Models for IMPATT Diode Analysis and Optimization

ALEXANDER ZEMLIAK

Department of Physics and Mathematics

Autonomous University of Puebla

Av. San Claudio s/n, Puebla, 72570

MEXICO

Abstract: - Some of nonlinear models for high-power pulsed IMPATT diode simulation and analysis is

presented. These models are suitable for the analysis of the different operational modes of the oscillator. Its

take into account the main electric and thermal phenomena in the semiconductor structure and the functional

dependence of the equation coefficients on the electrical field and temperature. The first model is a precise

one, which describes all important electrical phenomena on the basis of the continuity equations and Poisson

equation and it is correct until 300 GHz. The second approximate mathematical model suitable for the analysis

of IMPATT diode stationary operation oscillator and for optimization of internal structure of the diode. This

model is based on the continuity equation system solution by reducing the boundary problem for the

differential partial equations to a system of the ordinary differential equations. The temperature distribution in

the semiconductor structure is obtained using the special thermal model of the IMPATT diode, which is based

on the numerical solution of the non-linear thermal conductivity equation. The described models can be

applied for analysis, optimization and practical design of pulsed-mode millimetric IMPATT diodes.

Its can be also utilized for diode thermal regime estimation, for the proper selection of feed-pulse shape and

amplitude, and for the development of the different type of complex doping-profile high-power pulsed

millimetric IMPATT diodes with improved characteristics.

Key-Words: - Semiconductor devices, microwave, numerical methods, thermal model, analysis, optimization.

Received: September 14, 2021. Revised: April 18, 2022. Accepted: May 16, 2022. Published: June 30, 2022.

1 Introduction

IMPATT (IMPact Avalanche ionization and Transit

Time) diodes are principal active elements for

use in millimetric pulsed-mode generators.

Semiconductor structures suitable for fabrication of

continuos-mode IMPATT diodes have been well

known for a long time [1-2]. They have been

utilized successfully in many applications in

microwave engineering. The possibilities of using

the same structures for pulsed-mode microwave

generators are very interesting because the pulsed-

mode IMPATT-diode generators can successfully

operate at high current densities without

deterioration of reliability. The cross section of the

pulsed-mode IMPATT diode may be larger than

that of the continuous-mode diodes. Therefore, the

pulsed-mode oscillator can provide a larger power

output. Considering, that the increase of the output

power of millimetric generators is one of the main

problems of microwave electronics; it is important

to optimize the diode's active layer to obtain the

generator maximum power output.

One of the main singularities in the operation of

high-power IMPATT-diode pulsed-mode generator,

is the large variation of the diode admittance during

the pulse. This variation is significantly during each

current pulse due to the temperature changing of

the diode's semiconductor structure.

Therefore, diffusion coefficients, ionization rates

and charge mobility experience large variations

during the pulse. These changes strongly affect the

amplitude and phase of the first harmonic of the

diode's avalanche current. Therefore, the admittance

value also changes. This results in the instability of

generator's output power and frequency within each

generated microwave pulse.

Pulsed-mode IMPATT diodes that are utilized in

microwave electronics are, most frequently, the

single-drift and double-drift structures similar to

continuous-mode ones [1-5]. The typical diode

structure is shown on the Fig. 1 by curve 1, where

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N is the concentration of donors and acceptors, l is

the length of diode active layer.

Fig. 1 Doping profile for two types of IMPATT diodes:

1 - constant doping profile; 2 - quasi-Read-type

doping profile.

In this type of diodes, the electrical field is strongly

distorted when the avalanche current density is

sufficiently high. This large space charge density is

one of the main reasons for the sharp electrical field

gradient along the charge drift path. Because of this

field gradient, the space charge avalanche ruins

itself and consequently the optimum phase relations

degrade between microwave potential and current.

This factor is especially important when the

IMPATT diode is fed at the maximum current

density, which is exactly the case at the pulsed-

mode operation.

The idea to use a complex doping profile

semiconductor structure for microwave diode was

originally proposed in the first analysis of IMPATT

diode by Read [6]. This proposed ideal structure has

never been realized till now. However, a modern

semiconductor technology provides new

possibilities for the fabrication of sub micron

semiconductor structures with complex doping

profiles. This stimulates the search for IMPATT-

diode special structure's optimization for pulsed-

mode operation.

The proposed new type of IMPATT diode

doping profile is shown on the Fig. 1 by the curve 2.

This type of semiconductor structure can be named

as quasi-Read-type structure. This type of doping

profile provides a concentration of electrical field

within the p-n junction. This measure helps to

decrease the destruction of the avalanche space

charge and therefore permits to improve the phase

stability between the diode current and voltage.

Historically, many analytical and numerical

models have been developed for the various

operational modes of IMPATT diodes [1, 7-20].

However, they are not adequate for very high

current density values and different temperature

distributions inside the structure, which is exactly

the case for the pulsed-mode IMPATT-diode

oscillator. For this reason, we have developed a new

complex numerical model of the IMPATT diode

that is composed of the advanced thermal model and

the modified local-field model. The thermal model

provides the exact theoretical temperature

distribution along the diode active region. The

local-field electrical model calculates the functional

dependence of equation coefficients from electric

field and temperature, and using all these data

finally derives the IMPATT-diode dynamic

characteristics.

2 Numerical Models

Three numerical models are described in this

section. Two different electrical models useful for

the precise analysis and internal structure

optimization describe all important phenomena into

the semiconductor structure. The thermal model

describes the temperature distribution by means of

the thermal conductivity equation solution.

2.1 Precise Numerical Model

The numerical model developed for the analysis of

various generator operation modes. This model is

based on the system of continuity equations for

semiconductor structure:

(

)

(

)

( ) ( )

( ) ( ) ( ) ( )

∂

∂α α

∂

∂α α

n x t

t

J x t

xJ x t J x t

p x t

t

J x t

xJ x t J x t

n

n n p p

p

n n p p

, , , ,

= + +

= − + +

(1)

( ) ( )

(

)

( ) ( ) ( )

J x t n x t V D n x t

x

J x t p x t V D p x t

x

n n n

p p p

, , ,

= +

= −

∂

where n, p are the concentrations of electrons and

holes;

J

n p

,

are the current densities;

α

n p

,

are

the ionization coefficients; V V

n p

, are the drift

velocities;

D

n p

,

are the diffusion coefficients.

Ionization coefficients, drift velocities and diffusion

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coefficients are functions of two arguments; the

spaces coordinate x and the times coordinate t.

This model differs from the previously proposed

models described in [13-25] in that in this model the

ionization coefficients are functions of the electric

field and temperature at all points of the

semiconductor structure. The dependence of these

coefficients on temperature can be approximated

using the approach described in [26]:

( )

(

)

[

]

( )

[ ]

( )

[ ]

α

n

T E

TE

T E

E T

e E

,

. .

. . .

. .

. . /

=

⋅ < ⋅

⋅ ⋅ < < ⋅

⋅ > ⋅











− ⋅ + ⋅ ⋅

2 6 10 2 4 10

6 2 10 2 4 10 5 3 10

5 0 10 5 3 10

61 410 1 310 5

51 0510 1 310 5 5

50 9610 1 310 5

6 3

( )

(

)

[

]

( )

[ ]

α

p

T E

TE

E T

e E

,

. . .

. .

. . /

=

⋅ ⋅ < < ⋅

⋅ > ⋅











− ⋅ + ⋅ ⋅

2 0 10 2 0 10 5 3 10

5 6 10 5 3 10

61 9510 1 110 5 5

51 29610 1 110 5

6 3

The temperature T is expressed in o

C

and

electrical field E is expressed in V/cm.

The boundary conditions for the system (1) can

be written as follows:

(

)

(

)

(

)

(

)

( )

n t N p l t N l

J l t J J t J

D A

n ns p ps

0 0

0

0 0

0

, ; , ;

, ; , .

= =

(2)

where

J

ns ps

,

are electron current and hole current

for inversely biased p-n junction;

(

)

(

)

N N l

D A

00

,

are concentrations of donors and acceptors at two

space points x = 0 and x =

l

0

, where

l

0

is the

length of the active layer of semiconductor

structure.

Electrical field distribution into semiconductor

structure can be obtained from Poisson equation. As

electron and hole concentrations are functions of

the time, therefore, this equation is time dependent

too and time is the equation parameter. Poisson

equation for this problem has the following form:

(

)

(

)

( ) ( )

∂

E x t

x

U x t

xN x N x p x t n x t

D A

, , , ,=− = − + −

2

2 (3)

where

(

)

(

)

N x N x

D A

, are the concentrations of the

donors and acceptors accordingly, U(x,t) is the

potential, E(x,t) is the electric field. The boundary

conditions for this equation are follows:

( )

U

t U l t U U sin mt

m m

m

M

000 0

1

, ; ,= = + +

=

∑

ω ϕ

(4)

where

U

0

is the DC voltage on diode contacts,

U

m

is the amplitude of the harmonic number m,

ω

is

the fundamental frequency,

ϕ

m

is the phase of

harmonic number m, M is the number of

harmonics.

Equations (1)-(4) adequately describe the

physical processes in the IMPATT diode in a wide

frequency band. However, numerical solution of

this system is very difficult because of the sharp

dependence of equation coefficients on electric

field. The evident numerical schemes have poor

stability and require a lot of computing time for the

good calculation accuracy obtaining. It is more

advantageous to use non-evident numerical scheme,

that has a significant property of absolute stability.

The computational efficiency and accuracy are

improved by applying space and time coordinates

symmetric approximation.

After the approximation of the functions and its

differentials, the system (1) is transformed to the

non-evident modified Crank-Nicholson numerical

scheme. This modification consists of two

numerical systems, each of them having the three-

diagonal matrix. These systems have the following

form:

(

)

(

)

(

)

( ) ( )

( )

− − + + − + =

+ − + + − +

⋅ ⋅ + ⋅ ⋅ − +

⋅ ⋅ − ⋅ ⋅ −

−

+ +

+

− + + −

+ −

a b n a n a b n

a n a n a n b n n

V n r D n n

V p r D p p

n n i

k

n i

k

n n i

k

n i

k

n i

k

n i

k

n i

k

i

k

n n i

k

n i

k

i

k

p p i

k

p i

k

i

k

1

1 1

1

1 1 1 1

1 1

1 2

α τ

(5)

(

)

(

)

(

)

( )

− + + + − − =

+ − + − − +

⋅ ⋅ − ⋅ ⋅ − +

⋅ ⋅ + ⋅ ⋅ −

−

+ +

+

− + + −

+ −

a b p a p a b p

a p a p a p b p p

V p r D p p

V n r D n n

p p i

k

p i

k

p p i

k

p i

k

p i

k

p i

k

p i

k

i

k

p p i

k

p i

k

i

k

n n i

k

n i

k

i

k

1

1 1

1

1 1 1 1

1 1

1 2

α τ

i = 1, 2, ...

I

1

−

; k = 0, 1, 2, ...

∞

where aD

h

n p

,

=

τ

2

2, b

V

h

n p

,

=

τ

4

, r

h

=

τ

2

,

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i is the space coordinate node number, k is the

time coordinate node number, h is the space step,

τ

is the time step,

I

1

is the space coordinate node

number.

The approximation of the Poisson equation is

performed using the ordinary finite difference

scheme at every time step k:

(

)

U U U h N N p n

i

k

i

k

i

k

D i A i i

k

i

k

− +

− + = − + −

1 1

2

2 (6)

The numerical algorithm for the IMPATT diode

characteristics calculation consists of the following

stages: 1) The voltage is calculated at the diode

contacts for every time step. 2) The initial charge

distribution is calculated. 3) The electric potential is

calculated at every space point from Poisson

equation by the factorization method [27]. The

electrical field distribution along the diode active

layer is calculated. 4) The ionization coefficients

and drift parameters are calculated in numerical net

nodes for the current time step. 5) The system (5) is

solved by matrix factorization method and electron

and hole concentration distribution is calculated for

the new time step. After this, the calculation cycle

is repeated for all time steps from the beginning to

the step 3. This process is continued until the

convergence is achieved. The current of the

external electronic circuit is determined. Then all

harmonics of external current are calculated by the

Fourier transformation

(

)

(

)

J J J j

m m m0;exp=

φ

;

the admittance is calculated for harmonic number

m

(

)

mmm UJ

Y/= and the power characteristics

for all harmonics can be calculated by the

following formulas:

(

)

(

)

00

2/;Re5.0 UJPUYP mmmmm =−=

η

2.2 Approximate Numerical Model

Other numerical model is more suitable for the

previous analysis and for the diode internal

structure optimization. This model can reduce the

total computer time of the structure optimization

process.

The numerical method for the solution of the

system (1) is based on the classical Fourier series

utilization. This approach transforms of the

boundary problem for the system of differential

partial equations to an ordinary differential equation

system. The model describes the physical processes

in IMPATT diode by the stationary-operation mode

and provides the possibilities to reduce the demands

for a computer time that is necessary for the output

parameters calculation.

Let us assume that all functions of the system (1)

can be presented in a form of Fourier series:

( )

n x t n x jm t

m

, ex p= ⋅

= − ∞

∞

∑

ω

;

( )

p x t p x jm t

m

, exp= ⋅

= −∞

∞

∑

ω

;

( )

J x t I x jm t

n n m

m

, exp= ⋅

= −∞

∞

∑

ω

;

( )

(

)

( )

J x t I x jm t

p p m

m

, exp= ⋅

= −∞

∞

∑

ω

;

( )

α

α ω

n n m

m

x t x jm t, exp= ⋅

=−∞

∞

∑;

( )

(

)

( )

α

α ω

p p m

m

x t x jm t, exp= ⋅

=−∞

∞

∑;

( )

V

x t v x jm t

n n m

m

, exp= ⋅

=−∞

∞

∑

ω

;

( )

(

)

( )

V

x t v x jm t

p p m

m

, exp= ⋅

= −∞

∞

∑

ω

;

( )

D x t d x jm t

n n m

m

, exp= ⋅

=−∞

∞

∑

ω

;

( )

(

)

( )

D x t d x jm t

p p m

m

, exp= ⋅

=−∞

∞

∑

ω

.

In such a case the principal system (1) can be

reduced to a system of the ordinary differential

equations for the complex charge density and

for the current amplitudes:

(

)

( )

∑

∞

−∞

=

−

∞

−∞=

−





















−













=



















+













−=

kk

p

km

p

km

k

p

m

kk

n

km

n

km

k

n

nm

d

I

p

d

v

dx

dp

d

I

n

d

v

dx

dn

(7)

(

)

( ) ( )

{ }

( )

( ) ( )

{

}

∑

∞

−∞= −

−

∞

−∞= −

−

++−=

+−=

k

km

p

k

p

km

n

k

nm

m

p

k

km

p

k

p

km

n

k

nm

m

n

IIpjm

dx

Id

IInjm

dx

Id

ααω

m

=

∞

1

2

,

.

,

where

(

)

(

)

α α

nmp

m

, are the electron and hole

ionization coefficient amplitudes,

(

)

(

)

v v

nmp

m

,

are the electron and hole velocity amplitudes,

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(

)

(

)

d d

nmp

m

, are the electron and hole diffusion

coefficient amplitudes,

n

p

m m

,

are the electron and

hole concentration amplitudes,

(

)

(

)

I I

nmp

m

, are the

electron and hole current amplitudes.

A number of harmonics m in these series can be

reduced down to the number M, which defines

the accuracy of the solution and necessary computer

time. The system (7) can be presented in matrix

form as:

Y

A Y

'

=

(8)

The charge diffusion and sharp dependence of

the ionization coefficients on the electrical field

determine the great module of eigenvalues of the

matrix A. For this case, a shooting method, which

reduces a boundary problem to Cauchy problem, is

not suitable because coordinate basis degenerates in

the solution process and therefore is not stable. The

boundary problem (8) is solved on the basis of the

functional matrix correlation [28] :

(

)

(

)

(

)

B x Y x G x

t

=

(9)

where

B

t is the factorization matrix; G is the

boundary condition vector. The unknown matrixes

of equation (9) are satisfied in the

following differential equation system:

B

A

B

t

'

+

=

0

(10)

G

'

=

0

The fundamental matrix

F

is used to obtain the

process stability of the integration of equations (10).

This matrix is determined as

( )

(

)

{

}

F x A x h

t

k k

=exp , where k

h is the space

step. Transition to the next coordinate node is made

using the term

(

)

(

)

(

)

B x h F x B x

k k k k

+ = . The

degradation of coordinate basis B can be overcome

using the Gram-Schmidt ortogonalization procedure

for equation (9) on each integration step.

The algorithm for the analysis of IMPATT diode

includes the following steps: 1) the initial charge

distribution in the diode is calculated; 2) the electric

field harmonics are determined from the Poisson

equation; 3) ionization and drift parameters are

determined from the Fourier analysis, and the

matrix of the system of equations on the coordinate

net is formed; 4) the boundary problem is solved for

the system of continuity equations. Charge and

current amplitudes are determined. The harmonics

of the external circuit current are calculated. After

this, the calculation cycle is repeated from the

beginning to point 2) until the external current is

determined with sufficient convergence. Then all

output parameters of the IMPATT diode are

determined.

The main advantage of this harmonic method is

the reducing the total computer time for the

calculation of stationary mode of the IMPATT

diode. In Fig. 2 are shown computer time Tc in

relative units and relative error Er as the functions

of the harmonic number M.

Fig. 2 Computer time Tc in relative units and

relative error Er as the functions of the harmonic

number M.

These data are corresponded to the nonlinear modes

with average level of the non-linearity. For this case

we determine error as the relative difference of the

diode admittance value that we obtain by this

harmonic method and by more precise numerical

method of the section 2.1. It is clear that the

harmonic number M more than 12-15 is sufficient

to obtain a good accuracy of the diode parameters.

At the same time we have a significantly reducing

of the total computer time. Computer time for one

probe of diode analysis is the principal

characteristic of the optimization procedure. That is

the main reason why this approximate model is

elaborated. For example the total computer time for

the diode analysis by precise numerical model is

corresponded to the number of harmonic M = 40.

2.3 Thermal Model

The data of the temperature distribution that is

necessary for the calculation of the local-field

electrical model may be obtained from IMPATT

diode thermal model. This model determines the

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temperature distribution in all points of the diodes

active layer, for any given moment in time.

The IMPATT diode thermal model is based on

the numerical solution of the non-linear thermal

conductivity equation for silicon crystal, contact

planes and heat sinks. It determines the

instantaneous semiconductor structure temperature

at any point within the device for any "long" time

moment

′

t

. The thermal equation is solved in the

region that is shown on the Fig.3.

Fig. 3 The schematic diode construction with heat

sink.

R

d

- the diode radius,

R

hs

- the heat sink

radius.

The thermal equation has the following form:

( )

∂

∂ ρ ρ

T

t

k

CTCQ x t T

′= + ′

∆1, , (11)

where

′

t

is the time coordinate (this time scale

differs from the scale in the system (1) ), r is the

radial coordinate, x is the longitudinal coordinate, T

is the Kelvin temperature;

ρ

is the material density,

C is the specific thermocapacity, k is the

thermoconductivity coefficient, and

Q

x

t

T

(

,

)

′

is

the internal heat source that, in the general case, has

a dependency on the electrical field, current density,

and temperature,

∆

is the two-dimensional

Laplace operator and for the cylindrical coordinate

system has a form:

∆TT

x r rrT

r

= + ⋅













∂

2

1. The equation

(11) is solved within a volume that includes the

silicon crystal; the gold contact plane deposited on

the crystal; an integrated thermal contact and the

semi-infinite copper heat-sink. The boundary

conditions for the system (11) are follows:

∂

T

r=0

on the vertical axis of symmetry,

( )

∂

∂λ θ

T

rT= − − on

all vertical boundaries facing the air,

( )

∂

∂λ θ

T

xT= − −

on all horizontal boundaries facing the air,

(

)

∂

T

x

q t

k

= − ′ on the internal boundary with a semi-

infinite copper heat sink. The variable

λ

is the heat

transmission coefficient on the metal-air boundary,

θ

is the air temperature,

(

)

q t

′

is the thermal flux

entering semi-infinite copper heat sink.

The principal difference between the equation

(11) and the system (1) is that: the function T in

(11) depends on two spaces coordinates x and r. On

contrary, all functions of the system (1) depend only

on one space coordinate x. The dependence of all

functions of system (1) on r can be neglected,

because of approximations which result in

negligible error. However, the same dependence can

not neglect for equation (11), because it

corresponds really to the two-dimensional case (Fig.

3). We need to determine the functional

dependency of the internal heat source

Q

x

t

T

(

,

)

′

on the temperature to elaborate the IMPATT-diode

thermal model. This model may be simplified

significantly by the following important

approximations: the role of some metal layers (e.g.,

chromium, gold, palladium) in the diode thermal

balance and the influence of gold contact wire and

of ceramic housing of IMPATT-diode crystal may

be neglected. Also, the heat exchange between

diode elements and the atmosphere may be

neglected. These simplifications do not seriously

affect the accuracy of the model. The internal heat

source is defined for all points within the model

volume as follows:

∫′′

=

′

π

ϕϕϕ

π

2

0

),,,(),,(

2

1

),,( dTtxETtJTtxQ (12)

where

ϕ

ω

=

t

,

J

t

T

(

,

)

ϕ

′

is the instantaneous

IMPATT diode structure current density value,

E

z

t

T

(

,

)

ϕ

′

is the electric field intensity in the

point x at the time

′

t

;

This model is essentially different from the

model described in [29] because the heat source is

described now as the function of the electric field

intensity inside the diode structure (12). This

improvement is especially important for increasing

the accuracy of the temperature distribution

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calculation of the active layer of the IMPATT

diode.

The numerical solution of equation (11) is

performed by the finite difference method. Equation

(11) is solved by the alternating direction iteration

method for each coordinate direction. The second

order of the numerical approximation scheme is

used more frequently in this case. The alternating

direction implicit method can be expressed in

compact form as:

T T k

CT T CQ

ij

s

i j

s

ij

s

ij

s

j

s

+

−= +









+

1

2

1

2

1

τ ρ ρ

Λ Λ

(13)

T T k

CT T CQ

ij

s

i j

s

ij

s

ij

s

j

s

++

++ +

−= +









+

1

2

1

2

112

1

τ ρ ρ

Λ Λ

i = 1,2, ...

I

2

1

−

; j = 1,2,...

J

−

1

; s = 0,1,2,...

∞

;

where i, j are the space coordinate numbers, s is

the time coordinate number,

Λ

1

is the partial

numerical Laplace operator on the direction r ,

Λ

2

is the partial numerical Laplace operator on the

direction x. Two of these operators are

defined in the standard five-points numerical

pattern:

2

1

,1,,

1

,1,1

1

2

1

h

TTT

h

TT

ih

Tjijijijiji

ij

−+−+

+

−

+

−

=Λ ,

2

1,

,1,

2

h

TTT

Tjijiji

ij

−+

+

−

=Λ . The numerical scheme

(13) has the second approximation order only. In

this case, it is necessary to develop the numerical

net with a large number of cells to obtain sufficient

accuracy. That is the reason why the total computer

time that is necessary for the solution of the

optimization problem is too great. In this work, we

propose the other type of thermal equation

numerical approximation scheme for the

acceleration of the thermal equation solution and for

the reduction of the computer analysis time. The

total analytic Laplace operator

T

∆

can be

approximated with the numerical Laplace operator

ij

TΛ as:

ijij T

hh

T











ΛΛ

+

+Λ+Λ=Λ 21

2

1

21 12

(14)

In that case, we can approximate the right part of

equation (11) by the following numerical formula:

jij Q

h

E

C

T

C

k











Λ

++Λ 2

2

12

1

ρρ

(15)

where E is the identity operator. The operator

Λ

is

defined in the nine-point numerical pattern. The

approximation (15) is more complicated, but it has

the fourth approximation order. In such a case, we

can use the numerical net that is significantly more

thin to obtain accuracy that is equal to the scheme

(13) described above. For the solution of the

principal equation (11) by approximations (14)-

(15), we used one modification of the Peaceman-

Rachford numerical scheme that had been

developed by [30]:

s

j

s

ij

s

ij

QEbTbkE

TbkE

)())((

))((

2222

2

1

11

Λ+⋅⋅+Λ+⋅⋅+=

Λ−⋅⋅− +

χτχτ

χτ

(16)

2

1

22

2

1

22

)())((

))((

++

+

Λ+⋅⋅+Λ+⋅⋅+=

Λ−⋅⋅−

s

j

s

ij

s

ij

QEbTbkE

TbkE

χτχτ

χτ

where )/1( Cb

ρ

=

, )12/( 2

2,12,1 h=

χ

. We solve the

system (16) by the tridiagonal algorithm for radial

and longitudinal directions. This numerical scheme

provides a significant gain of computer time in

comparison with the scheme (13).

The group of the models presented in sections

2.1-2.3 serves as a basis for the precise and

complete analysis of the IMPATT diodes with the

different doping profiles for the various operation

modes.

3 Numerical Results

The models described above have been utilized for

the investigation of temperature distribution in

pulsed mode IMPATT diode having different

doping profiles. Also, the diode admittance

characteristics have been analyzed. This analysis

has been performed for two types of diode

structures: for the diode having a traditional

constant doping profile, and for the diode having

the new special complex doping profile named the

quasi-Read type structure. Both of these structures

are made of silicon. The first structure has the

doping value N cm

0

17 3

165 10

=

⋅

−

. for active layer.

The n region length is 0.4

µ

m

; the p region length

is 0.36

µ

m

. The second structure has two

levels of the active layer doping profile:

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317317100.2;102.1 −− ⋅=⋅= cmNcmN maxmin and the n,

n+, p, p+ region's lengths are 0.20

µ

m

, 0.18

µ

m

,

0.18

µ

m

, 0.16

µ

m

respectively. The diode

numerical simulation has been performed for the

following operational parameters: electrical current

pulse has a square form, pulse duration

τ

=100

nsec, period T=10

µ

sec

. The calculated

temperature distribution along the diode active

layer obtained for pulse current density of 100

KA cm/2 is shown on a Fig. 4 by the solid line

for an IMPATT-diode having an even doping

profile. The results obtained for the quasi-Read-type

structure are also presented in Fig. 4 by the dashed

line.

Fig. 4 The temperature distribution.

The data obtained for the diode having complex

doping profile demonstrate that the inner part of this

diode is hotter than that of the constant doping

profile diodes. This occurs because in the complex

profile diode electric field intensity maximum is

located further from the diode contacts than in the

constant doping profile diode. It means that the

heat source is located further from the contact

regions too and for a such case the complex-

doping-profile structure thermal flow dissipates

slower than in the first structure. This also explains

the existence of a larger temperature gradient along

the active layer in the complex doping profile diode.

For the structures studied, the temperature

difference between the p-n junction and the

contacts has been 18,8°C for the complex profile

diode and 14,4°C for even doping profile one.

Data on the diode active layer temperature,

obtained for all time step from 0 to 100 nsec has

been used for the calculation of the diode dynamic

admittance characteristics employing the nonlinear

electrical model. Several examples of the calculated

admittance characteristics are shown in Fig. 5.

Fig. 5. Admittance frequency characteristics

These characteristics obtained for the same two

types of IMPATT diode doping profiles (solid lines

for the even doping profile structure and dashed

lines for the quasi-Read structure), and for the same

operational modes as described previously. These

diagrams provide complete information on diode

admittance variation with time, during the feed

pulse. During the first 15 nsec the diode admittance

varies very significantly. This variation is due to the

strong temperature dependence of the physical

parameters of silicon, in the temperature range of

100-170 °C. During the initial part of the feed

pulse, the instantaneous active layer temperature

falls exactly within this temperature range. During

the next 85 nsec, the diode admittance has a stable

value. It is obvious, that it is impossible to obtain

adequate conditions for a stable microwave

frequency during the initial part of the current feed

pulse, if only special frequency stabilization

methods are not utilized. During this initial period

of the feed pulse, it is possible to obtain some

acceptable frequency stability only by utilizing a

special external passive circuit, or by the

synchronization of the signal. During the period

from 20 to 100 nsec, the imaginary part of diode

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admittance Im(Y) is changing more rapidly than the

real part Re(Y). Relative variations of Im(Y) are

around 50% of its average value. However,

variations of Re(Y) are only within 5% of the

average value. This demonstrates that the main

cause of the generator's signal instability is the

violation of reactive energy balance during the feed

pulse. Concurrently, the active energy balance is

conserved to a large degree of accuracy.

In order to provide some means for improving

the frequency and amplitude stability of the pulsed

IMPATT diode generator, it may be of interest to

investigate the influence of the feed current pulse

shape on phase relationship between diode current

and voltage. Using the present model, the optimum

shape of the feed pulse, and the initial and the final

feed current density values can be determined for

the particular case as functions of the diode

material, doping profile, diode structure diameter,

etc. However, the comparison of the admittance

characteristics for the two types of IMPATT diodes

leads to some important conclusions. The data on

the diode admittance characteristics presented in the

Fig. 5 show that the variations of admittance value

during the pulse are much less for the second

structure, (dash lines) than for the first (solid lines).

This can be observed in Fig. 5, where the

admittance curves for the quasi-Read structure lie

closer to each other, than the curves displaying the

traditional even-profile diode. This can be

explained by the larger electric field intensity in the

central part of active layer. The larger field intensity

results in more compact and dense n and p

avalanches propagating in the semiconductor

structure and, therefore, in better phase stability

between current and voltage. Therefore, the use of

the quasi-Read structure, secures better frequency

and amplitude stability of the generated

electromagnetic oscillations. This is the principal

and very important advantage of the new quasi-

Read structure in comparison to the even doping

profile structure.

4 Conclusion

The numerical models of the IMPATT diode, which

are presented in this work, have important

advantages when compared to other models. The

electric models together with the thermal model

take into account the temperature distribution in the

semiconductor structure and the dependence of all

principal physical parameters of the semiconductor

structure on temperature and the electrical field.

The other important advantage is the high stability

of the calculation process by means of the non-

evident difference scheme that is used for the

solution of the main system of equations.

The approximate non-linear IMPATT diode

model can be used successfully for the internal

structure optimization. In such a case a great

acceleration of the optimization process can be

obtained.

The proposed thermal and local-field electro-

dynamic models for pulsed mode IMPATT-diode

analysis increases the accuracy of diode internal and

external characteristics calculation. The method

presented here can be applied for practical design

of pulsed-mode millimetric IMPATT diodes. It can

also be utilized for diode thermal regime estimation

and for the selection of feed current-pulse shape and

amplitude. The method is suitable for the design of

IMPATT-diode based oscillators, amplifiers and

mixers. The most promising application of the

models is the development of the pulsed-mode

complex-doping-profile high-power millimetric

IMPATT diodes with improved characteristics.

Comparative analysis of IMPATT diode thermal

and electro-dynamic properties performed for two

types of the different doping profiles shows that

diodes with complex quasi-Read doping profile

have better perspectives for the pulsed feed current

modulation mode. This special semiconductor

structure has better phase correlation between

current and voltage and has the smaller variations in

the diode admittance. Therefore the complex-

doping-profile diodes have improved frequency

stability in pulsed-mode operation compared to the

traditional IMPATT diodes having a constant

doping profile.

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Volume 21, 2022

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