1. Introduction

Impatt (Impact Avalanche Transit Time) device is a class of rf negative resistance power generator which can cover frequencies from X-band to sub-mm-wave band (10-5000 GHz). Any form of junction fabricated from any semiconductor can operate in Impatt mode. Advance research work is being carried out round the globe with the objective of realizing high power at high efficiency and with low noise even at mm- and sub-mm-wave frequencies[1]. With the advancement of device technology and with the motive of furtherance of device rf performance, possible replacement of homo junction of single elemental semiconductor with hetero junction or hetero structure junctions from lattice matched pair of elemental semiconductors is being explored as supported by recent reports[2-6]. Si and Ge are twoconventional group IV semiconductors having tolerable lattice mismatch. Homo junctions from Si and Ge have become a reality in producing Impatt Power since decades. Hetero junctions from Si and Ge for other purposes have already been reported[7].

The authors have studied a new form of junction in the form of n-Si-Ge/p-Ge-Si hetero structure junction when both the p- and n- layers contain two materials. The band gap of Ge is 0.66 eV where as for Si, it is 1.12 eV. The carrier ionization rate causing avalanche breakdown remains high for low BG semiconductors. In fact the carrier ionization rate for Ge is higher by an order of magnitude compared to that for Si at the same magnitude of electric field. The order gap in values of carrier ionization rate between Ge and Si further widens if Ge is placed in high field zone close to junction and Si placed away from junction. With this idea in mind, the authors proposed Si-Ge-Si hetero structure junction of the form n-Si-Ge/p-Ge-Si when Ge impurity bumps are introduced near the junction interface on either sides of conventional Si homo junction leading to above referred hetero structure junction. The high values of electron and hole ionization rate of Ge in the high field zone near junction plane, thins the avalanche zone to less than 10% (3% in X-band) of the depletion zone on either side of the junction. This phenomenon makes this structure creditable as both the efficiency and diode negative conductance would be pushed to sufficiently high values as compared to both Si and Ge homo junctions. The authors have analysed the new diode structure following a sophisticated three tier computer algorithm. The diode has been designed and optimized for atmospheric window frequencies in X-band (15 GHz) and mm-wave band (94 GHz). The results observed are quite interesting indicating this device to be promising candidate for high power generation with high efficiency. The efficiency of this new proposed diode mounts to near optimum value of 29% for the X-band hetero structure diode as against 12-14% for homo junctions together with the diode negative conductance being doubled for the hetero structure. These characteristics would help in pushing the rf power delivery from the diode. Further, since the avalanching process is augmented and ionizing layer thinned down drastically, the avalanche noise from the hetero structure may remain quite low.

2. Computer Simulation Method

The typical general Impatt diode structure is shown in figure 1 for both the frequencies of design. The minority carriers say holes, under the influence of reverse breakdown voltage (field) crosses from n side to p side, passing through the high field zone around the junction. The hole charge concentration grows due to impact avalanche ionization process, thereby the reverse saturation hole current J_ps at n side edge grows to J_p on the p side extreme. Similarly the minority electron moves from p-side to n-side growing from J_ns to J_n. The charge growth up to the extent of 95%, takes place within the avalanche region (x_A). Then the grown charge drifts through drift region (holes in d_p) and (electrons in d_n) with respective saturated carrier drift velocity.

Figure 1. Schematic diagram of typical double drift n-Si-Ge/p-Ge-Si- based Impatt diode p-n Junction at 15.0 and 96.0 GHz

2.1. Analysis under Static Conditions

Two separate realistic self-converging iterative computer simulation programs have been framed for DC analysis and small signal analyses. Material parameters like the ionization rate, drift velocities etc., of all the materials have been taken from the recently reported published literature[8-10]. The diode active layer width is divided into large number of space steps of 1 nm per micron or less. The DC analysis is carried out by solving simultaneously three non-linear device equations namely Poisson’s equation, the carrier continuity equation and the space charge equation using a double iterative field maximum initiated simulation program[7] subject to fulfilment of boundary conditions in electric field and carrier currents[11].The basic equations for DC analysis are,

Poisson’s equation,

(1)

and the current continuity equation is given by,

(2)

where

(3)

Taking the drift velocity ∼ field variation to be represented by,_{n, p} = _{sn, sp[}1-exp (-_n,p E/ _sn,sp)]

Combined space charge equation can be written as,

(4)

Where

K=

is regarded as correction factor and can be evaluated from velocity ~ field characteristics in any semiconductor. The equations (1), (2) and (4), which consider only drift current, can be modified to incorporate diffusion and tunnel currents[12]. The dc analysis gives drift voltage drop (V_D), normalized avalanche zone (x_A/W), breakdown electric field and voltage (E_m & V_B), qualitative value of device efficiency (η) etc.[13]. The edges of the depletion zone are also determined accurately from this analysis. The DC electric field profiles, carrier current profiles, breakdown voltage etc. are obtained from this analysis. We have considered a one-dimensional model with doping distributions of the form n⁺npp⁺. The data obtained from the DC analysis are used as input for the high frequency analysis of the diode. The next phase of the computation has been designed for computation of RF properties of the optimized diodes over a wide band of frequencies. Implicit second order device equations on diode resistance and susceptance are framed based on Gummel-Blue approach[14].

2.2. High Frequency Analysis

The ac model takes into account the contribution of negative resistance from each space point and effectively determines the device performance parameters such as negative conductance (-G), susceptance (B) and negative resistance (-Z_R) of the diode. The variations of these values with frequency are also computed with the help of another double iterative computer program[15]. In the high frequency analysis, it is possible to find out the spatial variation of negative resistance (R) and reactance (X) in the depletion layer which would provide a clear idea regarding intensity of microwave oscillation produced along different region of the diode. The unperturbed diode impedance Z₀ (without considering diffusion current) is separated into its real and imaginary components (i.e. Z₀ = R₀+iX₀) to obtain the second order non-linear implicit differential equation in small signal resistance (R₀) and reactance (X₀) as,

(5)

(6)

With H(x) =

The equations (5) and (6) are solved simultaneously by modified Runge-Kutta method[16]. Finally the small signal diode resistance (R) and reactance (X) due to drift, diffusion and tunnel currents are obtained from the expression including perturbation part,

and

The integrated values of resistance and reactance give Z_R and Z_X using the relation,

and

Using the values of Z_R and Z_X, the diode conductance (G), susceptance (B) and the quality factor (Q) are calculated using the relation

and

The series positive resistance (R_S) of the device, which is one of the important limiting factors responsible for limiting power output and device efficiency, can also is determined. The RF power is estimated from the following[17]:

(7)

where, V_RF can be approximated as V_B/2 for 50% modulation. G_p is the device negative conductance at peak frequency. A is the area of the diode is taken as 10^-9 m² for 94 GHz and 1-5x10^-8m² for 15 GHz as obtained from experimental report. The high frequency analysis also gives the negative resistance distribution, R(x), profile, along the depletion zone which indicates the intensity of RF oscillation from individual space step. The avalanche phase delay (θ_a) and transit phase delay (θ_t) can also be obtained from the R(x) profile. The devices can be subjected to fine structure optimization through variation of diode structural parameters in order to optimize the value of RF negative resistance which can be achieved when the values of θ_a and θ_t are close to π/2 and the peaks of the R(x) profile in the drift zones are located close to the centre of the drift zones in both ‘n’ and ‘p’ regions. The results of DC and microwave performance analyses for n-Si-Ge/p-Ge-Si hetero structure double drift IMPATT diodes (n⁺npp⁺) have been studied and presented in this paper.

3. Computer Algorithm for DC and RF Analysis

Computer solutions become essential for understanding the properties of the devices, as analytical methods are not feasible to provide accurate information regarding the DC and high frequency parameters of these devices as the equations are complicated.

3.1. DC Computer Method

First the value of the field maximum E₀ and its location x₀ are suitably chosen for the diode. Then the Poisson’s equation and the carrier continuity equations are solved simultaneously through numerical approach. The space step width is taken to be very small i.e. of the order of nms (1000 steps per micron). The numerical solution is progressed from the point x=x₀ and moving first towards the right side and then towards left hand side of the field maximum till the field/carrier current boundary conditions are satisfied at x=x_R,L. A four track rapidly converging logic is framed to get the boundary condition satisfied at the edges of the depletion layer. The breakdown characteristics, efficiency are thus obtained from final solution[7].

3.2. RF Analysis

Table 1. Physical and Material parameters of Ge and Si

A fast converging and double iterative computer program is framed, which performs iteration over the initial values of R₀ and X₀ at the left hand edge (x = x_L) till the boundary condition is satisfied at the other edge (x = x_R). The iteration over initial choice of R₀ and X₀ at the beginning of each iteration involves a four-way adjustment possibility for which logic is framed to guide the iteration to the converging track. The final solutions at various frequencies for a particular diode have been studied to assess the RF properties. The high frequency analysis gives the bandwidth (BW), peak frequency (f_p), negative device conductance (G_p), negative resistance (Z_p), negative resistance profile, avalanche phase delay and other related parameters. The bandwidth for RF generation, the negative conductance, positive series resistance, and avalanche phase delay etc. also can be calculated from the final solution[16]. Further the negative series resistance generation profile across the depletion zone can be obtained for the analysis to give some interesting features.

4. Results and Discussions

Si/Ge hetero diode structure having the form n-Si- Ge/p-Ge-Si has been designed and optimized for frequencies of operation around 15 and 96 GHz. The materials parameters of both the materials are taken from recent experimental reports[8-10] are shown in Table 1. The electron/hole ionization rate (α, β) are obtained using exponential equations of the form,

α, β = A_n,pexp(b_n,p/E)^m, m=1.0 for Si and Ge

The structural and design parameter for the diode are presented in table 2. The realistic doping profile across the junction is considered through use of an appropriate exponential function. A thin and highly doped Ge layer of width 120 nm and doping, 1×10²³/m³, (as compared to Si layer, width= 3.5 micron, doping= 3×10²¹/m³) is considered near the junction plane. The presence of highly doped Ge layer pushes the breakdown electric field maximum value (3.17×10⁷ V/m).

Table 2. Structural parameters of n-Si-Ge/p-Ge-Si Hetero-structure based IMPATTs

Table 3. DC and Microwave properties of Hetero-structure DDRs

Table 4. DC and Microwave properties of Si and Ge Homo-structure DDRs at 94 GHz

The ionization rate in Ge is observed to be higher by an order of magnitude than Si at the same electric field. The gap in the values of carrier ionization rate in Si and Ge further widens as the Ge layer remains near the high field junction plane where as Si layer is considered away from the junction. The results of dc/RF analyses are shown in Table 3. The avalanche zone in case of 15 GHz hetero structure diode becomes only 3.34 % of the total depletion width (for Si or Ge homo junction it is 47%-48%). The efficiency of the device has been computed to be very high i.e., 28.9 % as compared to 12-14% for Si and Ge homo DDRs. Even for 96 GHz operation, the efficiency for such Si/Ge hetero structure DDR becomes 24.1%. The efficiency of Si or Ge homo DDR has been seen to be around 9-12 % and even for Si Low-high-low complicated multilayered diode, it is around 14.1%. If one sees the values of break down voltage, for Si/Ge hetero structure diode at 94 GHz, it is computed to be 13.2 V as against only 9.88 V for Ge diode shown in table 4. However the breakdown voltage has been found to be 18.6 V for corresponding Si DDR. The value of peak negative conductance (-G_P) for the n-Si-Ge/p-Ge-Si DDR structure are 6.99×10⁵ S/m² and 5.00×10⁷ S/m² at peak frequency 15 and 96 GHz respectively shown in figure 2. An admittance plot of n-Si-Ge/p-Ge-Si DDR (n⁺npp⁺) IMPATT diode for 15 GHz is shown in figure 3 and for 96 GHz diode shown in figure 4. The variation of the impedance of the diode with frequency is shown in figure 5. This is also evident that the values of both negative resistance and reactance decreases as the operating frequency increases.

Figure 2. Frequency Vs Negative Conductance for the IMPATT Diode (n-Si-Ge/p-Ge-Si) at both 15 and 96 GHz Peak Frequency

Figure 3. Admittance plot of n-Si-Ge/p-Ge-Si DDR (n+npp+) IMPATT diode (Peak frequency= 15 GHz)

Figure 4. Admittance plot of n-Si-Ge/p-Ge-Si DDR (n+npp+) IMPATT diode (Peak frequency= 96 GHz)

Figure 5. Plots of the variation of impedance of DDR Hetero structure Impatt diode

It has also been observed that the value of negative conductance and negative resistance are higher in case of these hetero structure diodes and these values are sufficiently higher compared to corresponding Ge homo DDR at 94 GHz operation. Thus the high efficiency and high values of negative conductance/negative resistance in case of hetero structure would help in realizing high microwave power. The RF power of this particular diode is more of about 6.447 W at 15 GHz and 1.189 W at operating frequency 96 GHz. The inspection of negative resistance profiles (not shown) exhibit the usual double bump character and the magnitudes of the peaks in both the drift zones are observed to be considerably higher for Si/Ge/Si hetero structure as compared to homo junction diodes. The authors have also computed the values of negative conductance at second, third and fourth harmonic frequencies in case of 15 GHz diode. It is interesting to note that the peak value of negative conductance at respective peak frequencies of harmonic bands (at 30, 48 and 68 GHz) remains almost at same level as of fundamental band. In case of Si or Ge Homo DDR, the values of negative conductance fall appreciably in harmonic peak frequencies. The efficiency of the diode remains the same (29%) both in fundamental and harmonic bands. Thus in case of Si/Ge/Si hetero structure, this diode can be operated at higher harmonic frequencies without much loss of microwave power as the negative conductance and efficiency remain at nearly same level even at harmonic peak frequencies. The analysis also has been extended to the complementary hetero structure diode having structure, n-Ge-Si/p-Si-Ge (Ge/Si/Ge). The performance of this diode remains close to Si homo junction diode. The presence of low ionising Si near the junction plane causes the loss of advantage of hetero structure.

5. Conclusions

Thus the hetero structure diode of the form n-Si-Ge/p-Ge-Si would provide the best rf performance amongst above cited the homo junction diodes and complementary hetero structure diode.

ACKNOWLEDGEMENTs

Prof. S. P. Pati wishes to express his sincere thanks to AICTE, New Delhi for grant of Emeritus Fellowship in his favour.