Anode domain transient processes in supercritical Gunn diodes

Solid-Stare Elecwmics Vol. 27, No. Printed in Great Britain.

3, pp. 233239,

ANODE

1984 0

DOMAIN TRANSIENT PROCESSES SUPERCRITICAL GUNN DIODES

003sllOlp4 1984 Pergamoo

$3.00 + .xl Press Ltd.

IN

0. A. KIREEV,M. E. LEVINSII’IRINand S. L. RUMJANTSEV A. F. Ioffe Phys. Techn. Inst. 194021 Leningrad, Polytechnicheskaya 26, U.S.S.R. (Received 28 April 1982; in revised form 8 June 1983) Abstract-Using a computer Gunn-diode model the transient processes connected with a supercritical anode domain are investigated. Anode domain formation, domain response to step-bias variation and anode domain annihilation are studied. It is shown, that transient processes in a diode with an anode domain are characterized by considerable larger times, than in a diode with a travelling (Gunn) domain with the same sample length, carrier concentration and bias voltage. Anode formation and transformation can be described as a process of nonlinear domain capacitance C, charging via a nonlinear resistance R, which is determined by the field outside the domain. Simple stationary equations can be used to find all parameters of these transient processes. In some practically important cases the anode domain transformation process can be described analytically. It is also shown that anode domain annihilation is characterized by a strong dependence of the annihilation time on the bias. The annihilation time increases indefinitely with the increasing bias up to the sustaining threshold.

NOTATION differential domain capacitance cd electron charge E electric field E, maximum domain field E, outside domain field E, threshold field J current density J, = en@, threshold current density L sample length boundary anode concentration 4 n cr minimal concentration of anode domain formation equilibrium electron concentration active region differential resistance for the sample with a domain low field resistance time u, domain annihilation voltage domain voltage ud u&r stable domain voltage uo bias voltage u, threshold voltage electron velocity V V,= 1.78 . lo5 m/s threshold velocity cc0 dielectric constant ICI low field mobility differential mobility pd time constant of the domain formation and transformation time constant of the domain annihi7, lation 7,, = C,R, time constant of the domain formation and transformation from simplified calculation

amplifiers[5-81. Stable high-field domains are of great importance in GaAs FET[9-121. Among other things it is necessary to point out that the equilibrium electron concentration no in millimeter-wave Gunn generators is to be about 1Or6- 1Or7cm-‘[13, 141.Under such doping level the interaction between Gunn and anode domain modes can have a determining influence on the amplifier and generator properties[8, 151. After all the static negative resistance phenomenon in anode domain mode is of great interest[ 15, 161. As for dynamic properties of the anode domain, they are practically unstudied up to the present. In this paper all the main transient processes of anode domain dynamics are investigated using a computer Gunn-diode model. Anode domain formation, domain response to step-bias variation and anode domain annihilation are studied. 2. SIMULATION CONDITIONS

mode in supercritical Gunn diodes can exist only in samples with high concentration levels, n, > n, > 2 -- 3 x lOI cm-‘[2]. In this situation the field model can pretend strictly speaking to qualitative description only. However numerous calculations[2, 17-221 show that the field model actually describes the anode domain mode in very delicate details. In this paper the Gunn diode is described by one-dimensional equations of the field model assuming field-independent diffusion coefficient, D = 200 cm2/s. The field dependence of the diffusion coefficient, D(E), can effect the quantitative characteristics of the anode domain. But the exact shape of the D(E) dependence is unknown today even for GaAs (see for example[23]). Also it was shown in Ref. [16] that relaxation effects become important for the same concentrations at which the D(E) dependence must be really taken into account. Anode domain

1. INTRODUCTION mode in a supercritical Gunn diode was discovered by Thim in 1971 [l, 21. Since that time more than 40 theoretical and experimental papers have been published in this field. Such interest is due to the following reasons. First of all, the Gunn diode in anode domain mode is one of the most promising semi-conductor switches[3,4]. Secondly, the supercritical diodes with anode domain can be used as power and wide-band microwave The anode

domain

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Therefore it is, strictly speaking, inconsistent to take into consideration the D(E) dependence in calculating the domain parameters in the frame of the field model. Anyway, it is well known that the field model gives a correct qualitative and semiquantitative description of the anode domain behaviour even with D = const[l8,21]. The field dependence of the electron drift velocity u(E) is approximated by the relation given by Thim [24]

v=

(1)

Diodes lengths L N 48 pm are typical for all experimental and computer studies [4, 19,2629]. The maximum value of n, is determined by two points. Firstly, it is shown below that for no > 8 x 10’5-10’6cm -3 the corresponding transient times are less than 1-2 x lo-i2s. These times are about he time of o(E)dependence formation. So for larger values of no the field model isn’t correct. Secondly the bistable switching diodes have maximum concentration about 8 x 10’5-10’6cm-3 [18, 19,211. Most experimental papers deal with samples of doping level about 3-5 x lOi cm- 3[1,4,26].

l+ p

3. STATIC J-U CHARACTERISTICS AND STABILITY CONDITIONS

where p, = 0.7 m2/V . s is low-field mobility, u, = 0.85 x lo5 m/s is high-field saturation velocity, E” = 3.8 kV/cm. It corresponds to a threshold field threshold velocity E, = 3.34 kV/cm and a v, = 1.78 x lo5 m/s. The cathode contact is described by a narrow region with doping level ten times higher than in the active region[25]. The doping level falls linearly from the contact to the active region over a length of 0.2-0.5pm. There is a “notch” of 0.51% near the cathode. The anode boundary condition assumes a constant concentration, n,,. Such a condition gives the opportunity to describe a wide variety of J-U curves characterized by different values of threshold current, threshold voltage, current drops etc. (see below). Transient processes for two samples of n = 8 x 10’Scm-3 (L =4pm) and n,=4x lii5 crn3 (L = 7.5 pm) are studied in detail. Anode domain mode stability for minimum concentrations near nC,- 2 x lOI cn-3 are also explored.

For fixed doping level in the active region, n,,, J-U characteristics strongly depend on boundary anode concentration, n,. In Fig. 1 the J-U curves are shown for four different values of n, and no = 8 x lOI cm 3 (L. = 4 pm). In good agreement with Ref. [22] increase of n, leads to a decrease of threshold voltage and threshold current (Fig. 1). The difference between the threshold voltage of anode domain formation, U,, and the domain annihilation voltage, U, decreases, as does the current drop, AJ. When the value of n, is rather large the switching effect disappears (Fig. 1, curve 3). Further increase of n, leads to disappearance of the negative differential resistance region (curve 4). Our calculations show that for n,
E4'

0

E(kV/cm)20 16

/

1=8@s)

Fig. 1. Steady-state J-U characteristics for anode domain mode. q, = 8 lOI cmm3. L = 4 pm n, (cm-‘): (1) 8. lOIs, (2) 1 1016,(3) 1.2. 10i6, (4) 1.6. lo’! Inset shows anode domain formation process for no = 8. 1015cn-3, n, = lOI IX-~, U, = 1.3 V. For t = 8 ps the anode formation process is practically finished.

Processes in supercritical Gunn diodes

While n, falls to nC,N 2 x 1Ou cme3 the field outside the anode domain tends to its thrdhold value E, Therefore even small doping fluctuations (“notch” in this model) lead to appearance of Gunn oscillations and to suppression of the anode domain mode[4,6, 18,291. However, with increasing bias the field outside the domain reduces, so the anode domain mode can be realized for n, 2 ncr and U, > U,. In agreement with several authors[7,20] our calculations show that further increase of bias leads again to the disappearance of the anode domain; Gunn generation taking place again at U, $ U, and n, 3 ne,. For no - ncr2 2.2-2.4 x 10’5cm-3 one can obtain the following picture of instability. A large amplitude travelling domain mode (Gunn oscillation) arises in a sample for the bias which is just above the threshold value. With increase in bias up to the critical value, U,,,, the anode domain appears. Large amplitude oscillations disappear but travelling domains of very small amplitude remain in the sample. Further increase of U, makes the diode perfectly stable. The region of stability is rather large. But as the bias increases still further the small-amplitude travelling domains appear again. At the same time very large amplitude anode domains continue to exist. The amplitude of the travelling domain grows rapidly with further increase of U, up to the second critical value U,,,. For U,,2 U,,, there is no anode domain and the usual Gunn oscillations take place.

4. ANODE DOMAIN FORMATION

There are two possibilities for the formation of a static anode domain. Firstly the travelling domain or accumulation layer can appear near the cathode. When this domain (or layer) approaches the anode the steady-state anode domain forms. Secondly, the anode domain can grow just near to anode (see inset

T

235

of Fig. 1). It depends on many factors: the concentration level n,, doping fluctuations (the depth and width of the “notch”), boundary conditions, the rate of bias application dU,dt and steady-state U,, magnitude after switching. Our calculations give the main relations as follows. The probability of domain formation near the anode increases with n, and n, and with decrease of the cathode field, dU,/dt rate and steady state value of U, after switching. The domain formation process has been simulated as follows. First a bias U,,g 0.94.95 U, was applied. Then a small voltage step AU, was applied so that U, + AU, > U,. The inset in Fig. 2 shows the field of parameters (hatched region) where the domain formation near the anode occurs (for n, = 8 x lOI crn3). For n, > 1.15 x lOi cme3 there is no current drop on J-U characteristic (see Fig. 1, curve 3). For each value of n, less than 1.15 x lOI cme3 there is a certain critical value of AU,. While AU, -CAU,, the domain formation occurs near the anode. If AU, > AU,, the domain formation takes place near the cathode. This phenomenon can be easily understood. During the first moment after the voltage step the space-charge distribution in the sample is conserved. So the field increase in each point of the active region is the same and equal to AE = AU,/L. If the voltage step amplitude AU,, is rather small, the field near the cathode is lower than its threshold value, E, (see. inset in Fig. 1). However, for n, > n,,, the field near the anode is larger than E,, so the domain formation takes place near the anode. If AU,, is large enough the field near the cathode exceeds the threshold value of E, and domain formation near the cathode is possible. Further growth of AU,, value leads to domain formation near the cathode because of the greater increment value of space-charge waves.

(PSI

Fig. 2. Time dependence of domain voltage U,during the formation process n, = 8. lOI cme3, L = 4 pm. n, (cm-‘): 1, 2-9. lOI’, 3, 4, 5-lo”‘, 6, 7-1.1 . 1016.The voltage drop across a completely formed domain UAV): (lw.3, (2)-0.33, (3)-0.25, (4)-0.32, (5)-0.42, (6)-0.50, (7)--0.55. 1’ and 2’ curves are calculated according to eqn (3).

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Anode domain formation is investigated in this paper for the case of its formation near the anode only. The process of domain formation near the cathode was described in detail earlier[25,30,31]. Figure 2 shows the time dependence of domain voltage, U,, during the domain formation process for various anode concentrations, n,, and voltage values across a completely formed domain, U,. The voltage across the domain, U,, is calculated as follows: U, = U,, - E,L - U,

(2)

where E, is the field outside the domain, U, is the voltage drop across the anode region before switching (see inset in Fig. 1. In all cases studied below, U” 4 U&J). There are two regions on the curves 1, 2 in Fig. 2. One of them describes the delay stage where the domain voltage, U,, grows very slowly. The second region is characterized by a faster U,, growth. Curves 3-7 have practically no delay stage. For these curves the time dependence of U, is closely described by:

Relation (3) is well known to be true for the process of travelling Gunn domain formation as we11[31]. Gunn domain formation is also characterized by some delay stage[29], but it is much shorter than in case of the anode domain. The physical reason of the slow growth of U, during the delay stage is as follows. If the anode concentration n, is about n, (curves 1, 2) the maximum field near the anode at the early stage of domain formation is only slightly over the threshold

field E,. Therefore the growth increment which is determined by the negative differential mobility is small. After the delay stage is over the Ud(t) dependence is represented by eqn (3). The values of T may be chosen in such a way that eqn (3) describes curves 3-7 very closely. These values of T coincide rather well with the domain characteristic time constant z. = Rd. C,[31, 321, where R,= L/en,&E,) is the active region differential resistance, and C,= cc,E,,,/2U, is the differential domain capacitance. Figure 3 shows the dependences of Z, T,, and travelling (Gunn) domain time constant on the domain voltage U,. Hatched region gives the T values fitted to the best agreement between (3) and computed curves Udt) (Fig. 2) for different boundary conditions. The points obtained by the calculations according to expression z. = C,R, (Necessary parameters E,, E,,, and U, are calculated from the simple equations of the steady state problem[l& 221). One can see that T and T,,values coincide very closely for n,= 8. 10’5cm-3. For n, =4. 10’5cm-3 the agreement is somewhat worse. It’s also seen that the Gunn domain time constant is considerably less than that of the anode domain. Note that for n, = 8 . 1O’jcme3 the anode domain characteristic time constant T is about the electron energy relaxation time in GaAs T, = 2 ps. Anode domain dynamics calculations for higher concentration levels evidently need the use of models dealing with finite electron scattering and the rethermalisation times. The dashed line in Fig. 3 shows, that the characteristic time constant of the travelling Gunn domain obtained using the field model for n, = 8 1O’jcm-’ is less than r<. Therefore this curve is only illustrative.

3-

Fig. 3. The dependence of domain formation (and transformation) time constant 5 on the voltage drop across the completely formed domain V,(n, = 8. lOI cn-‘). Dashed line represents the r(U,) dependence for Gunn domain[31]. Inset shows the same dependences for sample with n, = 4. lOI cm -3, L = 7.5 flrn.

Processes in supercritical Gunn diodes The travelling domain dynamics calculation for n, 2 5 . lOi cm -’ needs a model dealing with relaxation effects. To calculate r0 as a produce of C, and Rd some points are to be taken into account. For Gunn domain the field outisde of the domain E, is practically always substantially less than the threshold field E,. Thus the differential mobility in the field outside of the domain p,, is simply the low field mobility ~1,. Therefore for the Gunn domain & is equal to the low field resistance &. In the case of anode domain for realistic n, values the magnitude of E, is near to the threshold field E,. That’s why /J,,can be much less than p, and hence R, can be considerably greater than &. The anode domain capacitance is also higher than that of the Gunn domain with the same V, and 5. There are two reasons for that. Firstly, in case of anode domain the outside field E, is significantly higher than in case of the Gunn domain. It leads to a lower anode domain voltage V, = V,, - E,L.. Therefore the anode domain capacitance which grows with decreasing V, is found to be larger, than for the Gunn domain. Secondly the anode domain capacitance is larger even for equal V, values. The domain capacitance is about in inverse proportion to the domain width d = 2V,/E,,,. The Gunn domain width is determined by the sum of accumulation and depletion layers widths. At the same time the anode domain width is determined by the accumulation layer only. Thus the time constant of the anode domain formation is much larger than for the Gunn domain with the same sample parameters and bias values (Fig. 3). Calculations show that the time constant T weakly depends on boundary conditions and with n, iixed is determined only by the voltage drop across the stable domain V, (Fig. 3. Compare also curves 1, 2 and 4 in Fig. 2). It should be partcularly emphasized that the results presented here give the opportunity to calculate completely the transient process of domain formation using the parameters V,, Em and E,. All these parameters can be determined from the simple equations of the steady state problem (see for example[l8,22]).

5. ANODE DOMAIN RESPONSE STEP-BIAS VARIATION

TO

Anode domain transformation process is conveniently described by the time dependence of voltage across the domain, V, if the bias, V,,, changes very abruptly (step-bias variation). Results obtained show that the anode domain transformation process, as well as the transformation of the Gunn domain[32], is well described by equation

(4)

237

where

Here V, is the voltage across the domain before the bias change, V, is the voltage across a completely formed domain after the bias change, C, is the after the transformation domain capacitance (C, = cer,E,,,,/2V,), and R, is the active region differential resistance after the transformation. As well as in the previous section results obtained give the opportunity to describe analytically the transient process of domain transformation. The parameters Vd,, V,, C, and RdLcan be calculated from solution of the stationary problem[l8,22]. Moreover, when the bias V,, before and after the bias step, is appreciably larger than the threshold value of V,(V, 2 (1,5 t 2)V,), the results presented in Ref. [22] provide a fully analytical description of the transformation process. If V, 3 (1.5 t 2)V, the current density in a diode with an anode domain is quite near to its minimum value J,,+ determined by the concentration level no. According to[22]

J,,=Jfl+(JJ]. Here Js = enovs, where trS is the very high-field drift electron velocity (see relation (l)), A is a constant determined by v(E) parameters. For GaAs A = 3.3 x 10’5cm-3. Equation (5) is valid for n, 3 4 x lOI crn3. Knowing Jminit’s easy to find from eqn (1) the value of E, and hence the mobility p,,(Er) and the voltage drop across the domain V, = V, - E,L.. On the other hand in case of practical interest when no 6 10’6cm-3 there is a simple relation connecting V, and E,,,[22]:

Equation (6) determines the maximum domain field E,,, with known V,. Domain capacitance C, is calculated as C, z q,ld z c+!$,,/2V,. Thus, all quantities, which are necessary to calculate the process of domain transformation in accordance with eqn (4) are determined analytically. Figure 4 demonstrates very good agreement between computer and analytically calculated Vkt) dependences during the domain transformation.

6. DOMAIN ANNIHILATION AT THE BLAS BELOW MINIMUM SUSTAINING VOLTAGE

One can see from Fig. 1 that anode domain is characterized by a domain annihilation voltage V, which is less than the threshold voltage of domain

238

0. A.

0

I

I

I

I

2

4

6 tkJs9

6

KIREEV et

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Fig. 4. Time dependence of domain voltage lJ, during the transformation process. (1) computer calculation; (2) analaccording to ytical calculation eons (4H6) n 0 =4~10’5cn~3,L=7..5~m, U,,,=6 V, Uo2=4V.

al.

3). The same region is obtained for Gunn domain annihilation process as wel1[33]. It may be explained by fast annihilation of the small-amplitude domain when the field in the most part of the domain is less than the threshold value E,[33]. If n, > n, the region of fast decrease is absent (curve 4) because even after the annihilation process is ended, in stationary conditions, there is a region near the anode where the field amplitude E is rather large (see inset in Fig. 1, curve for t = 0). If U,, < ZJ, the process of annihilation can be well described by a simple exponential function: U, = U&e 'ira. The same expression is valid at the beginning of the annihilation process if U,z U,. The inset in Fig. 5 shows the dependence of 7. on the bias U,,during the annihilation process (to be compared with the same dependence for Gunn domain[33]). 7. CONCLUSION

formation U,. The difference between these two values in anode domain mode was noted in [3,6, 181. For the Gunn domain annihilation process a strong dependence of annihilation time on the bias voltage, U, < U,, was observed. The annihilation time indefinitely increases with the bias increasing up to the sustaining threshold. When U,-+O, the time of annihilation becomes much less than the characteristic times of domain formation and transformation[33]. Our calculations show that anode domain annihilation is characterized by the same dependences. Examples of domain annihilation for different values of U,< U, are shown in Fig. 5. It is evident that the annihilation time substantially increases with increasing U,. For n, = n, and U,zsU, a characteristic region of abrupt collapse is to be noted in Ud(f) curve (curve

0.0

The investigation of anode domain transient processes shows that anode domain formation and transformation can be described as the process of nonlinear domain capacitance C,, charging via a nonlinear resistance, R, = L/en,pAE,).The value of Rd can be considerably greater than the low-field diode resistance, R,,. The anode domain capacitance, C,, is much higher than the capacitance of a Gunn domain with the same n,, L and U, values. Thus the characteristic time constant of anode domain formation and transformation t0 = RdCd is much larger than the corresponding value of 7 for a Gunn domain. (2) Anode domain transient processes of formation and transformation can be described rather well without using the complete system of partial differential equations. There are simple analytical relations for these processes and all required param(1)

To/T IO

0.6 -

6

0.7 -

6

0.6 -

4

I2

0.5

Fig. 5. Time dependences of domain voltage Ukt) during the annihilation process. l-3 curves n, = no = 8. 10’5cm-‘. U,/U,. (lp.78, (2w.89, (3v.94, (4) n, = lOI cm-r; U,/U,= 0.94.Inset shows the dependence of the characteristic time constant of annihilation process on bias U, n,= q; n, (cm-‘); 1-8 10r5, 2-4. 1015.

Processes in supercritical Gunn diodes

eter can be determined using the simple system of ordinary equations describing the stationary problem.

(3) When the bias is high enough (U, 3 1.517,) the domain transformation process can be calculated analytically.

(4) When the bias falls below the minimum taining

voltage,

U,,

is characterized,

the anode domain

as well as Gunn

lation, by a strong dependence time on the bias, U,,.

The annihilation

sus-

annihilation

domain

annihi-

of the annihilation

-time indefinitely

increases with

the bias increasing up to the sustaining threshold, U,. Acknowledgement-It is a pleasure to thank M. Dyakonov and A. Furman afor many encouraging and useful discussions. REFERENCES

1. 2. 3. 4. 5.

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