The consideration of space charge in the analysis of resonant tunneling devices leads to a substantial modification of the current-voltage relationship. The region of negative differential resistance (NDR) is shifted to a higher voltage, and broadened along the voltage axis. Moreover, the peak value of current prior to NDR is reduced, leading to a reduction in the predicted peak-to-valley ratio. An approach is presented to include space-charge effects, and a recently fabricated GaAs-Alx Gal _ x As structure is analyzed, to underscore the importance of a self-consistent electrostatic potential in theoretical calculations

Cahay, M.

Datta, D.

Lundstrom, Mark S

McLennan, M.

English

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Purdue UniversityPurdue e-PubsDepartment of Electrical and ComputerEngineering Faculty PublicationsDepartment of Electrical and ComputerEngineering1987Importance of Space-Charge Effects in ResonantTunneling DevicesM. CahayPurdue UniversityM. McLennanPurdue UniversityD. DattaPurdue UniversityMark S. LundstromPurdue University, lundstro@purdue.eduFollow this and additional works at: https://docs.lib.purdue.edu/ecepubsPart of the Electrical and Computer Engineering CommonsThis document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.Cahay, M.; McLennan, M.; Datta, D.; and Lundstrom, Mark S., "Importance of Space-Charge Effects in Resonant Tunneling Devices"(1987). Department of Electrical and Computer Engineering Faculty Publications. Paper 79.http://dx.doi.org/10.1063/1.98097Importance of space-charge effects in resonant tunneling devicesM. Cahay, M. McLennan, S. Datta, and M. S. LundstromCitation:  50, 612 (1987); doi: 10.1063/1.98097View online: http://dx.doi.org/10.1063/1.98097View Table of Contents: http://aip.scitation.org/toc/apl/50/10Published by the American Institute of PhysicsImportance of space-charge effects in resonant tunneling devices M. Cahay, M. McLennan, S. Datta, and M. S. Lundstrom School of Electrical Engineering, Purdue University, West Lafayette, lndiana 47907 (Received 5 November 1986; accepted for publication 6 January 1987) The consideration of space charge in the analysis of resonant tunneling devices leads to a substantial modification of the current-voltage relationship. The region of negative differential resistance (NDR) is shifted to a higher voltage, and broadened along the voltage axis. Moreover, the peak value of current prior to NDR is reduced, leading to a reduction in the predicted peak-to-valley ratio. An approach is presented to include space-charge effects, and a recently fabricated GaAs-Alx Gal _ x As structure is analyzed, to underscore the importance of a self-consistent electrostatic potential in theoretical calculations. Since the pioneering work ofTsu and Esaki, 1,2 there has been a growing interest in double-barrier resonant tunneling devices. Structures grown by both molecular beam epitaxy (MBE)3 and metalorganic chemical vapor deposition (MOCVD)4 have been reported with improving peak-to-valley ratios, exhibiting negative differential resistance (NDR) at room temperature. It has often been suggested,-7 that the presence of electrons could substantially affect the shape of the electrostatic potential in the devices, since the well acts as a dynamic trap for the tunneling electrons. Some authors have estimated that modifications of 10 meV or 0.1 eV could occur :in the conduction-band energy profile.5 ,6 Very recently, a quantitative calculation illustrating the ef-fect of space charge on the current-voltage (l- V) character-istic has been reported,S However, as discussed later, our results are significantly different; this is possibly because of two assumptions made in Ref. 8 that are different from our model. In this letter, we investigate the space-charge effects in resonant tunneling devices and perform a fully self-consis-tent calculation of an J- V characteristic. We compare our results to the usual approach, in which space-charge effects are completely neglected, and the application of an external bias is assumed to result in a linear voltage drop across the device. Hereafter, the latter approach will be referred to as "Hatband theory." In equilibrium, the inclusion of self-consistency is achieved as follows. The conduction-band profile is initially assumed to be the "fiatband" profile, including only the var-iations due to band-gap discontinuities for the differing ma-terials. Electron density can be calculated by solving the Schrodinger equation; this result modifies the net charge density in the device, and in turn, leads to a new solution for electrostatic potential from the Poisson equation. The calcu-lations for electron density and electrostatic potential are performed iteratively, until the electrostatic potential con-verges to a final solution. For structure under bias, a linear voltage drop applied to the equilibrium solution serves as an initial guess, and iteration is continued until self-consistency is established. The funy self-consistent potential is then used to calculate current density. Following Vassell et al., 7 we calculate the electronic cur-rent of a device by solving the Schrodinger equation, with the usual ':,0undary conditions for plane-wave solutions. I The effective-mass variation across the device is included by re-quiring everywhere the continuity of the wave function, and its first derivative divided by the electron effective mass. The transmission coefficient T is then obtained using the trans-fer-matrix technique,2.7 from which the current density can be deduced using em"" i'" loo J = -A dE! dE,T(E1,E,) [feE) - feE + eV»), 2n-fr 0 0 (1) where E[ and E, are, respectively, the longitudinal and trans-verse component of the electron total energy E; fCE) is the Fermi-Dirac distribution function; V is the bias applied across the structure; and m~ is the electron effective mass in the contact. This is the same formula used in the fiatband theory; however, we perform the calculation of electrostatic potential, and hence the transmission coefficient, self-consis-tently, To save computation time, we follow Vassell et aU and assume that the transverse energy E, is equal to its ther-mal average, k sT. Hence, the transmission coefficient be-comes a function oflongitudinal energy only, and the inte-gration over transverse energy can be performed analytical-ly. This same assumption also applies in the derivation of electron density. The electron density is calculated by considering two streams of electrons impinging from the contacts. For elec-trons incident from the left, we have a density given by r + oodk n 1 ~r(z) = _zl¢~';T(Z)12if-r(k_), .10 2rr Z' B G wh.ere (Yl-r(kz) (2) _ m~kBT [ (Ej-Ec(O) -f?k;/2m~)] - 2 In 1 + exp , .,m ksT / (3) E c (0) being the bottom of the conduction band in the left contact, which is hereafter taken as a reference point, i.e., Ee (0) = O. Following a similar derivation, the charge den-sity associated with electrons impinging from the right is given by an expression similar to Eq. (2); however, the wave function must be calculated for the opposite flow of elec-trons, and the factor (Y(kz )must be referenced to the conduc-tion-band edge in the right contact, Ec (L). The modification of the electrostatic potential due to the electron density is then calculated by solving the Poisson equation between the two contacts: 612 Appl. Phys. Lett, 50 (10), 9 March j 987 0003-6951/87/100$12-03$01.00 @ 1987 American Institute of PhYSics 612 !(E(Z) ~<P(z») = - q[N ~ (z) - N A (Z) - n(z)], (4) allowing for a position-dependent dielectric constant ECz), In Eq. (4), ¢(z) is the electrostatic potential, and N A' N Ii are the ionized acceptor and donor impurity concentrations throughout the device. As many as 15 iterations are some-times required between Eqs. (2) and (4) to obtain current densities with three significant figures. 9 Figure 1 (a) shows the cross section of a resonant tun-neling device fabricated by Ray et al., 4 which we have chosen to examine. The structure is typical of those proposed or fabricated, with heavily doped contacts (2 X 1018 em- 3 ) for a large resonant tunneling current, and undoped "spacer" layers on either side of the double barrier region. The inclu-sion of spacer layers has several advantages. First, the spacer layers tend to reduce the migration of impurities from the contact regions to the resonant tunneling region, thereby reducing impurity scattering in the barrier and well regions. Second, a greater degree of symmetry in the conduction-band energy profile is maintained, since an applied bias is dropped across a longer, undoped region. As Ricco and Az-bel pointed out, [0 asymmetry in the conduction-band profile degrades the peak in resonant transmission, reducing the resonant tunneling effect. Finally, the presence of spacer lay-ers pronounces the upward shift of the conduction-band pro-file in the undoped region, reducing the component of ther-(0) Structure !\nolyzed L...z=a contacts r-=·.::J spacer ioyers rzzT13 barriers tL2.:::J well () 250 750 1000 1250 Positior; (Angstroms) (b) Equil:b,ium Conduction Band Profile u c o .D c o :;::; () :::> u c o U 0,5 0.4 0.3 0.2 O. : 0.0 1------- - -~-------~-- self-consistent 'l - - - flatboiCd r ~ I j I quos:-bound I' J" states: ...... 1 .. ·· .Self-consisteJt L ............. ··flatbar,d c.J _~_~-~:':::""""'-l \ \ I , o 250 500 750 1000 1250 Positior, (Angstroms) FIG. 1. (a) Structure fabricated by Ray eta!. (Ref. 4). Contact regions are GaAs doped 2X 10'" cm -3 (Te); spacer regions are undoped GaAs; har-riers are undoped Al" .• ,GlIo"As; and the well is undoped GaAs. (b) Equi-librium conduction·band profiles for self-consistent and ftatband calcula-tions. 513 Appl. Phys. Lett., Vol. 50, No.1 0, 9 March 1987 mionic emission current over the top of the barriers. This upward shift is explicitly taken into account in our self-con-sistent calculations, and is illustrated for the equilibrium case in Fig. l(b). The current-voltage characteristics calculated for the structure of Fig. 1 (a) are presented in Fig. 2, showing both self-consistent and flatband results. Note that the position of NDR is higher along the voltage axis for the self-consistent case. The consideration of space-charge shifts the conduc-tion-band edge upward in the undoped regi.on, pushing the quasi-bound state in the well farther from the conduction-band edge in either contact [see Fig. 1 (b) ] . Since the voltage at which NDR is observed depends on this distance, an up-ward shift of the quasi-bound state causes a translation of NDR along the voltage axis. Moreover, the magnitude of this translation wiU be approximately twice that of the quasi-bound state shift, since it requires a bias of approximately twice the height of the quasi-bound state above the conduc-tion-band edge in the contact to observe NDR. For the struc-ture of Fig. 1, the upward shift of the quasi-bound state due to the consideration of space charge is approximately 0.042 e V, leading to a translation in the NDR region of nearly 0.1 V, relative to flatband calculations. Returning to Fig. 2, we note that the inclusion of self-consistency has broadened the NDR region. In the self-con-sistent calculation, current density reaches a maximum when the quasi-bound state level is well above the conduc-tion-band edge in the contact. This is illustrated in Fig. 3, which presents plots of the conduction-band profile at biases corresponding to current maxima (points P and Q of Fig. 2) for (a) fiatband and (b) self-consistent calculations. After the maximum current is attained, therefore, a larger addi-tional bias is required in the self-consistent case to pull the resonant level below the conduction-band edge in the con-tact, and reach the point of minimum current. The result is a broadening of the self-consistent NDR region, compared to the flatband solution. Finally, the peak curent of the NDR region is reduced O,() l 0.06 O.C~ 004 , Q) 0.0:.5 L L u 0.02 0 01 C).no FIG. 2. Current-volta!;.:: characteristics (both selt:comistent and Hatband results) for the structure of rig. 1, at 300 K. Note that the inclusion of self. consistency has shifted the position ofNDR to a higher bias, and broadened the characteristic. In addition, the peak current is reduced for the self-con-sistcn! calculation, Cahayeta!. 613 (0) !Io:bond C'Jrlduc-::!orl Rend Profile O.S >, 0.'1 Ul , G) D.) , l.u D.2 U c O. , , 0 en O.D c o -C.l ., U -0.2 "J U c -0.5 o U o 250 500 750 1000 f-Jo:sition (I\ng:st r orn:s) 1250 (b) Self Con~;iste~,; Conduct:or'l Bond Profiie r, 0.0 ::;> <lJ I"""~ - -"-- -'---" 0.4 >-CJ) 0.3 L <lJ c-Ld 0.2. j I v c- 0.1 0 m t ..................  D.O c 0 " 0.1 .;:; () ::J <:) -0.2 c 0 -0.3 U ,-_._-I-i -t--- ;---------1--·--, -----------,-.--o 250 SeQ 7:)0 ,aoo 12::'0 F'osdion (Ar'gstroms) FIG. 3. Conduction-band profile for biases of current maxima, for (a) fiat-band analysis (point P of }<'ig. 2) and (b) self-consistent analysis (point Q of Fig. 2). The level of the quasi-bound state is well above the conduction-band edge in the contact, for the self-consistent case. for the self-consistent calculation. Since the onset of NDR occurs at a higher bias in the self-consistent case, the trans-mission coefficient is severely degraded. Current is maxi-mized when the product of a degrading transmission peak and an increasing flux ofincident electrons is maximized. As mentioned above, this occurs when the quasi-bound state energy is well above the conduction-band edge in the con-tact. For the structure of Fig. 1, the vaney currents in both self-consistent and flatband analyses are nearly equal. The reduction in peak current, therefore, leads to an overall re-614 Appl. Phys. Lett., Vol. 50, No.1 0,9 March 1987 duction in the peak-to-valley ratio for the self-consistent re-sult. For the ftatband analysis, the peak-to-valley ratio is 10.6:1, this is reduced to 4.95:1 for self-consistent calcula-tions. Similar calculations have been published quite recent-ly,R leading to similar conclusions. However, there are two assumptions made in Ref. 8 that are different from ours. First, the electron density is calculated quantum mechani-cally only in the quantum well and the barriers; outside the barriers, such as in the buffer regions and in the contacts, the electron density is calculated classically. In our calculations, the electron density is calculated quantum mechanically everywhere in the device. Second, in Ref. 8 the transverse energy E, of the electrons is assumed to be zero while we replace E, by its thermal average k B T. On applying our tech-nique to the device considered in Ref. 8, we find a peak cur-rent density lower by a factor of 3. Furthermore, the upward shift in voltage of the self-consistent ND R region, compared to the fiat band result, is reduced by a factor of 4. This high-lights the sensitivity of J- V calculations, with respect to space-charge effects. Although the inclusion of self-consistency improves agreement between theory and experiment, meaningful comparison must await a more precise knowledge of device parameters ( e.g, doping densities, interface and bulk charges, contact resistances, etc.), and a realistic treatment of carrier scattering. It has recently been suggested that se-quential tunneling, rather than coherent resonant tunneling, is the relevant mechanism for the operation of quantum-well diodes. 11 However, an analysis of this effect requires a prop-er treatment of carrier scattering within the device, and re-mains a goal of future research. Nevertheless, the conclu-sions of this work emphasize the importance of space-charge effects in the analysis of resonant tunneling devices. This work was supported by the Semiconductor Re-search Corporation under contract 86-07-089. 'L, Esaki and R. Tsu, IBM J. Res. Dev. 14,61 (1970). 2R. Tsu and L. Esaki, App\. Phys. Lett. 22, 562 (1973). 3T. J. Shewchuk, P. C. Chapin, P. D. Coleman, W. Kopp, R. Fisher, and H. Morko", AppL Phys. Lett. 46, 508 (1985). 's. Ray, P. Ruden, V. Sokolov, R. Kolbas, T. Boonstra, and J. Williams, App!. Phys. Lett. 48,1666 (1986). 5n. Jogai, K. L. Wang, and K. W. Brown, J. Appl. Phys. 59, 2968 (1986). "J. N. S"hu\man, App!. Phys. Lett. 49, 690 ( 1986). 7M. O. Vassel!, Johnson Lee, ami H. F. Lockwood, J. App!. Phys. 54, 5206 (1983). RH. Ohnishi, T. Inata, S. Muto, N. Yokoyama, and A. Shibatomi, App!. Phys. Lett. 49, 1248 (1986). "Detailed results will be published elsewhere. tCB. Ricco and M. Va. Azbcl, Phys. Rev. B 29, 1970 ( 1984). "S. Luryi, App!. Phys. Lett. 47, 490 (l9gS). Cahayetal. 614 

Importance of Space-Charge Effects in Resonant Tunneling Devices

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Importance of Space-Charge Effects in Resonant Tunneling Devices

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