A multibarrier heterostructure GaAs/AlAs current switching diode has been investigated by Monte Carlo method. The switching phenomenon is based on electron tunnelling and thermoemission from the GaAs wells to the AlAs barriers, electron drift across the thin AlAs barriers followed by a subsequent impact ionisation in the undoped GaAs layers. The calculated switching voltage is close to 100 V for the diode involving four AlAs barriers at 300 K lattice temperature. The estimated switching time is in the order of 10 ps

Grigaliūnaitė, G.

Reklaitis, A.

English

Nonlinear Analysis: Modelling and Control

Nonlinear Analysis: Modelling and Control, Vilnius, IMI, 1998, No 3MONTE CARLO STUDY OF MULTIBARRIER HETEROSTRUCTURE SWITCHA. ReklaitisSemiconductor Physics Institute, A.Go š tauto 11, 2600 Vilnius, LithuaniaG. GrigaliûnaitëVilnius Gediminas Technical University, Saulëtekio  al.11, 2040 Vilnius, LithuaniaAbstractA multibarrier heterostructure GaAs/AlAs current switching diode has beeninvestigated by Monte Carlo method. The switching phenomenon is based onelectron tunnelling and thermoemission from the GaAs wells to the AlAsbarriers, electron drift across the thin AlAs barriers followed by a subsequentimpact ionisation in the undoped GaAs layers. The calculated switchingvoltage is close to 100 V for the diode involving four AlAs barriers at 300 Klattice temperature. The estimated switching time is in the order of 10 ps.The switching voltage of the most developed devices is in the order of severalvolts [1-7]. On the other hand, high-speed switches exhibiting high switching voltage areof great importance. In [8, 9] a high voltage switching diode composed from amultibarrier GaAs/AlAs heterostructure is proposed. In the present paper this structure,is investigated by Monte Carlo simulations. The principle of operation of the voltage-controlled switch is based on the electron tunnelling process through the heterojunctioninterfaces, followed by subsequent interband impact ionisation and confinement of thecreated electrons and holes in GaAs wells.A schematic cross section of the proposed four-barrier AlAs/GaAs switch isdepicted in Fig. 1. The structure consists of the adjacent to the cathode, heavily dopedGaAs layer followed by four unit cells. Each unit cell involves an undoped AlAs barrierfollowed by an undoped GaAs layer, and finally, by a doped GaAs layer. The donorconcentration of 5x1018 cm-3 for the cathode and anode contacts is accepted. It isassumed that electrons in the contacts are in thermal equilibrium with the lattice. Themodels of GaAs and AlAs include G, L and X nonparabolic valleys for the conductionband, and heavy, light and split-off nonparabolic subbands for the valence band. Allbasic phonon scattering processes as well as ionised impurity scattering are taken intoaccount for the electrons and holes. The material parameters of GaAs are assumed to bethe same as the ones collected in [10]. The material parameters of AlAs are taken from[11]. Band offset between the G valleys of GaAs and AlAs is assumed to be 1.03 eV [12].GaAs, L = 0.25 mm, Nd = 1017cm-3AlAs, L = 0.10 mm,  undopedGaAs, L = 0.65 mm,  undopedAlAs, L = 0.10 mm,  undopedGaAs, L = 0.65 mm,  undopedGaAs, L = 0.25 mm, Nd = 1017cm-3AlAs, L = 0.10 mm,  undopedGaAs, L = 0.65 mm,  undopedGaAs, L = 0.25 mm, Nd = 1017cm-3AlAs, L = 0.10 mm,  undopedGaAs, L = 0.65 mm,  undopedGaAs, L = 0.25 mm, Nd = 1017cm-3Anode: GaAs, Nd = 5x1018 cm-3Cathode: GaAs, Nd = 5x1018 cm-3GaAs, L = 0.25 mm, Nd = 1017cm-3Fig. 1. A schematic cross section of the diode.The thermionic emission and tunnelling across GaAs/AlAs heterointerfaces wereconsidered for electrons and holes. Depending on the carrier wave vector, either atrapezoidal or a triangular potential barrier was considered for the tunnelling of eachindividual Monte Carlo particle impinging the barrier. The tunnelling probabilities usedin the simulations were derived by using the transfer matrix approach and taking intoaccount the discontinuity of the carrier effective mass at the heterointerfaces [8, 11].The tunnelling probability for the trapezoidal barrier is given by Eq. (1). Thethree layers composing trapezoidal potential barrier are characterised by indexes 1 (layerof incident electron), 2 (barrier layer), and 3 (the final layer after the tunnelling). Theobtained transmission tunnelling probability for a trapezoidal barrier is given byDkk Mxx= ´4 13212p,                                                                                                (1)where k1x and k3x are normal to the heterointerface wave vector components, incidentand final, respectively,M A B A BkkA B A Bi i d i d ixxi d i i i d20 0310 02= -+ -{}[ ( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )]/ // /h h h hh h h h+ -+ -{}[ / / / /( ) ( ) ( ) ( )][ ( ) ( ) ( ) ( )]m km kA B A Bm km kA B A BFxi i i d ixFi i d i d i12 10 0 02 310 02h h h hh h h hAi and Bi are Airy functions, Ai' and Bi' are their derivatives, andkmE eFd kx32 12 12= + -h( ) / / ,k m eFF = ( / )/2 22 1 3h ,h 0 0 2= -k U E eFF x( )/   ,hd F xk U E eFd eFd= - -( )/0 2   ,E E k mx II2 12 222= - h / ,m1 and m2 are electron effective mass in GaAs and AlGaAs layers, respectively, U0 is thebarrier height, F is electric field strength in the barrier, d is the barrier width, E1 is thekinetic energy of the incident carrier, and k // is the carrier wave vector component,parallel to the heterointerface.The tunnelling probability across a triangular barrier is given byDAi Bi Bi Ai=+ + -42 2np h n h h n h[ ( ) ( )] [ ( ) ( )]/ / ,                                          (2)wheren =m km kFx12 1,k m eFF = ( / )/2 22 1 3h ,h = -k U E eFF x( ) /0 2  .The other quantities in Eq.(2) have the same notations as in Eq.(1).The details of the Monte Carlo simulation are described in [6,10,13,14]. ThePoisson equation is solved on a spatial mesh with 2000 nodes. The potential distributionis updated at each time step Dt = 1 fs. The short time step was chosen in order to avoidthe numerical instabilities associated with high carrier concentration. A spatial mesh witha large number of nodes was employed in order to evaluate precisely the shape of thepotential distribution at the heterointerfaces, which shape is necessary for the accurateestimation of the tunnelling probabilities of each particle impinging the barrier. Resultswere obtained with 20,000 simulated particles. Calculations carried out with differentnumbers of particles showed that 20,000 simulated particles is sufficient for theachievement of high accuracy results.The impact ionisation initiated by the electrons and holes was considered in thesimulations, taking into account the secondary carriers created by impact ionisation [15].The final states of the carriers after each ionisation event were determined as randomquantities in such a way as to satisfy both energy and momentum conservation. Theconventional probability distribution of the incident carrier states before and afterionisation,  S( , ) ( )k k k k¢ µ - ¢ -4   was considered. The procedure is developed in such away that the possible final states of the carriers cover completely the allowablemomentum space for each individual ionisation event. The Keldysh formulaP P E E Et t= -02 2( ) /   was accepted for the dependence of the ionisation probabilityagainst the energy E of the incident carrier. The values of the ionisation thresholdenergy Et for electrons and holes were evaluated from energy and momentumconservation. The values of the adjustable coefficient P0 for electrons and holes areestimated by ensuring the best agreement of the calculated ionisation coefficients inbulk material with the experimental data [16]. The impact ionisation in AlAs wasdisregarded in the calculations because the experimental results [16] demonstratedionisation coefficients rapidly decreasing with the increase in the mole fraction x ofAlxGa1-xAs.The I-V characteristic of the diode was calculated taking into account anexternal circuit. The series connected load resistance Ra = 50 W  was accounted for self-consistently with the Monte Carlo simulations of the diode. The calculations werecarried out for the device area S = 5000 mm2. The behaviour of the circuit was governedby the equationR CdUdtU U R Iadd a d0 + = - ,                                                             (3)where C0 is the geometric capacitance of the diode, Ud is the diode bias voltage, U is thebias voltage applied to the circuit, and Id is the diode conduction current. This equationwas numerically solved simultaneously with the Monte Carlo particle simulations of thediode.0 50 1 0 0 1 5 00501 0 01 5 0Current  (A)Voltage  (V)Time  (ps)0.00.51.01.5Fig. 2. The response of diode current and diode voltage (solid curves) to a ramp ofcircuit bias voltage raised linearly from U(0) = 0 to U(t) = 150 V (dashes) during t = 150ps.The calculated circuit response to the ramp of the circuit  bias voltage U(t) isreported in Fig. 2. The results presented in Fig. 2 were obtained for U(t) linearly raisedwith time from U = 0 up to U = 150 V during 150 ps. As seen from Fig. 2, the diode isswitched from the low conductance (“closed”) state into the high conductance (“open”)state when the diode bias voltage is close to Ud = 100 V. The I-V characteristic reportedin Fig. 3 was obtained from the results presented in Fig. 2. The I-V characteristic exhibitsa pronounced S-type dependence of the current upon bias voltage. The current-voltageinstability associated with the S-type negative differential resistance Rd is damped by theexternal circuit because the condition Ra-1 + Rd-1 < 0 necessary for the development of theinstability was not satisfied. The switching time is in the order of 10 ps as it is seen fromthe results presented in Fig. 2. The calculations carried out for the cross section area S=10000 mm2 demonstrated essentially the same behaviour of the I-V characteristic. Theswitching time is raised to approximately 15 ps in a diode with the area S = 10000 mm2.The analysis of the calculated results shows that the scenario of diodeswitching from the closed state to the open state is as follows: The calculated potentialdistributions corresponding to the closed and open states are reported in Fig. 4. In theclosed state, the majority of the electrons are confined in the highly doped GaAs layersadjacent to the AlAs barriers.0 50 1000.00.51.01.5Diode current  (A)Diode voltage (V)Fig. 3. Calculated current - voltage characteristic of the diode.The electrons occupy the lowest G valley because the electric field in the dopedGaAs regions is weak. Electron tunnelling from the doped GaAs layers into the AlAsbarriers is responsible for current flow in the closed state. The electron energy isinsufficient for the carriers to surmount the heterojunction barriers by thermionicemission due to the large band offset (1.03 eV). Intense impact ionisation does notdevelop in the low conductance state. Although the electric field is sufficiently high forinterband impact ionisation in the undoped GaAs layers just before switching, theintense carrier multiplication does not originate because this region is depleted ofelectrons. As the bias voltage is further increased, the electric field in the AlAs layers israised, and the potential barrier at the heterointerface becomes thinner and morepermeable for electron tunnelling. Enhanced tunnelling current leads to the increase ofthe electron concentration in the AlAs barriers. These electrons, after crossing the AlAsbarriers, enter the undoped GaAs layers, where the electric field is very high. As a result,the avalanche carrier multiplication develops due to intense impact ionisation in theundoped GaAs layers. The intensity of electron impact ionisation is additionallyenhanced by the conduction band offset between AlAs and GaAs [17,18]. Furthermore,the electrons and holes generated by the ionisation are partially confined by theconduction and valence band offsets at the GaAs/AlAs interfaces. This leads to anadditional increase of the electric field at the interfaces, and, consequently, thetunnelling current grows as a consequence. As a result, the diode is switched to theopen state. The potential is redistributed in the open state as shown in Fig. 4. In thisstate, the high electric field in the AlAs barriers and the enhancement of the carrierenergy by the band offset between AlAs and GaAs is still sufficient for maintainingimpact ionisation in the open state.0 1 2 3 4020406080100Open stateClosed stateEnergy (eV)Posit ion (mm )Fig. 4. Potential (the edge of G valley and the top of heavy hole subband)distributions in the low and high conductance states. The results were obtainedunder the same conditions as in Fig. 3 and correspond to the time moments t = 90ps for the low conductance state, and to t = 130 ps for the high conductancestate, respectively. The upper and lower arrows indicate the regions in which anintensive impact ionisation is initiated by electrons and holes, respectively, in theopen state.The stability of the diode, both in the closed and open states, was investigatedby Monte Carlo simulations under the conditions of a constant bias voltage applied tothe circuit. The results of the calculations show that the diode is not switched to thehigh conductance state at least during 300 ps when the circuit bias voltage is just belowthe switching voltage U = 90 V. Also, the diode is sustained in the open state for at leastduring 300 ps when the constant bias voltage U=130 V is applied to the circuit. Noevidence of diode switching from the one state to the another was observed from MonteCarlo simulations during 300 ps.Because the switching process is initiated by the avalanche carriermultiplication in the undoped GaAs layers, the value of the switching voltage isdetermined by the length and number of unit cells. The condition for the onset of theavalanche multiplication initiated by electrons is given by M > 1, where [19]M x x x dx dxxL= - -òòa a b( ) exp ( ) ( ){ }[ ]/ / /00                                                (4)Here a and b are electron and hole ionisation coefficients, respectively, and L is thelength of the region in which the impact ionisation takes place. The electric field strengthF in undoped GaAs layers is nearly uniform just after switching. According to theresults of the calculations, F is close to 300 kV/cm in the undoped GaAs layers when thediode bias voltage corresponds to the switching voltage. The values of the ionisationcoefficients in GaAs at F = 300 kV/cm are [16]: a = 7300 cm-1  and b = 4300 cm-1. The totallength of the undoped GaAs layers in the device is L = 2.6 mm. By substituting thenumerical values of a, b and L into Eq. (4) we obtain M = 1.3, thus the condition for thedevelopment of the avalanche multiplication is satisfied.We note that the switching voltage is proportional to the number of unit cells.Simulations of two-barrier and three-barrier diodes demonstrated switching voltages of50 V and 75 V, respectively. It can be expected that a further increase in the number ofunit cells above the four used in this simulation will result in the increase of theswitching voltage to above 100 V. Also, one can anticipate that the switching time willnot rise due to internal current amplification which is conditioned by carriermultiplication induced by impact ionisation. The switching time is determined by the rateof space charge redistribution inside the unit cells. Because the potential barriers of theheterojunctions are sufficiently high, most of the created carriers are confined at theheterointerfaces. Therefore, the switching time can be estimated roughly as the timeduration of carrier drift across the GaAs layer of the unit cell. By assuming electron andhole drift velocity to be 107 cm/s, we obtain a switching time of 9 ps which is in closeagreement with the Monte Carlo results.The simulation of the circuit response to a circuit bias voltage which isdecreasing from U = 200 V to U = 0 V during a period of 200 ps showed that the closingtime of the device is on the same order as the opening time. According to the results ofsimulation the intensity of impact ionisation in the GaAs layers at the heterointerfacesbecomes insufficient to maintain the staircase shape of the potential distribution (Fig. 4,open state) when the diode bias voltage is reduced to a value below 50 V. The electronsand holes confined at heterointerfaces are extracted due to tunnelling andthermoemission processes. As a result, the potential distribution becomes linear, and thedevice is switched to the closed state.In conclusion, a new multibarrier heterostructure switch is proposed. It wasinvestigated by Monte Carlo particle simulations. The switching phenomenon is basedon electron tunnelling and thermoemission from the GaAs wells to the AlAs barriers,electron drift across the AlAs barriers, followed by subsequent impact ionisation in theundoped GaAs layers. The calculated switching voltage is close to 100 V for a diodeinvolving four AlAs barriers at 300 K lattice temperature. The estimated switching time isin the order of 10 ps.This work was supported by the US Ballistic Missile Defence Organisation, bythe US Air Force's European Office of Aerospace R&D under contract F61775-98-WE011and by Lithuanian Science Foundation.REFERENCES1. K. Hess, T. K. Higman, M. A. Emanuel, J. J. Coleman, J. Appl. Phys. 60, p.3775(1986). 2. J. N. Baillargeon, K. Y. Cheng, J. Laskar, J. Kolodzey, Appl. Phys. Lett. 55, p.663(1989). 3. C. Song, K. P. Roenker, J. Appl. Phys. 72, p.4417 (1992). 4. W. C. Hsu, W. C. Liu, D. F. Guo, W. S. Lour, Appl. Phys. Lett. 62, p.1504 (1993). 5. W. C. Liu, D. F. Guo, S. R. Yih, J. T. Liang, L. W. Liah, Appl. Phys. Lett. 64, p.2685(1994). 6. A. Reklaitis, R. Stasch, M. Asche, A. Krotkus, E. Schoell, J. Appl. Phys. 82, p.1706(1997). 7. D. F. Guo, Appl. Phys. Lett. 71, p.1219 (1997). 8. A.Reklaitis, Abstracts of 20th Workshop on Compound Semiconductor Devicesand Integrated Circuits, Vilnius 1996, p.95. 9. A. Reklaitis, J.Appl. Phys. (in press). 10. A. Reklaitis, J. Appl. Phys. 80, p.1242 (1996). 11. S. Adachi, J. Appl. Phys. 58, R1 (1985). 12. C. N. Yeh, L. E. McNeil, L. J. Blue, T. D. Race, J. Appl. Phys. 77, p.4541 (1995). 13. A. Reklaitis, Proc. Third ELEN Workshop (edited by C. Clayes and E.Simoen,Leuven, Belgium, 1996), p.108. 14. A. Reklaitis, L. Reggiani, J. Appl. Phys. 82, p.3161 (1997). 15. A. Reklaitis, J. Phys. Chem. Solids 42, p.891 (1981). 16. V. M. Robbins, S. C. Smith, G. E. Stillman, Appl. Phys. Lett. 52, p.296 (1988). 17. R. Chin, N. Holonyak, G. E. Stilman, J. Y. Tang, K. Hess, Electron. Lett. 16, p.468(1980). 18. F. Capasso, W. T. Tang, A. L. Hutchison, G. F. Williams, Appl. Phys. Lett. 40, p.38(1982). 19. S. M. Sze, Physics of Semiconductor Devices ( Wiley , New York, 1981).

Monte Carlo study of multibarrier heterostructure switch

https://www.journals.vu.lt/nonlinear-analysis/article/download/15262/14247

Monte Carlo study of multibarrier heterostructure switch

Abstract

Similar works

Full text

Available Versions

Nonlinear Analysis: Modelling and Control