Linear and nonlinear analysis of submicron n+nn+ diodes for microwave generators

MicroelectronicEngineering 19 (1992) 417-420 Elsevier

417

Linear and nonlinear analysis of submicron n+nn+ diodes for microwave generators V. Gruzhinskis a, E. Starikov ~, P. Shiktorov ~, L. Reggiani b, L. Varani b and T. Kuhn c aSemiconductor Physics Institute, Lithuanian Academy of Sciences Goshtauto 11, 232600 Vilnius, Lithuania bDipartimento di Fisica e Istituto Nazionale di Fisica della Materia, Universit£ di Modena, Via Campi 213/A, 41100 Modena, Italy CInstitut fiir Theoretische Physik, Universit~t Stuttgart, Pfaffenwaldring 57, 7000 Stuttgart 80, Germany. Abstract

A theoretical investigation of the electrical behavior of submicron n + n n + diode microwave generators is presented. Generation frequencies up to 800 G H z are predicted for InP made devices working at 300 K with an active-region length of 0.2 micrometer. 1. I N T R O D U C T I O N The modeling of microwave generators is an essential step to project and realize devices with o p t i m u m performances [1]. The aim of this communication is to propose an original h y d r o d y n a m i c approach which combines the advantage of requiring low-cost computational resources with the property of being physically sounded. As application, we will estimate the m a x i m u m and optimum frequencies attainable by submicron n + n n + diode structures made of InP, and evaluate the efficiency of the generator when it is placed in a series resonant-circuit. 2. T H E O R E T I C A L

MODEL

The theoretical model is based on a one-dimensional geometry and follows a three step procedure. First, we determine the following five static-characteristics of the bulk material as a function of the electric field E chosen along the z direction: ( m - 1), (v), (~/, (6v2), (6vge). Here rn -1 is the carrier reciprocal effective-mass along the field direction, v its drift velocity, e its kinetic energy and brackets have the meaning of average over the distribution function. These quantities are provided by Monte 0167-9317/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved.

+

418

+

V. Gruzhinskis et al. / Subrnicron n nn diodesjor microwave generators

Carlo simulators and include the main features of the energy band-structure [2]. Second, for a given voltage applied to the diode Ud(t) we determine the local space-time behavior inside the diode of: (i) the carrier concentration n ( z , t ) , (ii) the average drift velocity v(z, t), (iii) the average carrier energy e(z, t), (iv) the electric field E ( z , t), (v) the small-signal impedance at frequency u Z'(u), (vi) the generation band diagram and (vii) the o p t i m u m frequency for microwave generations. To this end we propose an original h y d r o d y n a m i c approach which is coupled with a self-consistent calculation of the electric field. In the third step, we simulate the series resonant-circuit to evaluate its generation efficiency r/(u) for a given load resistance R (measured in ~ m2), inductance L and applied voltage U,~. This is obtained by calculating both: the power which is generated and dissipated by the diode in the presence of a large harmonic signal of the applied field and, the electrical-behavior of the diode in the resonant-circuit. The set of equations describing steps two and three are below reported. Ot --

(nv)

(1)

Ov Ov 0--[ = e g r n - ' - u.v - V Oz Oe 0--7 =

1 0 n Oz (nQ~) OE

eEv

-

c -

) -

v Oz

10 n

(nQ~)

(2) (3)

E ( z , t) = E(O, t) + Ep(z, t)

(4)

Ep(z,t) -

(5)

e

[n(z',t) - ND(z')]dz'

CO~r

1[ S'

E(O,t) = 7

Ud(t) --

Ep(z,t)dz

(6)

Z'(')= fo°° l [dud(t)]

(7)

d -~Ud(t)-

(8)

-~Itot(t)d

1 A~e.

[ I t o t ( t ) - Ic(t)]

= -L1 [Ua - Rigor(t) - Ud(t)]

(9)

Here e is the electron charge, ¢0 the vacuum permittivity, ~. the static dielectric constant, eth the carrier thermal-energy, ND the donor concentration, l the total length of the diode, AI0 a small current superimposed as a step function to the stationary value I0, A the cross-sectional area of the diode, Itot the total current flowing in the circuit and It(t) = ( e A / l ) f : n v d z the conduction current. The h y d r o d y n a m i c approach contains five parameters which depends only on e(z,t). They are: m - 1, the velocity relaxation rate v., the energy relaxation rate u~, the variance of velocity fluctuations

V. Gruzhinskis et al. / Submicron n + nn + diodes f o r microwave generators

200

n+= 10"em -~

n +

In P / T=300 K /

419

:2

1~o

C9 (D

n=l+lO0 / x lO~Sem-7

i00

Or' r~.

50 i

0.05

0.1

0.1+1.0

; 0,1

0.15 (#m)

Uth I

2

Fig. 1 - Schematic of the diode structure

I

I

i

4 6 8 Applied voltage (V)

I

10

considered as applied to the case of InP

Fig. 2 - Generation-band diagram of the InP diode of Fig. 1 with n = 1017 c m - 3

for the reported ranges of doping concen-

and d = 1 # m . Urn is the threshold volt-

trations and lengths of the active region.

age for generation.

Q. = (gv 2 ) and the covariance of velocity-energy fluctuations Q, = ( g v & ) . Their values are directly obtained from the knowledge of the five static characteristics of the bulk material reported in the first step. Finally, as boundary conditions we have taken those of zero spatial derivative at the source contact and linear spatial derivative at the drain contact.

3. R E S U L T S A N D C O N C L U S I O N S Results for the case of an InP diode, whose structure is schematically shown in Fig. 1, are reported in Figs. 2 to 4. Once the diode structure is fixed, for a given Ud we determine the frequency ranges where the real part of the small-signal impedance is negative, since in these regions there exists the possibility for having microwave generation. By varying Ud, these frequency ranges are collected to construct the generation-band diagram on a frequency-voltage plane. This diagram is shown in Fig. 2 for the case of n = 1017 c m - 3 and an active length d = 1 #m. Then, for a given applied voltage we define the minimum, m a x i m u m and o p t i m u m frequency of generation. These correspond, respectively, to the appearance, disappearance and m a x i m u m negativity of the small-signal impedance. The above quantities are reported in Fig. 3 as a function of the diode active-length. Results show that submicrometer InP diodes can in principle generate in a frequency range from 100 up to about 800 GHz, the m a x i m u m frequency value occurring for d = 0.2 p r o . Figure 4 shows the efficiency of the series resonantcircuit for the case of d = 0.3 p r o , R = 1.5 × 10 l° ~ m 2 and Ua = 3.5 V when the

420

V. G r u z h i n s k i s et al. / S u b m i c r o n n + nn + d i o d e s J b r m i c r o w a v e g e n e r a t o r s

T

......

I

I

. . . . . . . .

•

I

5 -

800

4

/

"~

-

ttydrodynamie approach

-q-

600 o

h~

400 o~ o r~

200 i

OL O.

. . . . . .

,

i

i

_

±

0.2 0.4 0.6 0.8 l. Length of aet, ive region (;zm)

Fig. 3 - Characteristic generation frequencies as a function of diode length for an applied voltage of 2.5 V.

200

300

400

500

600

700

Frequency (GHz)

Fig. 4 - Efficiency of the series resonantcircuit as a function of frequency as obtained for a load surface resistance of 1.5 × 10- 10 fire2 and Ua = 3.5 V. Continuous curve refers to the hydrodynamic approach, dots to Monte Carlo simulations.

frequency is swept by varying L. The agreement found between the hydrodynamic approach and a full Monte Carlo simulation proves the physical reliability of the method here presented which has the advantage to require short computing times and to be easily inserted in CAD modeling. An analysis carried out for the case of GaAs shows similar characteristics but with a general trend of lower generation frequencies.

4. R E F E R E N C E S

1 M. Shur and T.A. Fjeldly Eds., Supercomputer Simulation of Semiconductor Devices" Elsevier Science Publisher, (Amsterdam, 1991). 2 C. Jacoboni and L. Reggiani, Rev. Mod. Phys. 55, 645 (1983).