1. Introduction
An impact avalanche transit time (IMPATT) diode is considered to be one of the most useful electronic power devices used for high frequency communication[1]. It enjoys the flexibility of being realized from any semiconductor material[2]. Both simulation and experimental researches have been undertaken to optimize the device performance of the IMPATT diode using different base materials. It has always been a tough task for any experimental study dealing with any new material. That is why one can find only a few reports about the experimental analysis of the IMPATT diode with base material other than Si and GaAs. Researchers have tried IMPATT diodes with different base material. The performances of wide bandgap semiconductor materials for IMPATT applications have been an interesting research area during the last decade as well as recent studies[1-5]. SiC has gained importance for realization of a high power IMPATT diode[6-10]. Almost all of the reports on the SiC IMPATT diode are based on simulation. The accuracy of the material parameters is very important for the quality and acceptability of the simulation results. These parameters also vary with electric field, doping and temperature. We may also find experimental and simulation reports for these parameters at different electric field and temperature. But mostly we do not find these data over the electric field and at the temperature of our interest. Further, the experimental and simulation results for these parameters may differ significantly leaving us in a dilemma over which data to choose for our simulation. We have to either use the available data or get the data, with some assumption/interpolation, from the available data for our simulation. In some way the simulation studies are compromised due to the non-availability of exact data or fitting equations for all the material parameters. So, we performed a sensitivity analysis to study the influence of a small variation in impact ionization rate data and saturation drift velocity on the performance of 4H-SiC and 6H-SiC based IMPATT diodes. We believe such a sensitive analysis will have a great impact on the simulation of any IMPATT device in future.
2. Design and material parameters
We have considered two different symmetrical DDR IMPATT diode structures based on 4H-SiC and 6H-SiC semiconductors having doping distribution of the form n+npp+ for the present study. The active region widths of the structures are obtained for a design frequency
Wn,p=0.5vsn,spfd, |
where vsn, sp are the saturation drift velocity of electron and hole respectively. The total active region width for 4H-SiC is considered to be 820 nm with 410 nm for each n-and p-region. The active region width for 6H-SiC is considered to be 720 nm with 360 nm for each n-and p-region. The n+ and p+ regions of the diode are heavily doped with each having a doping concentration of
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The most important material parameters for simulation study of IMPATT diodes are the ionization rates
Electron ionization rates:
αn=Anexp[−(Bn/E)m]. |
(1) |
Hole ionization rates:
αp=Apexp[−(Bp/E)m]. |
(2) |
The values of An, p and Bn, p are the constants for electron and holes and their values for wide electric field range in 4H-SiC and 6H-SiC are taken from experimentally reported data[11, 12], where E is the electric field. The electron and hole drift velocity versus electric field characteristics of both 4H-SiC and 6H-SiC have an empirical field dependence given by:
Drift velocity of electron:
vn=vsn[1−exp(−μnE/vsn)]. |
(3) |
Drift velocity of hole:
vp=vsp[1−exp(−μpE/vsp)]. |
(4) |
The saturated drift velocities of electrons
3. Simulation method
To study the sensitive analysis of SiC DDR IMPATT diodes, we have considered the diode of the one dimensional model to be consisting of several small space points with a space step of 1 nm. The sensitive analysis of the diodes is carried out with the simulation method. The simulation method starts from DC analysis. The dc analysis is carried out by solving simultaneously three important device equations namely Poisson's equation, the carrier continuity equation and the space charge equation using a double iterative DC simulation program. The simulation takes into account the contribution from each space point. It thus efficiently determines DC electric field profiles, carrier current profiles, breakdown voltage etc. for the proposed diodes. The results obtained from DC analysis are used for the small signal analysis of the SiC IMPATT diodes using a small signal computer simulation method. The small signal model also takes into account the contribution from each space point. The equations are again solved at each space point and it thus effectively determines the device parameters such as negative conductance
3.1 DC analysis
The Poisson's equation for a p-n junction semiconductor is given by:
∂E(x)∂x=qεs[ND−NA+p(x)−n(x)], |
(5) |
where NA and ND are the accepter and donor doping concentrations respectively.
The carrier continuity equation for electron and hole are given by:
∂n∂t=1q∂Jn(x)∂x+g(x), |
(6) |
and
∂p∂t=1q∂Jp(x)∂x+g(x), |
(7) |
where the avalanche carrier generation rate
g(x)=αn(x)n(x)vn(x)+αp(x)p(x)vp(x). |
(8) |
For static condition Eqs. (6) and (7) can be written as:
g(x)=1q∂Jn(x)∂x=−1q∂Jp(x)∂x. |
(9) |
Now using the relation
∂Jn(x)∂x=αnJn(x)+αpJp(x)=−∂Jp(x)∂x. |
(10) |
Using the expression for total current density as
The pair of equations in Eq. (10) will reduce to the combined carrier continuity equation,
∂P(x)∂x=(αn+αp)+(αn−αp)P(x), |
(11) |
and the space charge equation, which gives the mobile space-charge concentration
q∂(p−n)∂x=J(αnvn+αpvp)−q(αn−αp)(p−n)+K∂E∂x, |
(12) |
where K is a correction factor whose value depends on the nature of the carrier velocity-field characteristics in a semiconductor and it is given by:
K=Jpμpvp(1vsp−1vp)−Jnμnvn(1vsn−1vn). |
(13) |
The boundary conditions for the electric field at the edges of the depletion layer are given by:
3.2 Small signal analysis
An iterative and generalized computer simulation method is developed to study the small-signal behavior. This method has the advantage of its applicability to any diode structures with incorporating the full effect of mobile space charge. The DC results for any structure under particular operating conditions and doping profile obtained through the method described above are used as input for the small-signal analysis. The boundary edges, which are fixed by DC analysis, are taken as starting and end points for small signal analysis. A double iterative simulation over the initial choice of the values of R and X is described in detail and used to solve simultaneously the two second order Eqs. (14) and (15) subject to exact boundary conditions.
They can be written as follows:
D2R+(αn−αp)DR−2rωˉv2DX+[ω2ˉv2−H(x)]R−2¯αω¯vX=2¯α¯v∈, |
(14) |
D2X+(αn−αp)DX+2rω¯vDR+[ω2¯v2−H(x)]X+2¯αω¯vR=−ω¯v2∈, |
(15) |
where
Thus this equation gives rise to the right-hand side boundary condition as:
[iωvp+D]Z=1vp∈,x=xR. |
(16) |
Similarly, noting that at the nn+ interface the current is due to electron only and the current due to hole concentration is negligible, the left-hand side boundary condition can be obtained as:
[−iωvn+D]Z=−1vn∈,x=xL. |
(17) |
A solution of Eqs. (14) and (15) along with boundary conditions Eqs. (16) and (17) would give the diode characteristics.
Finally, Iteration over the values of R and X at
ZR=∫w0R(x)dx, |
ZX=∫w0X(x)dx, |
The maximum RF power output
PRF=V2RF(−Go)A/2, |
(18) |
where VRF is the amplitude of the RF swing and is taken as
3.3 Noise analysis
Noise in impact avalanche transit time (IMPATT) diodes originates from the random nature of the impact ionization process. This brings fluctuations in DC values of current and field giving rise to small-signal components to DC values even when no small-signal voltage is applied to the diode. The avalanche process is composed of a noiseless generation rate g and a noise generation rate
dJc=2qdfdJcA. |
(19) |
An amount of mean square noise current < d
⟨dj2c⟩=2qdfdJcA=2q2dfγ(x′)Adx. |
The noise source
vt(x′)=∫w0e(x,x′)dx, |
(20) |
where the integration has been taken over the entire width of the depletion layer. So the transfer impedance is defined as,
Zt(x′)=vt(x′)dIc(x′), |
(21) |
where
⟨v2⟩=2q2dfA∫|zt(x)|2γ(x′)dx′, |
(22) |
from which the noise measure (NM) can be obtained from the formula as given by:
NM=A⟨v2⟩/df4kT(−ZR), |
(23) |
where k is the Boltzmann constant, T is the absolute temperature, and ZR is the diode resistance.
In order to get these values we must first determine the values of
The current density equation may be written as,
J=Jn+Jp+∂∂t(∈Em), |
(24) |
where Em is the electric field associated with the mobile space charge. From Poisson's equation for mobile space charge it can be written as
Jpvp−Jnvn=∈∂Em∂x. |
(25) |
Eliminating Jn from Eqs. (24) and (25) and rearranging we get
p=1q(vn+vp)[J−∈(∂Em∂t−vn∂Em∂x)]. |
(26) |
Similarly, the expression for n can be written as:
n=1q(vn+vp)[J−∈(∂Em∂t+vp∂Em∂x)]. |
(27) |
The combined carrier continuity equation in the presence of noise can be written as:
∂∂t(p−n)=1q∂∂x(Jn−Jp)+2(g+γ), |
(28) |
where the noiseless avalanche generation rate under signal condition is given by:
g=αn(x)vn(x)n(x)+αp(x)vp(x)p(x). |
(29) |
Substituting Eqs. (26), (27) and (29) in Eq. (28) and simplifying we get a second order differential equation for the DC electric field Em of the form:
[D2−k2+(αn−αp+2rnk]D+2ˉαkEm=(2ˉα−k)J+2rpγ(x′)ˉv∈, |
(30) |
where
[D2−k2+(αn−αp+2rnk)D+2¯αk−H]e=(2¯α−k)j+2qrpγ(x′)¯v∈. |
(31) |
Here
γ(x′)=αn(x′)vn(x′)n(x′)+αp(x′)vp(x′)p(x′). |
(32) |
With
D2eR(x,x′)+[αn(x)−αp(x)]DeR(x,x′)−2rnω¯vDei(x,x′)+[ω2¯v2−H]eR(x,x′)−2¯α(x)ω¯vei(x,x′)=2rpqγ(x)¯v(E). |
(33) |
D2ei(x,x′)+[αn(x)−αp(x)]Dei(x,x′)+2rnω¯vDeR(x,x′)+[ω2¯v2−H]ei(x,x′)+2¯α(x)ω¯ver(x,x′)=0. |
(34) |
The boundary conditions are obtained by assuming that the current in the n-side edge is purely due to electrons and that the p-side edge is due to holes. So that
{[−iωvn+D]e(x,x′)=0,x=xL,[iωvp+D]e(x,x′)=0,x=xR. |
(35) |
The noise electric field is obtained by solving Eqs. (33) and (34) with the boundary conditions Eq. (35) by the computer simulation method.
4. Result and discussions
In this paper we have pictured out the influence of small variation (5% and 10%) in the ionization rate data and drift velocity data on the performance of 4H-SiC and 6H-SiC IMPATT diodes. The results obtained by the computer simulation of IMPATT diode, with 4H-SiC and 6H-SiC as base materials, are presented in this section and the simulation values of microwave characteristics of the respective IMPATT diodes without variation of material data are presented in Table 2. The breakdown voltage
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Fig. 3 depicts the variation of efficiency of 4H and 6H SiC IMPATT on the subsequent variation of material data of both 4H-and 6H-SiC materials. The variation in efficiency informed that material data is one of the most important parameters of IMPATT diodes one must take care of. It is noticed that when both the electron and hole ionization rates increase from 0 to 10%, up to 5% increasing the material data value the efficiency of 4H-SiC IMPATT increases and then after that it falls down with increasing in ionization rate. It is interesting to notice that a reverse trend of 4H-SiC is found in 6H-SiC IMPATT with increasing the ionization rate as shown in the figure. But when the ionization rate of electron and hole decrease in the case of the 4H-SiC IMPATT its efficiency falls down whereas in the 6H-SiC IMPATT it first increases and then falls down. As far as the saturation drift velocity is concerned, when both the drift velocity of electron and hole increase in both the diodes a reverse effect of efficiency is found as shown in Fig. 3. For the 4H-SiC IMPATT 0% to 5% variation of saturation drift velocity of both the electron and hole causes the efficiency to decrease whereas in the 6H-SiC IMPATT it increases. But from 5% to 10% variation of the material data, a reverse trend is found in both diodes. Negative conductance is one of the most important parameters of the IMPATT diode. Figs. 4(a) and 4(b) represent the effect of electron and hole ionization rate coefficients and saturation drift velocity on the negative conductance. It is noticed that with decreasing in material data value at 5% and 10% the negative conductance in 4H-SiC IMPATT diodes decreases whereas it increases in the 6H-SiC IMPATT Overall it is noticed that the negative conductance of both diodes increases from low value to high value of material data. However, there have been significant changes of negative conductance with the variation in ionization rate data as compared to that of saturation drift velocity data.
A maximum of 6% increase in the value of negative conductance can be seen for a 10% increase in electron and hole ionization rate in 6H-SiC whereas it is 5% with a 5% increase in ionization rate of electron and hole in 4H-SiC. However, the expected output power variation is not noticed suitably as stated by the variation of the material data. This confirms the need of an accurate ionization rate and drift saturation velocity data for simulation of IMPATT diodes. This type of sensitivity analysis should be done if the accurate material data is not available. The same can also be done when we find varying reports of material data. But as far as the breakdown voltage is concerned its value decreases with an increase in material data.
Fig. 5 shows the influence of material data on the peak operating frequency and it is noticed that the peak operating frequency
The noise analysis of the diode reveals the influence of change in ionization rate and drift saturation velocity data on the value of mean square noise voltage per bandwidth (MSNVPBW) and noise measure of the 4H-SiC and 6H-SiC IMPATT diodes at 220 GHz. These variation values have been presented in Fig. 6. From Fig. 6 it has been noticed that the noise measure increases with increase in the saturation drift velocity of electron and hole vsn and vsp respectively, whereas there is no significant change in the noise measure with the change in ionization rate coefficients of both electron and hole α and
5. Conclusion
The variation in ionization rate and drift velocity affects the device efficiency as well as breakdown voltage. Further the influence of these variations is also clearly visible on the device properties like negative conductance and power density. However ionization rate change has very small impact on the noise measure, whereas significant noise measure (NM) is observed due to variation of drift velocity. It thus compels us to conclude that one has to be very careful while using material data for predicting device performance through simulation