Effect of a Constant Electric Field in the Diode Base on the Lifetime of Minority Charge Carriers
Abstract Generated abstract
This paper analyzes how a constant electric field in the base of a semiconductor diode, produced by an impurity concentration gradient, affects the effective lifetime of minority carriers during turn-on transients. A one-dimensional planar diode model with a bounded base and a high-resistance region near the p-n junction is treated by solving the minority-carrier transport equations using Laplace transforms and boundary conditions at the junction, region interface, and ohmic contact. The resulting expressions relate the effective lifetime to electric-field strength and surface recombination velocity, and reduce to the known diffusion-only case when the field region boundary approaches the ohmic contact. The analysis indicates that, for the proposed model, surface recombination velocity has a stronger influence on the transient rate than the constant electric field.
Full Text
Influence of a Constant Electric Field in the Base of a Diode on the Lifetime of Minority Current Carriers
S. A. Zaidman
(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)
It is known that the transient process in a semiconductor diode can be affected by a constant electric field in the base, produced by a gradient in impurity concentration \[1, 2\]. Depending on the direction of the electric field, it can either decrease or increase the rate of the transient process. If the diffusion of donors into an $n$-type semiconductor takes place from the side of the rectifying contact, then the field is accelerating; if from the side of the ohmic contact, then the field is retarding. In the first case the lifetime of minority carriers in the base of the diode decreases, while in the second case it increases. In the present work a quantitative dependence is proposed that relates the lifetime of minority charge carriers $\tau$ to the electric field in the diode base $E$ and to the recombination velocity at the ohmic contact $S$.
Consider a model of a one-dimensional planar diode with an abrupt stepwise $p$—$n$ junction and a bounded base region, with the conductivity of the $p$ region assumed to be much greater than the conductivity of the $n$ region (Fig. 1).
Fig. 1. Analytical model of the diode
At some distance from the $p$—$n$ junction, at $x = x_1$, there is a concentration gradient of donors that creates a constant pulling electric field. The presence of a high-resistance region near the $p$—$n$ junction increases the reverse breakdown voltage. To solve the problem posed, we write a system of equations describing the motion of minority carriers (holes) when the diode is switched from the neutral state into the forward regime by a voltage generator.
In the region $0 < x < x_1$,
\[ \frac{1}{D_p}\cdot \frac{\partial p}{\partial t} + \frac{p - p_n}{L_p^2} = \frac{\partial^2 p}{\partial x^2}, \tag{1} \]
where, at $x = 0$,
\[ p(0) = p_n \cdot e^{\frac{qV_{pn}}{kT}} = p_0,\qquad t > 0, \]
where $p_n$ is the equilibrium concentration of holes in the $n$ region,
\(V_{pn}\)—the voltage drop across the \(p\)–\(n\) junction,
\(L_p\)—the diffusion length; in the region
\[ x_1 < x < d \]
\[ \frac{\partial^2 p}{\partial x^2} -\frac{qE}{kT}\cdot\frac{\partial p}{\partial x} -\frac{\Delta P}{L_p^2} = \frac{1}{D_p}\cdot\frac{\partial p}{\partial t}, \tag{2} \]
where, at \(x=d\),
\[ \frac{\partial p}{\partial x}-U_p\cdot E\cdot p(d) = -\frac{S}{D_p}\cdot p(d), \]
where \(U_p\)—the mobility of holes,
\(D_p\)—the diffusion coefficient of holes.
In addition, at the boundary of the regions, at \(x=x_1\),
\[ p_1(x,t)=p_2(x,t)\quad\text{and}\quad \frac{\partial p_1}{\partial x}=\frac{\partial p_2}{\partial x}. \tag{3} \]
The latter condition ensures continuity of the current at the point \(x_1\). Before the application of the forward pulse, the concentration of holes in the base is negligibly small,
\[ p(x,0)=0,\quad t=0. \tag{4} \]
Solving the system of equations (1) and (2) by means of the Laplace transform and the residue theorem, for the case of several roots, we obtain the distribution of the concentration of nonequilibrium current carriers. In the region \(0<x<x_1\),
\[ \Delta p_1(x,t)= \frac{ p_0\left[ \kappa'_2\cdot \operatorname{sh}\frac{x-x_1}{L_p} +\frac{1}{L_p}\cdot \operatorname{ch}\frac{x-x_1}{L_p} \right] }{ \kappa'_2\cdot \operatorname{sh}\frac{x_1}{L_p} +\frac{1}{L_p}\cdot \operatorname{ch}\frac{x_1}{L_p} } + \tag{5} \]
\[ +\sum_{n=1}^{\infty} \frac{ p_0\cdot \kappa_2^{1}\cdot x_1\cdot \sin\frac{\mu_n(x-x_1)}{x_1} -\mu_n\cdot \cos\frac{\mu_n(x-x_1)}{x_1} }{ \left(\frac{\mu_n^2}{x_1^2}+\frac{1}{L_p^2}\right) \cdot \left[(\kappa_2^{1}\cdot x_1+1)\cos\mu_n-\mu_n\cdot\sin\mu_n\right] } \cdot e^{-t/\tau_r}, \]
where
\[ \kappa'_2= \frac{qE}{2kT} - \sqrt{\left(\frac{qE}{2kT}\right)^2+\frac{1}{L_p^2}}, \tag{5′} \]
\[ \kappa_2^{1}= \frac{qE}{2kT} - \sqrt{\left(\frac{qE}{2kT}\right)^2-\frac{\mu^2}{x_1^2}}. \tag{5} \]
In the region \(x_1<x<d\),
\[ \Delta p_2(x,t)= \frac{ \frac{p_0}{L_p}\left[1-\kappa'_2(x-x_1)\right] }{ \kappa'_2\operatorname{sh}\frac{x_1}{L_p} +\frac{1}{L_p}\cdot \operatorname{ch}\frac{x_1}{L_p} } + \]
\[ +\sum_{n=1}^{\infty} \frac{ p_0\cdot \mu_n\left[1+\kappa'_2(x-x_1)\cdot e^{-t/\tau_r}\right] }{ \left(\frac{\mu_n^2}{x_1^2}+\frac{1}{L_p^2}\right) \left[(\kappa'_2\cdot x_1+1)\cos\mu_n-\mu_n\cdot\sin\mu_n\right] }, \tag{6} \]
where the reciprocal of the effective lifetime is
\[ \frac{1}{\tau_r}=D_p\cdot\left(\frac{\mu^2}{x_1^2}+\frac{1}{L_p^2}\right) =\frac{D_p\cdot\mu^2}{x_1^2}+\frac{1}{\tau_p}, \tag{7} \]
and \(\mu\) is the root of the transcendental equation
\[ \operatorname{ctg}\mu=-\frac{\kappa'_2\cdot x_1}{\mu}. \tag{8} \]
It should be noted that if the boundary separating the two regions of the base approaches the second contact, i.e. \(x_1 \approx d\), then, as is evident from the expressions for the concentrations (5) and (6), we arrive at the formula obtained for the case in which only diffusion is taken into account, when there is no impurity concentration gradient in the base [3].
\[ \Delta p(x,t)= \frac{ p_0\cdot\frac{1}{L_p}\cdot \operatorname{ch}\frac{d-x}{L_p} +\frac{s}{D_p}\cdot \operatorname{sh}\frac{d-x}{L_p} }{ \frac{1}{L_p}\cdot \operatorname{ch}\frac{d}{L_p} +\frac{S}{D_p}\cdot \operatorname{sh}\frac{d}{L_p} } + \tag{9} \]
\[ +\sum_{n=1}^{\infty} \frac{ \mu_n\cdot \cos\mu_n\frac{d-x}{d} +\frac{S\cdot d}{D_p}\cdot \sin\mu_n\frac{d-x}{d} }{ -\left(\frac{\mu_n^2}{d^2}+\frac{1}{L_p^2}\right) \left[\left(1+\frac{Sd}{D_p}\right)\cos\mu_n-\mu_n\cdot\sin\mu_n\right] }, \]
where, for complete agreement of formulas (5) and (9), in addition to \(x_1=d\), it is necessary to set
\[ \kappa_2=\frac{S}{D_p}. \tag{10} \]
On the basis of (5) and (10) we find the expression for the lifetime \(\tau\)
\[ \frac{1}{\tau_p}=\frac{s^2}{D_p}+\frac{q\cdot s}{kT}\,|E|. \tag{11} \]
Using further the relation between \(\frac{1}{\tau_r}\) and \(\frac{1}{\tau_p}\) from [7], we obtain the following formula for the effective lifetime \(\tau_r\), which characterizes the rate of the transient process:
\[ \frac{1}{\tau_r}=\frac{D_p\cdot\mu^2}{d^2}+\frac{s^2}{D_p}+\frac{s\cdot q}{kT}\cdot E. \tag{12} \]
Thus, in the presence of a high-resistance region near the \(p\)–\(n\) junction, the effective lifetime is inversely proportional to the square of the recombination velocity at the ohmic contact and inversely proportional to the electric-field strength.
Consequently, for the proposed diode model, the surface recombination velocity has a more substantial influence on the rate of the turn-on transient process than does the electric field.
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