please see:

V. P. Kesan, D. P. Neikirk, B. G. Streetman, and P. A. Blakey, "A New Transit Time Device Using Quantum Well Injection," IEEE Electron Device Lett. EDL-8, April 1987, pp. 129-131.

V. P. Kesan, D. P. Neikirk, T. D. Linton, P. A. Blakey, and B. G. Streetman, "Influence of Transit Time Effects on the Optimum Design and Maximum Oscillation Frequency of Quantum Well Oscillators," IEEE Trans. Electron Devices ED-35, April 1988, pp. 405-413.

Kiran Kumar Gullapalli, Master's thesis title: "Simulation of Electron Transport in Double Barrier Diodes," The University of Texas at Austin, 1991.

Vijay Reddy, PhD dissertation title: "Characterization of High Frequency Oscillators and Varactor Diodes Grown by Molecular Beam Epitaxy," The University of Texas at Austin, 1994.

also see our work on experimental performance:

Experimental (Microwave and DC) Operation of the Quantum Well Injection Transit Time (QWITT) Diode

Influence of Space Charge and Quantum Well Negative Resistances on Resonant Tunneling Diodes

V.P. Kesan, D. R. Miller, V. K. Reddy, K. K. Gullapalli, and D.P. Neikirk

Microelectronics Research Center, Department of Electrical and Computer Engineering,

The University of Texas at Austin, Austin, Texas 78712.

ABSTRACT

This paper discusses the small-signal and linear analysis of resonant tunneling diodes used as negative resistance oscillators. The analysis concentrates on the transit (i.e., space charge) effects associated with a depleted spacer layer outside the quantum well. Closed form expressions are derived for small signal (linear) specific negative resistance based on the quantum well current-voltage characteristics. Conditions under which transit effects dominate device characteristics are shown to hold for most practical cases. The analysis shows that the effective limit on the maximum oscillation frequency of quantum well oscillators is determined largely by transit effects, rather than the intrinsic characteristics of the quantum well.

Introduction

Tunneling through quantum wells has been the subject of much recent research [1-14]. Because of the negative differential resistance exhibited by quantum wells, and since tunneling is an inherently fast transport mechanism, these devices have been proposed for use in extremely high frequency oscillators. Encouraging preliminary experimental results for such oscillators have been obtained at millimeter-wave frequencies [11, 13]. Experimentally, the quantum well diode with the highest reported frequency of oscillation has required the use of moderately doped spacer layers approximately 0.5 micron thick on either side of the quantum well region [11, 13]. Such thick spacer layers are expected to produce major effects on device performance, since a significant portion of the spacer layer on the anode side of the quantum well may be depleted, resulting in a transit-time delay. Deliberate exploitation of transit-time effects in the anode side spacer layer should enhance the performance of quantum well oscillators [22, 23], and such devices have been termed quantum well injection transit (QWITT) diodes [23]. In fact, the existence of significant depleted regions in the best experimental quantum well oscillators suggests that these structures [11, 13] operate in a QWITT-mode. Small-signal analysis of the negative resistance of transit-time diodes is useful for establishing oscillation thresholds, and hence the structures needed to obtain the highest possible operating frequencies from these devices. This paper presents such an analysis for quantum wells with anode-side spacer layers (i.e. QWITT diodes), and presents results which indicate the importance of transit/space charge effects for device oscillation.

Small Signal Analysis

The physical structure under consideration is shown in Fig. 1(a). It is assumed that the length of the quantum well region, l, is much smaller than the depleted spacer layer length W. The depletion region transit-time is thus much greater than the transit-time through the quantum well, and so transport through the quantum well is considered to be instantaneous here. The quantum well is treated as an injecting cathode whose small-signal equivalent circuit is the 'cold' (geometric) capacitance of the well, in parallel with a conductivity which represents the tunneling current (see Fig. 1(b)). For small-signal analysis the quantum well is characterized by a normalized injection conductance sigma, which is given by


where JQW is the instantaneous current density, VQW is the instantaneous voltage, and Vo is the dc bias voltage, each across the quantum well region only, excluding the voltage dropped across the depletion region. At present there are no generally accepted, accurate theoretical models which predict the transport behavior (and hence sigma) of quantum well structures. However, it is possible to estimate the value of sigma using the dc J-V characteristics of quantum well diodes. As noted above the terminal J-V characteristics must be corrected for any voltage drops across depletion and contact regions. Examination of both experimental and theoretical results [30-33] indicates that for a wide variety of quantum well structures, when biased to produce negative differential resistance, the room temperature value of sigma lies between about -0.05 ohm-1cm-1 and -0.5 ohm-1cm-1, with an average value of about -0.3 ohm-1cm-1 . Note that quantum well structures with large peak-to-valley ratios but lower current density [30] have given lower values of sigma than that obtained from structures with higher current density but lower peak-to-valley ratios [11, 13, 32]. Thus, from a small-signal perspective, the magnitude of the current density is more important than the peak-to-valley current ratio for the quantum well. This large current density is related to the use of thin quantum well barriers; in some cases [11,13] the barriers are more than a factor of two thinner than other structures [30].


Figure 1: (a) Quantum well diode structure which exhibits significant transit-time effects. The GaAs spacer layer on the cathode side of the quantum well region is made thin to reduce parasitic series resistance. A thick, lightly GaAs spacer layer is used on the anode side to produce a depletion region of length W, much longer than the thickness of the quantum well region l. . The transit-time through this layer is much larger than that through the quantum well region, and space charge effects in this region of the device can significantly affect the current-voltage curve; this device forms a quantum well injection transit (QWITT) diode.



Figure 1: (b) Small-signal equivalent circuit for the structure shown in (a). ZQW is the specific impedance, ' is the effective dielectric constant, and sigma is the injection conductance, of the quantum well; Ztt is the specific impedance, and is the dielectric constant of the depletion region. Rprstc represents the parasitic series resistance due to any undepleted spacer regions, highly doped contact regions, and ohmic contacts.

Once sigma has been found, the specific impedance of the quantum well injection region at an angular frequency is given by



where ' is the effective dielectric constant of the injection region. The total small-signal specific impedance (excluding parasitic elements) of the QWITT diode is the sum of this specific impedance and that of the depleted spacer region Ztt



To obtain the specific impedance Ztt, charge transport through the depleted (drift) region must be considered. Here it is assumed that the depletion region electric field is high enough to cause injected charge to traverse the depletion region at a constant saturated velocity vs. Although this approximation will break down for extremely thin depletion regions in which transient transport effects may be significant, it should be adequate in establishing trends of device operation.

Use of a frequency-independent injection conductance sigma and a constant saturation drift velocity permits the application of analytical methods previously established for other transit-time devices [24-26]. Based on these results, the specific impedance at an angular frequency for this device is


where is the dielectric constant of the drift region, and is the drift (or transit) angle, given by . For simplicity the dielectric constant of the quantum well region and the drift region have been assumed to be equal in this case. The transit (space charge) specific negative resistance that can be obtained from the depleted spacer region is the real part of Eq. 4:



The specific resistance of the quantum well injection region given by the real part of Eq. 2 is



The total specific resistance of a QWITT diode (excluding parasitic resistance in the device) is given by the sum of Eqs. 5 and 6. In conventional transit-time devices the injection conductance is positive, yielding a positive injection resistance, which must be overcome by a drift region negative resistance to obtain oscillation. For a quantum well injection region sigma can be negative, and thus the possibility of a negative injection conductance is introduced. In such a case the resistance of both the transit-time region and the injection region may be negative.

As expected from a transit-time analysis, there is a specific length of the depletion region W which yields the maximum negative resistance for any given frequency, injection conductance, and saturation velocity, given by



When using Eq. 7 it is necessary to select the appropriate branch of the inverse tangent function. For sigma positive, Eq. 5 yields a negative resistance only for the (2pi -  pi/2, 2pi + pi/2) branch. For sigma negative, the (-pi/2, pi/2) branch of the inverse tangent function should be used. At any given frequency there is also an optimum value of injection conductance sigma which will maximize the transit-time negative resistance, given by



Substitution of Eq. 8 into Eq. 7 yields an optimum transit angle of 5pi/3 for positive sigma, and pi/3 for negative sigma. Thus, at a particular operating frequency there is a unique combination of injection conductance sigma and drift region length W that will yield the absolute maximum small-signal transit-time negative resistance for this device, which is

    (9)

As noted above, the quantum well region can be biased so that it also produces a negative resistance; this corresponds to the negative sigma case. Considering Eq. 6 it is clear that there is a specific value of injection conductance sigma which will maximize the negative resistance available from the quantum well region, given by

  (10)

Comparing this to Eq. 8, it is found that quantum well negative resistance is maximized at a value of |sigma| which is 31/2 larger than that which maximizes the transit-time negative resistance. The relative importance of the two regions to total negative resistance can be determined by comparing the transit-time and quantum well negative resistances for sigma chosen to maximize quantum well performance. Using Eqs. 6 and 10, the maximum specific negative resistance available from the quantum well region is

  (11)

For this value of sigma Eq. 7 yields an optimum transit angle of pi/2, and Eq. 5 then gives the negative resistances available from the transit-time region of the device as

 (12)

If transit time effects dominate the negative resistance of the device then |Rtt| >> |RmaxQW, which yields the condition

  (13)

Thus, at a given frequency, if the value of injection conductance is chosen to maximize the quantum well negative resistance, transit time effects will still dominate as long as the frequency is lower than 0.4vs/l. Assuming a conservative saturation velocity of 6x106 cm/sec and a typical quantum well length l of 10 nm, transit time effects dominate when the operating frequency is less than approximately 380 GHz. At this frequency a very high injection conductance of -2.7 (ohm-cm)-1 is required to satisfy Eq. 10.

If the value of injection conductance is fixed, rather than being allowed to vary in accordance with Eq. 10, transit time effects may dominate over an even wider frequency range than that indicated by Eq. 13. For a fixed value of sigma, we should consider two limiting cases for Eqs. 5 and 6, corresponding to  << and  >> . For low frequencies we have

  (14)

and

                                         (15)

The first term in Eq 15 is negative since sigma is negative. The second term is always positive and represents the space-charge resistance familiar in conventional transit-time device theory. The total Rd will be negative if . Thus, at low frequencies, the condition to ensure that |Rtt| >> |RQW| is

(16)

Using the typical values for vs and l given above, Eq. 16 requires that |sigma| << 3.4 (ohm- cm)- 1, which is easily satisfied for the all quantum well structures studied to date.

For high frequencies ( >> ), the limiting values of negative resistance are given by

 (17)

and

  (18)

Thus, at high frequencies, the condition to ensure that |Rtt| >> |RQW| is

 (19)

Again using typical values for vs and l , Eq. 19 leads to the requirement that the operating frequency be less than 760 GHz. Use of Eqs. 16 and 19 provides a more realistic constraint on the requirement that |Rtt| >> |RQW| than Eq. 13, since it appears very difficult to achieve values of |sigma| > 1 (ohm-cm)-1. Hence, a sufficient condition to ensure that transit time effects dominate the intrinsic quantum well negative resistance is that Eqs. 16 and 19 be satisfied. For frequencies below , the QWITT diode exhibits a constant, broad band negative resistance and for frequencies much greater than , the resistance falls rapidly as -3. Therefore, it is desirable to operate the device in the "low frequency" regime where the negative resistance is constant and avoid the -3 falloff at frequencies much greater than . Figure 2 shows the small signal negative resistance produced by a fixed injection conductance of -0.4 (ohm-cm)-1, illustrating the relative contributions of the quantum well and the depletion region, as well as their low and high frequency behaviors.



Figure 2: Small signal calculated negative resistance fro a QWITT diode (from eqs. 5 and 6).

Small-signal analysis indicates that transit effects associated with the anode side depleted spacer layer in quantum well oscillator/QWITT diodes dominate design considerations. At any particular frequency of operation, achieving maximum performance requires appropriate optimization of both the depleted length of the anode spacer region and the quantum well injection conductance. Appropriate optimization of these parameters is predicted to yield QWITT devices whose RF performance potential is superior to that of bare quantum wells. The structure of the best experimental quantum well oscillators strongly suggests that they are in fact operating in a QWITT mode; additional performance improvements should result from further systematic optimization as suggested by the analysis presented in this paper.

QWITT Linear J - E Model

The QWITT diode was originally conceived as a "conventional" transit-time device and the acronym stood for Quantum Well Injection Transit-Time. The operation of transit-time devices, in the conventional sense, usually requires that the phase delay due to carrier transit-time in the drift region be a significant fraction of the ac voltage cycle. For example, devices such as the impact-ionization avalanche transit-time (IMPATT) diode operate only in a narrow frequency band where the carrier transit-time is approximately half of the ac voltage cycle (i.e., the drift angle, W/vsat, is pi). In the QWITT diode when sigma > 0, the device exhibits narrow-band negative resistance similar to other transit-time devices only when the drift angle (for optimum sigma) is 5pi/3 [23]. Thus, in this regard, it is similar to a conventional transit time device.

However, when sigma is negative, as it can be in a QWITT diode, the situation is quite different. As shown above, for << , the optimum depletion layer thickness is independent of frequency, which is a unique feature not seen in positive injection conductance devices such as IMPATTs. This is true even if the transit angle (i.e., delay) is much less than the ac period. The utilization of the drift region in QWITT diodes is very different from the use of carrier "delay" effects in devices such as IMPATTs. The negative resistance enhancement in the QWITT diode is simply due to the fact that the injection conductance is negative and the electric field is allowed to drop over a drift region. However the carrier density modulation in the transit region prevents unlimited increase in the negative resistance of the device and counteracts the beneficial effects of negative injection conductance. Hence there exists an optimum W (Wopt, see Fig. 5, below) for maximum negative resistance for a given sigma. This in essence is the QWITT diode, and to highlight this fact and avoid confusion with the conventional notion of 'transit-time,' the word 'time' is dropped from the name. It should be pointed out that all transit time analysis for IMPATTs and QWITTs is valid as a function of frequency, regardless of the drift angle magnitude compared to 2pi. But, only if sigma is negative can the negative resistance increase even when the drift angle is much less than 2pi.

To illustrate the above discussion, a dc analysis that takes into account space-charge modulation effects can be used to find Eq. 15 and shed insight into the effect of the drift region on the J - V characteristics [34]. Consider again the geometry of Fig. 3. The device is divided into two regions. The region on the emitter (cathode) side of the device up to and including the double barrier quantum well structure is called the injector. The moderately doped thick spacer layer, of thickness W, on the collector (anode) side forms the drift region. The electric field in the drift region is assumed to be high enough to cause the injected electrons to traverse the drift region at a constant saturation velocity, vsat. The decoupling of the injector from the drift region should be a reasonable assumption since the injector characteristic should not depend strongly on the drift region design. It is, however, influenced by the thin spacer layers on the emitter side of the quantum well. A dramatic example of this is the zero-bias, multi-state DBRTD recently reported by Gullapalli et al. [35]. The advantage of lumping the complex physics of the quantum well into an injection conductance is that the drift region and the quantum well can be independently optimized and the effects of each region on the other can be conveniently studied.



Fig. 3 Layer schematic of AlAs/GaAs Quantum Well Injection Transit (QWITT) diode.

For purposes of analysis we decouple the semi-classical transport occurring in the drift region from the quantum interference effects occurring in the quantum well [36]. The influence of the drift region can then be understood by characterizing the injector with an injection characteristic. This characteristic, shown in Fig. 4, describes the current through the device as a function of the electric field at the boundary separating the injector and the drift region.



Fig. 4 Current density versus electric field (J - E) characteristics of the DBRTD quantum well. Under the assumption that the drift region carrier transport is at a constant saturation velocity, independent of the injection characteristic, the J - E curve can be extracted from the DBRTD J - V characteristic by accounting for the voltage drop due to the depleted spacer layers and ohmic contact resistance.

Since the device is biased in the NDR region, the quantum well injector can be characterized by a normalized injection conductance, sigma, which is given by

                                          (20)


where the NDR characteristic was assumed to be linear between the peak and valley.

We can now see how increasing the drift region or spacer layer width on the anode side should affect . Shown in Fig. 5 are the electric field profiles in the drift region corresponding to the peak and valley points of the J - E injector characteristic.



Fig. 5 (a) A qualitative illustration of the electric field profiles in the drift region corresponding to the peak and valley points of the injector characteristic of Fig. 4. W is the drift region length. The positive slope, from Poisson's equation, represents the case where the injected electron density necessary to support the current is less than the background doping. (b) The voltage is just the shaded area between the peak and valley electric field curves and is maximized at an optimum drift region length, Wopt.

The slope of the electric field is positive, which from Poisson's equation, represents the case where the injected electron density necessary to support the current is less than the background doping density, as shown below

                                      (21)


where J is the current density through the device and Nd is the background dopant concentration. The slope of the field at the valley is greater than that at the peak because the current density, and thus the electron concentration, is lower in the valley. The voltage , the difference in the peak and valley voltages, is just the shaded area between the peak and valley electric field curves. Therefore, by increasing W from the baseline value of 250Å, one can increase the .

Quantitatively, the magnitude of the electric field as a function of position, E1(z) at a constant current density J1 is:

                                  (22)


where Eo is the electric field at the injecting plane z = 0, the boundary between the injector and the drift region. With a perturbation in the form of additional current , the resultant electric field, E2(z) profile is given by

                          (23)


The corresponding change in voltage across the drift region to first order in is then

                       (24)


where W is assumed to remain the same before and after the current perturbation. Dividing both side by , one arrives at an expression for the differential resistance / across the drift region:

                              (25)


where

                                      (26)


Equation 25 is identical to Eq. 15, which was derived from the drift-diffusion transport equations. This resistance is the ratio of the change in voltage across the depletion layer to the change in current density.

Since it is desirable to operate the QWITT diode in the "low frequency" limit where << , one can attempt to maximize the negative resistance given by Eq. 25. For a given sigma and vsat, the optimum drift region length, Wopt, for maximum negative resistance is

                                          (27)


Wopt is the drift region length at which is maximum. From Eq. 27, we see there are two ways to increase Wopt: decrease sigma or increase vsat. By decreasing sigma, one reduces the characteristic frequency, , and thus the frequency range over which the negative resistance is constant. Increasing vsat is possible by adjusting the material composition of the drift region. However, lattice-matching requirements restrict the choice of drift region material composition. AlAs/GaAs and AlAs/In0.53Ga0.47As QWITTs require the drift region to be GaAs or AlGaAs and In0.53Ga0.47As, respectively. Therefore, vsat is fixed by the drift region composition, where the saturation velocity for GaAs and In0.53Ga0.47As is 6 x 106 cm/sec and 5 x 106 cm/sec, respectively [37, 38].

Figure 6 shows measured J - V characteristics for a baseline AlAs/In0.53Ga0.47As DBRTD (W = 300Å) and an AlAs/In0.53Ga0.47As QWITT with a drift region length of 1000Å. The emitter spacer layers and quantum well in both devices are nominally identical. The quantum well consists of 17Å AlAs barriers and a 50Å In0.53Ga0.47As well. The emitter spacer layers, beginning with the closest to the quantum well, are 50Å nominally undoped In0.53Ga0.47As (n-type 5 x 1015 cm-3), 100Å (4 x 1016 cm-3) In0.53Ga0.47As, and finally 100Å (2 x 1017 cm-3) In0.53Ga0.47As. The for the baseline DBRTD is only 0.5 V. But by increasing the drift length to 1000Å the is doubled to approximately 1.0 V. Since is increased while remains essentially the same, the specific negative resistance increases over that of the baseline DBRTD. This increased negative resistance relaxes circuit impedance matching constraints and allows the use of larger area devices. Therefore much higher RF output powers can be obtained with QWITT diodes.



Fig. 6 Measured J - V characteristics of baseline AlAs/In0.53Ga0.47As DBRTD with W=300Å and W=1000Å, AlAs/In0.53Ga0.47As QWITT. The quantum well in both devices consists of 17Å AlAs barriers and 50Å In0.53Ga0.47As well. The for the baseline DBRTD and QWITT are 0.5 V and 1.0 V, respectively. The injection conductance, sigma, for these devices is -0.5 (ohm-cm)-1. Since the has doubled while remains essentially the same, the specific negative resistance is effectively doubled over that of the DBRTD.

Sponsors

This work was supported by the Texas Advanced Technology Research Program, the Joint Services Electronics Program under contract AFOSR F 49620-86-C-0045, and the National Science Foundation under grant ECS-8552868.

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