The I-V curve is the most important parameter of power semiconductors (PSD), since power semiconductor output capacity and efficiency depends upon it to a great extent. Many publications are devoted to the analysis of the physical phenomena determining I-V curves of power semiconductor devices (e.g. the monographies [1, 2] and references in them). Thus, in practice, as the mathematical model of this characteristic, the following semiempirical formula ^{[3] }is widespread. \[\begin{equation} A + BI + Cln (I+1) + D \sqrt{I} \end{equation}\] This formula summarizes numerous researchers' results and describes I-V curves of devices of different types and purposes in a satisfactory way. This formula (1) describes a conductive state of PSD. In this formula, *I* is the forward current which is flowing through a semiconductor device, *V* is the voltage drop which is induced by this current, and the factors *A, B, C* and *D* are adjusted parameters of the model, which in practice are determined from a condition of the best fit to results of theoretical calculations and experimental data. It is considered that the back saturation of the current *I _{s}* flowing through the PSD in a closed condition is quite low. Therefore, the return branch of the I-V curve is usually not taken into consideration.

It is necessary to note that I-V curves, as well as other parameters of semiconductor devices, depend on temperature. In other words, factors *A, B, C* and *D* in the formula (1), as well as back current *I _{s}*, show this temperature dependence. Therefore, in specifications of PSDs, their I-V curves are given at two different values of temperature from the working range (usually it is 25 0С and 125 0С). Thus, these I-V curves are made on the basis of expression (1), where factors

*A, B, C*and

*D*are determined by way of experiment on the basis of results of two independent tests conducted at the above mentioned values of temperature. In case of necessity to determine the I-V curve of a definite device at some third temperature, one has to conduct a new test (to define

*A, B, C, D*and

*I*) or to be satisfied with interpolation of the curves available. However, the accuracy of this method depends only upon the level of experience and firmness of the researcher.

_{s}In this case, in expression (1), which describes a forward branch of the I-V curve, the temperature dependence of certain factors is examined depending on their physical characteristics. For this dependence, understandable analytical expressions are derived and it is demonstrated that it is possible to regard this dependence as a linear one with a high degree of accuracy.

In addition, the temperature dependence of the back current *I _{s}* flowing through the PSD in a closed condition is studied here. It is evident that in high temperature ranges the resistance of low alloy areas of the device has a significant effect on the value of this current. The proposed mathematically friendly model helps with a plotting of the curves that comply with an inverse branch of I-V curve on the grounds of experimental data.

__Temperature Dependence of the Forward Branch of the I-V Curve__

It is well known that under direct connection of a PSD, the voltage drop *V* on it (when connected) is developed from a voltage drop U_{pn} of one or several of the p-n transitions of the device, and a voltage drop* V _{R}* of low-alloy areas of its structure: \[\begin{equation} V = V_{pn} + V_R \end{equation}\] Thus, the voltage drop on the contacts of the device and on its high-alloy areas can be neglected without affecting the accuracy of the device. The voltage drop

*V*

_{pn}^{[4-6]}contributes significantly to the total voltage drop (2) at low currents

*I*flowing through the PSD. Taking into consideration that in operating modes the reverse saturation current is

*I*it is in this case possible to write down:\[\begin{equation} V \approx V_{pn} \approx c \frac {kT}{e} ln {(I+1)}{I_s} \end{equation}\] where

_{s}< < I,*k*is the Boltzmann constant,

*e*is the elementary charge,

*T*is the device temperature,

*c*is the proportionality factor (adjusted parameter of model), dependent on the number and technological features of manufacturing ot the p-n transitions.

The presence of *V _{pn}* is connected to the existence of a potential barrier

*V*between contiguous

_{0}*p-*and

*n-*areas. The height of this barrier (in volts) above low temperatures is determined by the concentration

*N*of donor atoms in the

_{D}*n*-area and the concentration

*N*of acceptor-impurity atoms in the

_{A}*p*-area

^{[4-6]}: \[\begin{equation} V_0 \approx \alpha \frac{kT}{e} ln \frac{N_DN_A}{n_i^2} \end{equation}\] where

*a*is an adjusted parameter of model,

*n*is the concentration of intrinsic carriers:

_{i}\[\begin{equation} n_i \approx 2 \Bigg ( \frac{2 \pi \sqrt{m_n m_p}kT}{h^2} \Bigg ) ^{3/2} exp \Bigg (- \frac{E_g}{2kT} \Bigg ) \end{equation}\]

where *m _{n}* and

*m*are the effective masses of electrons and of vacant electron sites, respectively,

_{p}*E*is the width of the semiconductor’s forbidden zone,

_{g}*h*is Planck's constants.

If a voltage greater than *V _{0}* is applied to a semiconductor device, the potential barrier between

*p*- and

*n*-areas of

*p-n*transitions vanishes, and the complete voltage drop on the device will be determined mainly by the addend in expression (2), i.e. by the voltage drop

*V*in low-alloy areas of its structure:

_{R}≈ RI\[\begin{equation} V \approx V_0 + RI \end{equation}\]

In power electronics engineering, the expression (6) is known as the linearized I-V curve of the PSD^{ [1, 6]}. The value *V _{0}* in this expression is called a pinch-off voltage, and

*R*is the dynamic resistance of the device. The experimental relation

*R(T)*is described well enough by the following expression

^{ [1, 6]}\[\begin{equation} R(T) = R_0 [1 + \beta (T - T_0)] \end{equation}\] where

*R*and

_{0}≡ R(T_{0}),*β*is the temperature coefficient of resistance determined by way of experiment [

*β = (1.5)×10*]. Note, that the formula (3) is true for

^{-3}К^{-1}*V*≤

*V*

_{0}. On the contrary, the expression (6) is realized for

*V*>

*V*

_{0}. So, for a forward branch of I-V curve of PSD, it is possible to write down the following interpolation formula applicable for all

*V*:\[\begin{equation} V \approx V_0 + RI + c \frac{kT}{e} ln \frac {(I+1)}{I_s}\end{equation}\]

The expression (8) allows us to bring to light a physical meaning and to find temperature dependence of factors *A*, *B* and *C*, which are included in the semiempirical formula (1).

Comparing (8) and (1), neglecting width change *E _{g}* of the semiconductor's forbidden zone, and taking into account the temperature dependence of the pinch-off voltage (4), dynamic resistance (7) and back saturation current (20), for moderate temperature areas we obtain:

\[ A(T) \equiv V_0 (T) - \frac{ckT}{e} ln I_s (T)\] \[\begin{equation} = \frac{(a + \gamma c) E_g}{e} - \frac{kT}{e} \Bigg [ \alpha ln \frac{4(2 \pi \sqrt{m_nm_p}kT)^3}{N_DN_Ah^6}+clnI_s (\infty) \Bigg ]\end{equation}\]

\[\begin{equation} B(T) \equiv R(T) = R_0 [1 + \beta (T - T_0)] \end{equation}\]

\[\begin{equation} C(T) \equiv \frac{ckT}{e}\end{equation}\]

Hence, with logarithmic accuracy, it is possible to consider the temperature dependence of factors *A*, *B* and *C* in the formula (1) as a linear one.

The expression (8) does not contain items of type *D√I* . This is explained by the fact that the correlation between voltage drop on low-alloy areas of the device and forward current strenghth was considered as linear as well, as is reflected in formulas (6) – (8). At the same time, it is well-known (see, for example, [1, 2, 4]), that the linear dependence *V*(*I*) can be observed only if the main mechanism of diffusion of uncombined charge carriers in a semiconductor is their diffusion on phonons and other heterogeneities of the grating, and the basic mechanism of their recombination is the linear recombination, for example, on the impurity (extrinsic) bands. The combination of effects such as electron-hole scattering (EHS) or non-linear Auger recombination, the decrease of emitter transitions' injection factors, as well as the heavy doping effect, can result ^{[1, 2]} in a non-linear increase of voltage drop *V* along with the increase of current* I. *

In expression (1), this fact is taken into account with the help of the addend *D√I*. It is necessary, however, to bear in mind that this addend is a totally adjusted one and describes the I-V curve of the PSD in the region of small non-linearity only approximately. Therefore, based on the physical meaning of this addend, it is practically impossible to obtain unique temperature dependence of the factor *D*. It will be determined by the specific type of non-linearity and in the present article is not examined.

This can be explained by the fact that non-linear effects become apparent only at very high level current densities *j ≥ 1 ÷ 3 kA/cm ^{2}* that cannot be reached during the devices' production-run work. For these PSDs, the linear dependence (6) of the isothermal I-V curve down to

*j ~ 1 кА/см*is confirmed not only by experiment, but also by theoretical studies, the results of which are represented in full in the monographies [1, 2].

^{2}In such a way, with a high-level of accuracy, the isothermal I-V curve of production-run PSDs can be described by expression (1), where *D ≡ 0*:

\[\begin{equation} V = A + BI + C ln (I+1) \end{equation}\]

The temperature dependence of factors *A*, *B* and *C* in this expression is described by the formulas (9) – (11), which can be presented properly in writing in the following form (convenient from the point of view of practical applications):

\[\begin{equation} A(T) = A_0[1 - \alpha (T - T_0)] \end{equation}\] \[\begin{equation} B(T) = R_0[1 + \beta (T - T_0)] \end{equation}\]\[\begin{equation} C(T) = C_0 \frac{T}{T_0}\end{equation}\]

where *A _{0} * ≡ A(T

_{0})

*R*≡ R(T

_{0 }_{0})

*≡ B(T*

_{0})

*, C*≡ C(T

_{0}_{0}), and

*α*and

*β*are constants determined by experiment.

The comparison of theoretical calculations carried out in the formulas (9) – (12) and experimental data I-V curves of a thyristor Т243-500 and a diode DL343-630-34, is presented in a fig. 1.

As this figure demonstrates, the suggested theory satisfactorily describes the measurement results. The observed dependence deviation *V*(*I*) from theoretical curves in the region of high *I* is explained by the fact that at such currents it is practically impossible to hold a constant temperature of PSD. Therefore, in the field of high* I, * devices heat up slightly during measurements. Compliance with the formulas (6) and (7) at given currents* I *resulted in higher values of* V* as compared to actual values which conform to the initial temperature of the devices.

__Temperature Dependence of the Inverse Branch of the I-V Curve__

At the reverse connection of PSD, its impedance *R _{s }*will be derived from resistance

*R*of back-dislocated

_{pn }*p-n*transitions and resistance

*R*of low-alloy regions of its structure: \[\begin{equation} R_s = R_{pn} + R\end{equation}\] Thus the back current which flows through the device, is usually saturated quite fast already at the low voltage applied

*V*(the weak dependence of the generation current from the reverse voltage [5] is neglected here), therefore hereinafter we will study the inverse branch of I-V curve of PSD only in the field of its saturation. The appropriate saturation current:

\[\begin{equation} I_s = \frac {V}{R_s} = \frac {V}{R_{pn} + R} \end{equation}\]

Usually at moderate temperatures, Rpn >> R and therefore \[\begin{equation} I_s \approx \frac {V}{R_{pn}} \end{equation}\]

This current is connected with diffusion through back displaced *p-n *transitions of minor charge carriers and is determined [4-6] by the diffusion length *L _{n}*, the life time

*t*

*and the equilibrium concentration*

_{p}*n*

_{p}_{0}of electrons in

*p*-areas and the diffusion length

*L*, the life time

_{p}*t*

*and the equilibrium concentration*

_{p}*p*

_{n}_{0}of vacant electron sites in

*n*-areas:

\[\begin{equation} I_s \sim e \Bigg ( \frac{L_n}{\tau_n} n_{p0} + \frac {L_p}{\tau_p} P_{n0} \Bigg ) = e \Bigg ( \frac{L_n}{\tau_np_{p0}}+ \frac {L_p}{\tau_pn_{n0}} \Bigg ) n_i^2 \end{equation}\]

where *p _{p}*

_{0}и

*n*

_{n}_{0}– equilibrium concentrations of vacant electron sites in

*p*-areas and of electrons in

*n*-areas accordingly. Thus, taking into consideration (5), the temperature dependence of a back current

*I*in the region of moderate temperatures can be described using the following expression:

_{s}\[\begin{equation} I_s(T) \approx I_{s0} exp \Bigg (-\frac{\gamma E_g (T_0 - T)}{kT_0T} \Bigg )\end{equation}\]

where , I_{s0} ≡ I_{s} (T_{0} ), γ – adjustable parameter of the model which is determined by experiment.

**Figure 1.** *I-V curve of a thyristor Т243-500 and a diode DA343-630-34, calculated on the formulas (9) – (12) (continuous curves) аnd found by experiment (marks) at two temperature values T. These experimental measurements were carried out for two samplings, of four production-run devices of the same type each.*

The resistance *R _{pn}* at return swith connection of PSD, as well as voltage drop

*V*at its direct connection (see section 2), is conditioned by existence of a potential barrier

_{pn}*V*

_{0}between contiguous

*p-*and

*n-*areas. As the formulas (4) and (5) demonstrate, at fixed value of the reverse voltage

*V*along with the rise in temperature

*T*of the device due to increase of its own charge carriers' concentration

*n*the height of this barrier may decrease so much, that the resistance

_{i}*R*will be less than the resistance

_{pn}*R*of low-alloy areas of PSD, which will be increasing at temperature rise in accord with (7). In such a way, there is the following in high-temperature region in compliance with the formula (17):

\[\begin{equation}I_s(T) \approx \frac{V}{R(T)} = \frac{V}{R_0[1 + \beta (T - T_0)]}\end{equation}\]

Applying expressions (20) and (21), on the ground of equation (17) it is possible to write down the following:

\[\begin{equation} I_s(T) \approx \frac {VI_{s0} exp \Bigg ( - \frac{\gamma E_g(T_0 - T)}{kT_0T} \Bigg )}{I_{s0}R_0[1 + \beta (T-T_0)] exp \Bigg ( – \frac {\gamma E_g 8T_0 -T)}{kT_0T} \Bigg ) + V} \end{equation}\]

The expression we came to represents an interpolation formula for temperature dependence of back saturation current and it is applicable at any value of *T*.

The diagrams of this dependence made for a production-run thyristor Т243-500 at 2 values of a reverse voltage *V* (0,5 and 1 V), are shown in fig. 2 (continuous curves). The point graph in this figure corresponds to the formula (20).

**Figure 2. ***Graphs of temperature-dependence of back saturation current for a thyristor *

**CONCLUSION**

The results presented in this article show that there are simple analytical expressions that adequately describe temperature dependence of I-V curve of PSD. So the forward branch of I-V curve of production-run devices with a high degree of accuracy is described by the expression (12) which factors as it stipulated in formulas (13)-(15), show linear temperature dependence. And the expression (22) is an opportune mathematical model of the reverse branch of these I-V curves.

The temperature dependence of back saturation current *I _{s}*(

*T*) is shown in fig.2. As this figure reveals low-alloy areas of PSD in high temperature region have limiting influence on values of this current. This influence is especially noticeable at low voltage

*V*applied to.

The above-mentioned factors should be taken into consideration when engineering and designing of PSDs, intended for use at high temperatures of a semiconductor structure.

**List of Literature**

1. V. Grigorenko, P. Dermenzhi, V. Kuzmin, T. Mnatsakanov. Simulation and design automation of power semiconductor devices – М.: Energoatomizdat, 1988.

2. A. Otblesk, V. Chelnokov. Physical problems in power semiconductor electronics. – L.: Nauka, 1984.

3. Thyristors. Information materials. ABB Semiconductors AG. – ABB Semiconductors AG, 1999.

4. G. Epifanov. Basic physics of microelectronics. – М.: High school, 1971.

5. A. Lebedev. Physics of semiconductor devices. – М.: Phizmatlit, 2008.

6. Yu. Evseev, P. Dermenzhi. Power semiconductor devices. – М.: Energoizdat, 1981

**For more information, please read:**

Modeling of Power Semiconductor Devices

PSD Surge Current Dependence on Current Impulses

V-I Characteristics – Varistors