 (1989). T. J. Tolmunen and M. A. Frerkhrg, Theoretial Effiieny of Multiplier Devies, Proeedings of the Seond nternational Symposium on Spae Terahertz Tehnology, edited by F. T. Ulaby and C. A. Kukkonen, Jet Propulsion Laboratory, California nstitute of Tehnology, Pasadena, CA, February 26, 1991, p P. D. Batelaan and M. A. Frerking, Quantum Well Multipliers, Conferene Digest for the 12th nternational Conferene on nfrared and Millimeter Waves, edited by K... Button (EEE, New York, 1987), p U. Lieneweg, B. R. Hanok, and J. Masexjian, BarrierntrisiN+ (BN) Diodes for NearMillimeter Wave Generation, ibid. (1987), p. 6 W. R. Frensley, Quantum Kineti Theory of Nanoeletroni Devies, Proeedings of the nternational Symposium on Nanostruture Physis and Fabriation, edited by M. A. Reed and W. P. Kirk (Aademi, Boston, 1989), p K. L. Jensen and F. A. Buot, J. Appl. Phys. 67, 762 (199). R. K. Mains, J. P. Sun, and G.. Haddad, Appl. Phys. Lett. 55, 371 (1989). W. B. Joye and R. W. Dixon, Appl. Phys. Lett. 31, 354 (1977); C. Kittle and H. Kroemer, ThermaZ Physis, 2nd ed. (Freeman, San Franiso, 1986). R. K. Mains, J. P. Sun, and W. L. Chen, Extension of the JoyeDixon Approximation for Semiondutor Devie Modeling within the ThomasFermi Approximation (unpublished). lo J. P. Sun and R. K. Mains (unpublished). G. H. Wannier, Rev. Mod. Phys. 34, 645 (1962). M. Rossmanith, J. Leo, and K. von Klitzing, J. Appl. Phys. 69, 3641 (1991). 13H. C. Liu, Superlatties and Mirostmtures 7, 35 (199). 14H Sakaki, T. Matsusue, and M. Tsuhiya, EEE J. Quantum Eletron. 2;, 2498 (1989) J. Appl. Phys., Vol. 72, No. 6, 15 September 1992 Sun et al. 2346