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JOURNAL OF APPLIED PHYSICS 100, 033718 2006

Transition between N- and Z-shaped current-voltage characteristics in semiconductor multiple-quantum-well structures
O. V. Pupyshevaa
Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan and Department of Low Temperature Physics, Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow 119992, Russia

A. V. Dmitriev
Department of Low Temperature Physics, Faculty of Physics, M. V. Lomonosov Moscow State University, Moscow 119992, Russia

A. A. Farajian, H. Mizuseki, and Y. Kawazoe
Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

Received 17 January 2006; accepted 17 June 2006; published online 14 August 2006 We study theoretically the vertical electron transport in semiconductor multiple-quantum-well structures, where sequential tunneling between neighboring wells takes place. The nonuniformity of electric field along the growth axis and charge redistribution among the quantum wells, as well as between the inner wells and contacts, are taken into account. A simple and efficient model of charged contact layers is proposed. The calculated I-V curves exhibit regions of conventional N-shaped negative differential conductivity and Z-shaped portions of intrinsic bistability, both arising due to the tunneling resonances. A general explanation of their formation mechanism is given, which is valid for any form of interwell transitions of resonant nature. The conditions of Nand Z-shaped curve observation and controllable transition between them are discussed. © 2006 American Institute of Physics. DOI: 10.1063/1.2234546
I. INTRODUCTION

The most interesting and important feature of semiconductor superlattices SLs and multiple-quantum-well MQW structures is that their current-voltage I-V characteristics usually possess a region of negative differential conductivity NDC . It arises due to the tunneling resonances between different quantum wells. In the structures with lowtransparent quantum barriers separating the wells, the formation of NDC regions is associated with nonuniformity of the electric field in the vertical direction along the growth axis . The formation of high-field domains in superlattices was considered as early as 30 years ago by Esaki and Chang.1 The feedback mechanism linking the changes in potential energy with carrier trapping in the quantum well was discussed by Ricco and Azbel.2 This leads to a necessity to solve the problem of mathematical description of the vertical electron transport in MQW structures self-consistently. Within the last 20 years, double-barrier resonanttunneling structures DBRTSs were intensively studied, both experimentally and theoretically. Bistability of their I-V characteristics, i.e., existence of two current values corresponding to the same voltage due to a resonance between the cathode emitter and the quantum well, was found experimentally by Goldman et al.3 It's intrinsic character was proposed and clarified in further discussion.4 The authors explained the obtained results using the sequential tunneling approach and taking into account charge accumulation in the quantum well, in agreement with the observed system behavior in magnetic field.5
a

Electronic mail: olga.pupysheva@rice.edu

These experiments immediately gave rise to numerous theoretical investigations of DBRTSs see, e.g., Refs. 6 and 7 . In particular, it was noticed7 that they may actually exhibit not bi- but tristability; however, the term bistability remains more common in the literature. The negative output resistance technique, proposed later,8,9 made it possible to directly observe continuous Z-shaped I-V curves of semiconductor double-barrier structures instead of conventional N-shaped ones. The problem of current instability and dynamical behavior arises here, as in every system manifesting a multistable I-V characteristic and/or NDC.10 Stability of the central branch of the Z-shaped region was studied in Refs. 11 and 12 within the sequential tunneling model. The general features of the current density front propagation in the systems with Z-shaped current-voltage curves are compared with those of more familiar S-type systems, for example, by Meixner et al.13 Interesting details regarding the lateral transport and lateral current density fronts in DBRTSs can be found, e.g., in Refs. 14 and 15. However, in the discussion below we assume no inhomogeneity along the layers of the structure and discuss only vertical charge transport. Behavior of relatively long SL and MQW structures under electric field is similar to that of DBRTSs. The contemporary situation and the latest achievements in the study of vertical electron transport in semiconductor SL can be found in recent comprehensive reviews.16,17 Time-independent models, reasonably describing multistable SL with electric field domains, were proposed by Bonilla et al.18 and Prengel et al.19 They used a combination of the rate equations, within the framework of the sequential
© 2006 American Institute of Physics

0021-8979/2006/100 3 /033718/9/$23.00

100, 033718-1

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tunneling approach, and the discrete Poisson equation. There have been many subsequent works modeling the sequential tunneling transport, in particular, time dependent, in weakly coupled semiconductor SLs.20­31 However, they use some specific assumptions, which make it difficult to generalize the obtained results. This concerns, first of all, the particular form of electric field dependence of the transition probabilities between the electron states localized in different quantum wells, or, alternatively, the electron drift velocity. In particular, the probability of reverse transition is often taken to be zero, while sometimes it is not negligible. As concerns the most commonly used boundary conditions, the electron concentrations n1 and nN in anode and cathode are either taken equal to the concentrations in the adjacent inner wells of the superlattice n 1 = n 2, nN = nN-1 19­22 or fixed at some constant levels not necessarily equal to each other .22­24 In the former model there is no electric field in the leftmost and rightmost quantum barriers of SL, while the latter does not account for the dependence of n1 and nN on the applied voltage and current density values. Both of these assumptions can be improved. Interestingly, the same problem was properly solved for DBRTS long ago,5 by applying the condition of electrical neutrality of the whole system including the accumulation and depletion layers of cathode and anode, respectively. But in studies of SL the global charge conservation26 ­29 or similar boundary conditions30 were utilized in only a few papers, and only one among them was devoted to the stationary transport.26 However, the I-V curves calculated in Ref. 26 contain few self-crossings, which is unusual and might be due to the approximations used in this work for the transition probabilities corresponding to the leftmost and rightmost quantum barriers. The current status of the contact modeling problem is discussed in detail in the reviews.16,17 In the present work we develop a rather general model for calculations of the stationary electron transport in MQW structures, with simple and efficient description of the contact effects. We do not consider dynamical phenomena, such as movement of the high-field domains and current oscillations, and restrict ourselves to the steady state. Solving the Poisson and kinetic equations self-consistently provides us with nonlinear electric field and carrier distribution in the MQW structure. We include the carrier concentrations in the contacts as variables to be calculated and use the global charge conservation for the whole system. We also take into account the continuity of the energy spectra of the contacts. This work gives a clear explanation of the occurrence of the N- and Z-shaped I-V curves. This explanation is independent of the particular form of the transition probabilities between the neighboring wells, as long as the transitions are of resonant nature. Unlike many other works, we do not neglect the reverse electron transitions between the quantum wells. Furthermore, the current-voltage characteristics of the structures with different numbers of quantum wells are compared. We derive the conditions of the transition between N- and Z-shaped curves with increase of the number of layers. This provides a possibility to control the appearance or disappearance of intrinsic bistability in MQW structures. Let us mention that our model does not use the period-

FIG. 1. Schematic model of the potential profile of MQW structure under electric field a , cathode b , and anode c .

icity of the MQW structure, and thus is applicable for the transport study in disordered systems.32 Although we consider only periodic structures in the present work, the discussed mechanism of the transition between N- and Z-shaped I-V curves is quite general and holds for disordered structures as well.
II. MODEL

The model of the potential profile of n-type MQW structure under electric field is schematically shown in Fig. 1 a . We consider a structure with thick quantum barriers of widths bk, much larger than well widths dk. Consequently, the potential variations in the quantum wells can be neglected, and the well potentials uk are taken to be constant. We assume for simplicity that only the wells are doped; the two-dimensional 2D concentrations of electrons and donors in the kth quantum well are nk and n0, respectively. As for the k quantum barriers, they are supposed to bear no electric charge, and thus the electric field in them is constant, and the potentials Uk = Uk x depend linearly on the coordinate along the growth axis. The low-transparent barriers allow us to consider the electron states as localized in the quantum wells and the vertical transport as sequential electron tunneling between the states localized in adjacent wells. We not only consider the charge redistribution among the inner quantum wells but also account for the electrodes cathode and anode , and, in particular, for charge accumulation and depletion layers in them. Following Refs. 19 and 24, we describe the contact layers as two additional, first and Nth, quantum wells of fixed widths d1 and dN, respectively. The total number N of wells in the structure includes Nw inner quantum wells and two contact layers. Notice that the energy spectra in the contact layers are continuous and the 2D electron concentrations n1 and nN enter our equations as unknown variables to be calculated. This distinguishes our model from those of Refs. 19 and 24.

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N

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The electron concentrations in the electrodes outside the charged areas are uniform and do not vary. We assume that there is no electric field in these outer, electrically neutral parts of the electrodes. The band bending effect in the charge accumulation and depletion layers, shown by thin solid lines in Figs. 1 b and 1 c , is effectively included in our model. Namely, a step function is substituted for the actual spacevariable potential in the contact layers. This function takes the values u1 and uN in cathode and anode, respectively. We assume that for any contact charge value, the electron chemical potential in the contact layer coincides with the chemical potential in the outer part of the same electrode, see Figs. 1 b and 1 c . Correspondingly, the voltage drop across the MQW structure under consideration is given by the equality V= u1 +
1

nk - n0 =0 . k
k=1

4

- uN - e

N

1 = e

N-1

With this assumption, the equalities similar to Eq. 2 are valid for both the contact layers k =1, N , with D0 = DN =0. As there are no quantum wells in the outer parts of the electrodes, u0 and uN+1 values are not defined in our model. In order to write the formal expressions analogous to Eq. 3 for 0 0 D0 and DN, we set u0 = u1, u0 = u0, and uN+1 = uN, uN+1 = uN. 0 1 The kinetic equation for the current density between any two adjacent wells, or a well and a contact, under dc conditions can be expressed via Pk and Rk, the integral probabilities per unit time of the direct and reverse electron transitions: I = e P kn k - R kn
k+1

uk +
k=1

1

- e

N

.

1

.

5

Here uk = uk - uk+1 is the potential drop across the kth quantum barrier, and 1 and N, the chemical potentials in the charged layers of contacts, are functions of electron concentrations n1 and nN and temperature, as follows from the Fermi statistics. This contact description is consistent with the assumption of rectangular shape of the inner quantum wells used in the present work and works well when the accumulation and depletion layers are thin. This is true in the case of heavily doped contacts, which we adopt here. Our model accounts for the most important features of the system, its electrical neutrality and a connection between the electrical charge in contacts and bias applied to the structure. At the same time it has the advantages of being simple and efficient for calculations, as compared with the more sophisticated models of contacts.5,26,30 These properties are especially important when a complicated structure with numerous layers is to be considered, as in the present work. Let us now discuss the equations we use to describe the vertical electron transport in MQW structure. The charge of the kth quantum well is connected with the electric field in the surrounding barriers by integrating the Poisson equation: n0 - nk = k 1 4e Dk - D
k-1

,

2

where e is the absolute value of the electron charge. The induction Dk of the electric field in the kth barrier determines the relative potential shift of the neighboring quantum wells, according to the formula Dk = u e
k+1

- uk - u bk

0 k+1

+u

0 k

.

3

0 Here u0 and uk+1 are the values of the well potentials in the k absence of the electric field and without charge redistribution in the structure, when all the wells are electrically neutral. As discussed above, there is no electric field to the left of the cathode accumulation layer and to the right of the anode depletion layer see Fig. 1 . This means that the whole MQW structure including the charged contact layers stays electrically neutral:

The transition probabilities depend on many factors, the most important being electron momentum distribution in the wells, energy spectra, and physical mechanism of interwell transitions. We assume that the carrier distribution is classical in the inner wells but may be degenerate in heavily doped contacts; so the distribution is described by the temperature and in contacts only by the Fermi energy. In our model the electric field does not influence the size-quantization energies in the wells, so that the electron states in each well shift together with the conduction band bottom on the energy scale. Hence the probabilities depend on the field only through uk, the potential shift of the energy levels in the adjacent wells. As concerns the interwell transitions, they can be accompanied by acoustical-phonon, optical-phonon, or impurity scattering. The choice of the structure material, temperature, and the layer parameters, presented in Appendix A, allows us to simplify the model by neglecting the electron-phonon interaction. The applicability of this approach, which is similar to the one used in Ref. 32, is discussed in Appendix B. Therefore, in the current work we consider an important mechanism of the electron interaction with charged donors in the quantum wells, although a generalization for any scattering mechanism is also possible. As the Coulomb matrix element is proportional to p 2 + / rD 2 -1/2, where p is the electron momentum change accompanying the transition and rD is the Debye screening radius, the corresponding probability reaches a maximum when the interwell transition with zero p is possible. The ionized impurity assisted transition is elastic, so this takes place when the energy levels in the neighboring wells coincide. In contrast, when an energy shift between the wells appears, a change of the in-plane momentum must accompany the transition, and hence the transition probability decreases see details in Appendix B . As a consequence, the probabilities Pk and Rk have a resonant form, i.e., they have maxima for such uk and uk+1 values when the energy levels in the neighboring wells coincide, that is, when they are in resonance. The transition probabilities used in our work are depicted in Fig. 2 as functions of uk for the structure parameters presented in Appendix A.

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FIG. 2. Decimal logarithm of direct solid line and reverse dashed line transition probabilities in 1/s of Appendix B vs potential drop in meV in the corresponding quantum barrier: transitions between two equivalent inner quantum wells a and transitions between the cathode and the adjacent inner well b . The structure parameters are described in Appendix A. Two maxima of the direct transition probabilities Pk are observed for uk 0, because there are two size-quantization levels in each quantum well, which can coincide either with the states localized in the neighboring wells or with the Fermi energy in cathode.

Let us stress that the formation mechanism of the observed I-V curve peculiarities to be explained below is valid for any form of the transition probabilities as functions of uk as long as they keep a resonant shape. So our explanation is rather general, which is its important advantage.
III. RESULTS AND DISCUSSION

The system of equations 2 ­ 5 can be rewritten as follows: nk = n0 + k 4e
2

uk-1 - u uk - u0 k - bk bk-1
k+1

0 k-1

,

6 7 8

I = ePknk - eRkn
N N

,

nk =
k=1 k=1

n0 = const , k

0 u0 = u0 = uN = uN =0 . 0

This nonlinear system consists of 2N -1 equations in 2N -1 variables uk and nk. For the n-type MQW structures studied here, the physically meaningful solutions are those with positive electron concentrations nk. As can be easily seen from the kinetic equation 7 , the condition nN 0 ensures nk 0 for all k. Therefore, the donor concentration in the anode depletion layer should be taken large enough to

provide the desired voltage value, in analogy to a plane capacitor. The system of equations 6 ­ 8 is solved selfconsistently for a given current value in the following way. For arbitrary electron concentration nk and potential drop uk, one can easily obtain uk-1 and then nk-1, successively using Eqs. 6 and 7 . After repeating this procedure for k = N , . . . , 2, that is, from anode to cathode, all the unknown variables are expressed through nN. To close the system, the global charge conservation, Eq. 8 , is solved numerically. All the possible nN values are found by means of shooting algorithm, and then for each nN concentration the other unknown variables are determined, including parameters of the two contacts and the total voltage drop across the MQW structure. Let us mention that if one repeats the procedure described above in the opposite direction, i.e., from cathode to anode, the accuracy of the numerical solution will decrease significantly. The reason is that, when finding the concentration nk+1 from Eq. 7 , one needs to divide a small value ePknk - I by another small value, eRk. Furthermore, the electron concentration in cathode n1 is larger than all the other electron concentrations, and a small relative change in its value would result in larger relative changes in all nk. The numerically calculated I-V curves of double- and multiple-quantum-well structures exhibit both Z- and N-shaped regions, as one can see from Fig. 3. Each of current maxima is observed when the energy states in a pair of neighboring wells are in resonance. It is worth mentioning that the form of the Z-shaped regions in Fig. 3 is quite similar to those obtained experimentally.8,9 There are two different types of these current maxima, those corresponding to the resonances in a pair of the adjacent inner quantum wells and those caused by the resonances between the cathode and the second well. The former type is related to the maximum of the transition probability Pk uk for k 1 shown in Fig. 2 a , while the latter is realized in the vicinity of a P1 u1 maximum depicted in Fig. 2 b . Due to the nonlinearity of the electric field distribution along the growth axis of the structure, the uk values increase with k see, e.g., stationary electric field profiles in Refs. 19­21 and 23­26 . As a result, for larger k, the resonance between kth and k +1 th wells and the corresponding current maximum are shifted to a lower voltage. So, for any of the curves in Fig. 3 the current maximum observed at the lowest voltage corresponds to the resonance between the rightmost inner quantum well and the previous one, i.e., to the maximum of PN-2 uN-2 transition probability; the next current maximum corresponds to the resonance between N -2 th and N -3 th wells, etc. The last two current maxima at higher voltages correspond to the P1 u1 transition probability maxima, occurring when the Fermi level in cathode is close to one of the two size-quantization levels in the adjacent inner well see Fig. 2 and its caption . The number of maxima on the I-V curve increases for the longer structures, i.e., for those with larger N. Namely, for the structures under consideration, which have two energy levels in each quantum well, one can expect totally N -1 current maxima. Nw -1 of them correspond to the

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FIG. 3. Color online Decimal logarithm of current density in A/cm2 vs voltage in V for the MQW structures with different numbers of layers. Each curve is labeled by the numbers of inner quantum wells Nw = N -2. The structure parameters and transition probabilities are given in Appendixes A and B. Boxed part of upper panel a is zoomed in a lower panel b .

resonances between the inner wells, and the last two are caused by the resonances between the cathode and the adjacent inner well, as discussed above. However, due to the low barrier transparence, some minor maxima in the low-field region cannot be clearly distinguished against the background of the fast growth of the curves see Fig. 3 . Moreover, if all the quantum barriers in the structure have the same width and height, then the probabilities of transitions through the leftmost and rightmost barriers, calculated as described in Appendix B, turn out to be much lower than all the others. This happens because of the relatively small number of electrons at the Fermi level of the corresponding contact, due to its continuous energy spectrum. In this case only two current maxima, which correspond to the maxima of P1 u1 , can be observed for any N. To make the other current maxima visible, one has to take the first and last barriers thinner than those inside the MQW structure, as it is done in the present work see structure parameters in Appendix A . Let us now compare the I-V curves for any pair of MQW structures with different numbers of quantum wells, N1 and N1 + N2 . As can be seen from Fig. 3, the maximum current densities, which are reached at the resonance in the same pair of adjacent wells, are almost equal in short and long structures. They are determined by the maximum, resonant values of the corresponding transition probability Pk. The corresponding voltages, however, significantly increase with N.

To explain this, we can roughly describe the longer MQW structure as a series connection of two shorter ones, containing N1 and N2 wells.33 It is very unlikely that two resonances could take place simultaneously in both of these parts of the long structure, because the electric field distribution along the growth axis is inhomogeneous. Therefore, when considering a resonance in the first part containing N1 wells, we can model the second part as having a regular, monotonously increasing I-V curve. Let the current I correspond to the voltages V1 in the first short structure and V2 in the second one, taking into account the actual electron redistribution in each of them.33 Then the same current value would be observed at the voltage V1 + V2 in the whole long structure. As a result, the shift of I-V curve along the voltage axis for the long structure, as compared with the short one, equals V2. Thus, the higher the current density, the higher is the V2 value, and hence the stronger the I-V curve deformation becomes for the long structure. In particular, this leads to a transition from conventional N-shaped current maxima to Z-shaped ones and to the elongation of the voltage interval where Z-shaped regions of the curve are observed. A similar transformation of I-V curves can be observed as a result of an increase of the doping level in the quantum wells.24,25 Our results are in agreement with those of a different model by Wacker et al.25 It is worth noting that the mechanism of such Z-shaped peculiarity is intrinsic. It should not be confused with an extrinsic bistability,3,4 which might arise from the series resistance of the Ohmic contacts34 or other external resistors.35,36 The intrinsic bistability discussed here originates from the behavior of various internal parts of the structure under consideration. It is based on the electrostatic feedback including the charge redistribution and accumulation in the quantum wells, similar to the mechanism proposed in Ref. 3 for DBRTS. The peculiarities of I-V curves can be also described analytically from a different point of view, using the following simplifying approximation. The electron current in the reverse direction from right to left in Fig. 1 quite often can be neglected as compared to the direct current: Rknk+1 Pknk. Then the kinetic equation can be rewritten in the form I ePknk . 9

Here the probability Pk is a function of uk and of the chemical potentials in the case of the degenerate electron distribution for k = 1 and N -1 , as discussed earlier. As concerns the electron concentration, nk = nk uk , uk-1 , according to Eq. 6 . This allows us to recursively calculate the derivatives of all the variables in terms of the single potential drop u j for some particular j. An extremum of I-V curve takes place when dI dI/d u j = =0 , dV dV/d u j that is, when either

10

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FIG. 4. Sample characteristics of current density vs electric field in the jth quantum barrier a , voltage drop across the whole MQW structure vs electric field in the jth quantum barrier b , and current density vs voltage drop across the whole MQW structure, possessing a Z-shaped region c .

dI =0 d uj or dV =. d uj

11

12

As can be seen from Eq. 9 , equality 11 can be fulfilled in the vicinity of an extremum of P j u j , i.e., when a resonance occurs between the jth and j +1 th quantum wells. As shown schematically in Fig. 4, with the increase of u j, the differential conductivity of the jth quantum barrier changes its sign and becomes negative. At the same time, for all the other, nonresonant barriers in the rest of the structure it remains positive. Thus, the voltage across the rest of the structure decreases as the current decreases. Together with the growing voltage across the jth barrier, this may lead either to an increase or to a decrease of the bias applied to the whole structure, according to Eq. 1 . In the former case a conventional N-shaped region of the I-V curve is formed, while the latter corresponds to a Z-shaped region, depicted in Fig. 4. Condition 12 is fulfilled when at least one of the functions uk u j , entering the sum in Eq. 1 , has an infinite derivative. Then d u j / d uk = 0, which can happen when the resonance occurs between the kth and k +1 th wells. The behavior of the functions I uk and V uk is analogous to those discussed in the previous paragraph, and the corresponding region of the I-V characteristic, again, can have either N-like or Z-like form. The electron concentration n j rises with the increase of u j and decrease of u j-1, according to Eq. 6 . If n j increases with u j while P j decreases, their product, which gives the current density by Eq. 9 , might grow monoto-

nously. Then none of conditions 11 and 12 is fulfilled. This means that for a very smooth P j u j maximum the corresponding part of the I-V curve may have no maximum. Such situation is realized when the jth barrier has relatively high transparence. Then the resonance between the jth and j +1 th quantum wells would not affect much the system behavior. One can see from this explanation that the quantities which control the occurrence of N- and Z-shaped portions of I-V curves are the two derivatives in the nominator and denominator of Eq. 10 . These derivatives are determined by the sharpness of the maximum of the transition probability functions P j u j and by the effective differential conductivity of the nonresonant part of the structure, correspondingly. In particular, a faster decrease of the transition probability with the potential drop u j in the resonant part of the structure results in a faster decrease of the current density I and of the voltage across the nonresonant part of the structure. Then the formation of Z-shaped current maxima is more favorable than for a smoother P j u j peak. Now it is clear that to observe both conventional NDC and intrinsic bistability, as well as transitions between them, the structures with relatively thick quantum barriers should be taken. One could already notice that the barrier widths in the MQW structures under consideration are taken larger than in many other works, where relatively thin, although still low-transparent, quantum barriers are studied. As a side effect, this decreases the current density values obtained in the present work in comparison with those of the other works, and shifts the current maxima away from each other. Decreasing the transparence of the MQW structure, one can easily cause the transitions between N- and Z-shaped current-voltage characteristics. This can be achieved by increasing either the width, or the height, or the number of quantum barriers, the latter case being demonstrated in Fig. 3. Let us emphasize that the explanation given above is valid for any functions Pk uk and Rk uk of resonant nature. For example, one can consider a MQW structure with both ionized impurity and optical-phonon scattering. This would lead to the formation of the Pk and Rk peaks for such uk values when the size-quantization levels in the neighboring wells either have the same energies or are separated by LO. Importantly, the corresponding current maxima of the I-V curve would have N- or Z-like shape, still determined by the factors discussed in the present work. Moreover, transitions from N- to Z-shaped I-V curves might take place not only in the semiconductor MQW structures but also in other quasi-one-dimensional systems with NDC, for example, in carbon nanotubes.37,38 Intrinsic bistability can be observed in such systems due to the defects or doping, decreasing the differential conductivity of their nonresonant parts. The transition mechanism discussed here does not depend on the origin of the negative differential conductivity.
IV. CONCLUSIONS

We studied vertical electron transport in semiconductor multiple-quantum-well structures within the sequential tun-

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neling approach and under assumption of resonant interwell transitions. The Poisson equation and kinetic equation for the current were solved self-consistently for the electrostatic potentials of the wells and electron concentrations in them. A simple and efficient model of contacts was proposed. The calculated dc I-V curves exhibit conventional N-shaped NDC regions as well as Z-shaped regions of intrinsic bistability, both arising due to the tunneling resonances. A general explanation of their origin was given, valid for any form of resonant electron interwell transitions. We observed a transition between N- and Z-shaped curves when the number of layers in the structure is varied and studied the corresponding conditions. We found that the structures with sharp transition resonances and low overall conductivity are favorable for the formation of Z-shaped portions of I-V curves.

Pk =
,

W

k, ;k+1,

E k, E k, , E E

k+1

,E

k+1,

fk E

k

1- f

k+1

k+1

2d2pkd2rk 2d2pk+1d2r 2 2 2 2
-1 k

k+1

fk E

k

2 d 2p kd 2r 2 2

,

B1

Rk =
,

W

k+1, ;k,

E k, E k, , E

k+1

,E

k+1,

f

k+1

E

k+1

1- f k E E

k

2d2pk+1d2r 2 2
k+1

k+1

2 d 2p kd 2r 2 2 .

k

f

k+1

k+1

2d2pk+1d2r 2 2

-1

B2

ACKNOWLEDGMENTS

The authors are deeply grateful to Dr. R. V. Belosludov, Dr. A. G. Mironov, Dr. M. A. Ormont, Dr. V. I. Pupyshev, and Professor I. P. Zvyagin for fruitful discussions and comments. The authors would like to express their sincere thanks to the crew of the Center for Computational Materials Science of the Institute for Materials Research, Tohoku University, for their continuous support of the SR8000 supercomputing facilities. This work was supported by Grant No. 1786.2003.2 of the Russian Foundation for Basic Research and a Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports, Science and Technology, Japan.

Here k =2, ... , N -2 ; indices and enumerate the sizequantization levels in the kth and k +1 th quantum wells, respectively; Ek, and Ek+1, are the energies of these levels; pk, rk, pk+1, and rk+1 are the electron in-plane momentum and coordinate vectors in the kth and k +1 th wells; Ek and Ek+1 are the full energies, Ek = E
k,

+

p2 k , 2m*

E

k+1

=E

k+1,

+

2 pk+1 , 2m*

APPENDIX A

In this work we consider gallium nitride based periodic MQW structures with the following parameters of the material and of the layers: LO-phonon energy LO = 92 meV; effective electron mass m* = 0.2m0; lattice parameter c = 5.17 å; dielectric constant = 12.2; well widths dk =12c for k =1, ... , N; barrier widths bk =24c for k =2, ... , N - 2 and bk =12c for k =1, N - 1; well potentials in the absence of the electric field and without charge redistribution in the structure, u0 = 0 for k =1, ... , N; barrier potentials in the absence k of the electric field and without charge redistribution in the structure, U0 = 100 meV for k =1, ... , N - 1; threek dimensional donor concentrations n0 / dk =1015 cm-3 for k k =2, ... , N - 1 and n0 / dk =1019 cm-3 for k =1, N; temperature k T = 10 K. There are two size-quantization levels in each inner quantum well of the structure, with the energies Ek,1 = 22.8 meV and Ek,2 = 81.0 meV k =2, ... , N -1 .

APPENDIX B

The integral probabilities of electron transition between the kth and k +1 th inner quantum wells are defined by the following general expressions:

of the electron states in the kth and k +1 th quantum wells; Wk, ;k+1, and Wk+1, ;k, are the probabilities per unit time of the direct and reverse transitions between the th level in the kth well and th level in the k +1 th well; and f k and f k+1 are the electron distribution functions in the kth and k +1 th quantum wells. When the barriers are wide enough, the interwell transition time is much greater than the intrawell relaxation time, one can treat f k as the equilibrium distribution function. The electron gas can be considered nondegenerate, provided that the electron concentration is much less than the effective density of states in the conduction band.39 This condition can be fulfilled for relatively low donor concentration in the inner quantum wells, like those mentioned in Appendix A. Then the interwell transition probabilities do not depend upon the electron concentrations, and Fermi factors can be eliminated from formulas B1 and B2 . As concerns the contacts, they are usually heavily doped, and the electron concentrations in them may be relatively high; consequently, the Fermi statistics must be used here. The electron energy spectra of the contacts are taken continuous, and the transition from summation to integration in Eqs. B1 and B2 gives the probabilities for k = 1 and k = N -1. All the information concerning the microscopic probability of a particular interwell transition, including energy and in-plane momentum conservation if applicable, is included within Wk, ;k+1, and Wk+1, ;k, . This may lead to a restriction of the integration interval. As an important example, in the present work we consider the probability of interwell transition under assumption that the main electron scattering mechanism is the interaction with ionized impu-

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rity. Indeed, at low temperature we can neglect the scattering by acoustical phonons, as well as the absorption of optical phonons. As for the LO-phonon emission, it becomes possible when an electron transition takes place between two levels either from the same quantum well or from two neighboring wells with energy difference equal to LO. However, the distribution function f k Ek , which enters Eq. B1 , is non-negligible only when the electron occupies the first size-quantization level of the kth well, with the in-plane kinetic energy p2 /2m* not more than a few meV. For the k transitions with such initial energy Ek Ek,1 + few meV , the optical-phonon scattering can be avoided as long as limited potential drops between the neighboring wells uk 32 eVmax LO are considered. In order to extend the uk limit, we consider material with relatively large LO-phonon energy, namely, gallium nitride see Appendix A . Importantly, each resonance between the kth and k +1 th wells k =1, ... , N -1 occurs when the potential difference between them is significantly smaller than LO of the chosen material see Fig. 2 . However, some uk values, calculated as described in Sec. III for relatively high current densities, may turn out to be still too high. So, the high-voltage regions of the curves in Fig. 3, which correspond to uk eVmax for at least one k, are presented formally. We keep them to provide a better illustration of the phenomenon under study, which is quite general for all the current maxima and independent of the particular uk values. However, one should keep in mind that, strictly speaking, the assumption of the absence of LO-phonon scattering is valid only for the low voltages. The probability of the interwell transition is proportional to the square of the impurity scattering potential matrix element. On one hand, this matrix element is determined by the overlap of the tails of the wave functions in the neighboring wells. On the other hand, due to the Coulomb scattering involved, it depends on the electron momentum difference between the initial and final states. One can roughly consider these two factors to describe the tunneling and scattering components of the interwell transition, correspondingly. We phenomenologically separate them by the following product: W
k, ;k+1,

One can estimate tk E as the transmission coefficient through a rectangular potential barrier. Parameters of the barrier are taken such that within the quasiclassical approximation its transparence is equal to that of the kth barrier of the MQW structure. Denoting the height of the rectangular barrier as U and its width as b and using standard quantum mechanics, one finds the following expression for the transmission coefficient: tk E =8 U - E E-u
k

E-u
k

k+1 k+1

cosh 2 b U - u

U-u
k+1

- U + uk -2E U + u +4 U - E E-u
k

-2E
k+1 -1

E-u

,

where = 2m* U - E / compare with the approximate expression in Ref. 3 . As for the dimensionless term Wks,k+1 in Eq. B3 , in the case of impurity scattering it reaches a maximum when the th level in the kth well is in resonance with the th level in the k +1 th well. When a finite level shift E = Ek, - Ek+1, appears, the electron in-plane momentum changes during the elastic transition, and hence the transition matrix element decreases, as described in Sec. II. For simplicity we take the probability dependence on the absolute value of E in the following form: W
s k,k+1

E k, , E

k+1,

=

Aq . A + Eq
q

Here the characteristic constant energy A and the dimensionless exponent q are model parameters, and E is the difference between the energies of the electron movement along the growth axis in the initial and final states. The numerical results shown in Figs. 2 and 3 are obtained for the transition probabilities calculated as described here, with A = 1 meV and q =1.
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2 3 1

E k, E k, , E
t k,k+1

k+1

,E

k+1,

=W

E k, , E

k+1, k+1,

W

s k,k+1

E k, , E
k+1

E k, E

.

B3

The Kronecker symbol equals unity for Ek = Ek+1, and zero otherwise. It accounts for the total energy conservation, as it is appropriate for the elastic impurity scattering. The tunneling term Wkt,k+1 E in Eq. B3 can be estimated as the probability current due to the resonant tunneling of an electron with energy E from the kth to the k +1 th quantum well. The probability is proportional to the corresponding transmission coefficient tk E : W
t k,k+1

E k, , E

k+1,

=
k+1,

2 E-u dk m .
*

k

tk E ,

where E = max E

k,

,E

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36 37

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