Electric Field in a Semiconductor Between Conductivity-Type Junctions

Yu. S. Ryabinkin

(Presented by Professor Doctor A. A. Vorob’ev)

With a uniform distribution of the concentration of excess impurity \((N=\mathrm{const})\) and in thermodynamic equilibrium, the value of the electrostatic potential \(\psi\) in a semiconductor between two junctions is constant; therefore the field \(E=-\dfrac{d\psi}{dx}=0\). Injection of nonequilibrium current carriers is associated with deformation of the potential curve, which leads to the appearance of a field. Because of considerable mathematical difficulties, the field is usually taken into account in a simplified way, by introducing a certain effective diffusion coefficient instead of the true one. Below, the distribution of the electric field is found and its influence is taken into account explicitly.

The Equation Describing the Field Distribution and Its Approximate Solution

To simplify the notation, we introduce dimensionless variables (all notation is conventional)

\[ \overline{N}=\frac{N}{n_i}; \qquad \overline{x}=\frac{x}{l_i}; \qquad \overline{\psi}=\frac{q\psi}{kT}; \qquad \overline{E}=\frac{q l_i}{kT}E; \]

\[ \overline{j}=\frac{l_i}{qD_p n_i}\,j. \tag{1} \]

Here \(n_i\) is the intrinsic concentration; \(l_i=\sqrt{\dfrac{\varepsilon kT}{4\pi q^2 n_i}}\) is the Debye length (in germanium at \(T=300^\circ\mathrm{K}\), \(n_i=2\cdot10^{13}\ \mathrm{cm}^{-3}\); \(l_i=10^{-4}\ \mathrm{cm}\));

\[ \frac{kT}{q l_i}=260\ \mathrm{V/cm}; \qquad \frac{qD_p n_i}{l_i}\simeq 1.5\ \mathrm{A/cm^2}. \]

In dimensionless variables the equations for the hole and electron currents and Poisson’s equation [1] have the form

\[ j_p=pE-\frac{dp}{dx}, \tag{2} \]

\[ -\frac{1}{b}j_n=nE+\frac{dn}{dx}, \tag{3} \]

\[ \frac{dE}{dx}=p-n+N \tag{4} \]

(the bar over dimensionless variables is omitted; \(b=\dfrac{\mu_n}{\mu_p}=\dfrac{D_n}{D_p}\)).

Let us consider the regime of stationary injection when the distance between the alloyed junctions is much less than the diffusion length (i.e., for \(j_p=\mathrm{const}\), \(j_n=\mathrm{const}\), \(N=\mathrm{const}\)). Taking the sum and difference of equations (2) and (3) and transforming

\[ \frac{dE}{dx}=-\frac{d^2\Psi}{dx^2} =-\frac{1}{2}\frac{d}{d\Psi}\left(\frac{d\Psi}{dx}\right)^2, \]

we arrive at the basic equation

\[ \frac{d^3\Psi}{dx^3} -\left[ C_1-j_- \cdot x+N\Psi+\frac{1}{2}\left(\frac{d\Psi}{dx}\right)^2 \right] =j_+, \tag{5} \]

where \(j_{\pm}=j_p\pm\dfrac{1}{b}j_n\) and the integration constant \(C_1=n+p+j_- \cdot x-N\Psi-\)

\[ -\frac{1}{2}\left(\frac{d\Psi}{dx}\right)^2. \]

We choose the zero of the potential and the coordinate at the point where

\[ \frac{d\Psi}{dx}=0 \]

(for example, the top of the potential barrier of the emitter–base in a \(pnp\)-triode or a \(pin\)-diode); therefore the value of \(C_1\) is equal to the sum of the electron and hole concentrations at this point: \(C_1=n_1+p_1\). It can be shown that

\[ C_1-j_- \cdot x \gg \left|N\Psi+\frac{1}{2}\left(\frac{d\Psi}{dx}\right)^2\right|. \tag{6} \]

Taking this inequality into account and putting \(y=C_1-j_- \cdot x\), we transform (5) to the form

\[ -\frac{d^2E}{dy^2}-\frac{y}{j_-^2}E=-\frac{j_+}{j_-^2}. \tag{7} \]

The general solution of this equation in terms of modified Bessel functions of the first and second kind \(I_\nu(\xi), K_\nu(\xi)\), with \(\xi=\dfrac{2}{3j_-}y^{3/2}\), has the form [2, 3]

\[ E=\sqrt{y}\left(AI_{1/3}+BK_{1/3}\right) -\frac{2}{3}\frac{j_+}{j_-^2}\sqrt{y} \left( I_{1/3}\int K_{1/3}\,d\xi - K_{1/3}\int I_{1/3}\,d\xi \right). \tag{8} \]

where \(A, B\) are arbitrary constants.

Using the asymptotic expressions \(I_\nu(\xi)\sim e^\xi/\sqrt{2\pi\xi}\), \(K_\nu(\xi)\sim \sqrt{\pi/(2\xi)}\,e^{-\xi}\) and the conditions \(E(0)=0\), \(E(w)=E_w<4\) at the boundaries of the region under consideration, we obtain, according to (8),

\[ E=\frac{j_+}{y}-\frac{1}{y^{1/4}} \left[ \frac{j_+}{C_1^{3/4}}\, \frac{\operatorname{sh}(\xi-\xi_w)}{\operatorname{sh}(\xi_0-\xi_w)} + \left( \frac{j_+}{y_w^{3/4}}-y_w^{1/4}E_w \right) \frac{\operatorname{sh}(\xi_0-\xi)}{\operatorname{sh}(\xi_0-\xi_w)} \right]. \tag{9} \]

The distribution of the volume-charge density has the form

\[ \begin{aligned} p-n+N=\frac{dE}{dx} &=\frac{j_+j_-}{y^2} -\frac{j_-}{4y^{5/4}} \left[ \frac{j_+}{C_1^{3/4}}\, \frac{\operatorname{sh}(\xi-\xi_w)}{\operatorname{sh}(\xi_0-\xi_w)} + \left( \frac{j_+}{y_w^{3/4}}-y_w^{1/4}E_w \right) \frac{\operatorname{sh}(\xi_0-\xi)}{\operatorname{sh}(\xi_0-\xi_w)} \right] \\ &\quad +\frac{1}{y^{1/4}} \left[ \frac{j_+}{C_1^{3/4}}\, \frac{\operatorname{ch}(\xi-\xi_w)}{\operatorname{sh}(\xi_0-\xi_w)} - \left( \frac{j_+}{y_w^{3/4}}-y_w^{1/4}E_w \right) \frac{\operatorname{ch}(\xi_0-\xi)}{\operatorname{sh}(\xi_0-\xi_w)} \right], \end{aligned} \tag{10} \]

where

\[ y(w)=y_w,\qquad \xi(0)=\xi_0,\qquad \xi(w)=\xi_w . \]

Stationary injection into the base of a \(pnp\) triode

Practically throughout the entire interval \(0<x<w\) (excluding a small neighborhood of the boundaries \(x=0\) and \(x=w\)),

\[ E=\frac{j_+}{C_1-j_-x}. \tag{11} \]

This is caused by the rapid attenuation of the hyperbolic terms in (9), owing to the large value of the difference \(\xi_0-\xi_w\).

Substituting (11) into equation (2) and solving it under the boundary conditions \(p(0)=p_1\), \(p(w)=p_2\), we find for the current-voltage characteristic the equation

\[ (C_1-j_-x)^{a+1}-2p_2(C_1-j_-w)-C_2C_1^a=0, \tag{12} \]

where \(C_2=n_1-p_1\), \(a=j_+/j_-\).

In alloy \(pnp\)-triodes, even at high injection levels, \(j_+\simeq j_-\simeq j_p\simeq j\) (total current). Therefore, according to (12),

\[ j=\frac{C_1-p_2-\sqrt{C_1C_2+p_2^2}}{w}. \tag{13} \]

Taking into account the inequalities \(C_1\gg 1\), \(\xi_0-\xi_w\gg 1\), from (10) we find that at the point \(x=0\)

\[ p_1-n_1+N=\frac{j}{\sqrt{C_1}}+\frac{3}{4}\frac{j^2}{C_1^2}. \tag{14} \]

Thus, the space-charge density at the top of the emitter–base potential barrier increases with increasing \(j\), while remaining positive. In this case, at first \(n_1>p_1\), and then \(n_1<p_1\). Therefore, with increasing \(j\) the constant \(C_1=n_1-p_1\) changes sign.

Eliminating \(C_2\) from (13) with the aid of (14) and taking into account that \(C_1\gg N\gg 1\) and \(w>1\), we obtain

\[ j=\frac{1}{w}\left[C_1-p_2-\sqrt{C_1N+p_2^2+\frac{p_2\sqrt{C_1}}{w}-\frac{C_1^{3/2}}{w}}\right]. \tag{15} \]

If here the quantity \(p_2\) is neglected, then under the condition \(\dfrac{\sqrt{C_1}}{wN}>1\) the expression under the radical is negative. Therefore this inequality may be regarded as a criterion for the transition to the saturation regime, when allowance for the quantity \(p_2\) becomes essential\(^1\). In a regime far from saturation, \(p_2\) is small and

\[ j=\frac{C_1-\sqrt{C_1N}}{w}. \tag{16} \]

The distribution of minority carriers between the junctions, with allowance for the field, has the form

\[ p=\frac{1}{2}(C_1-jx)-\frac{C_1C_2}{2(C_1-jx)}, \tag{17} \]

as follows from (2) and (11), with \(p(0)=p_1\).

Denoting \(j_E=pE\) and \(j_D=-\dfrac{dp}{dx}\), from (17) and (11) we find the ratio of the field and diffusion components of the current along the base,

\[ \frac{j_E}{j_D}=\frac{(C_1-jx)^2-C_1C_2}{(C_1-jx)^2+C_1C_2}. \tag{18} \]

Far from saturation, \(j_E/j_D<1\); with increasing current, under saturation \(j_E/j_D>1\) everywhere in the base.

The assumptions made above (validity of inequality (6); approximation of \(I_\nu, K_\nu\) by the first terms of the asymptotic expansions; approximation (9) by only the first term) are not difficult to justify by direct substitution of (16) into the corresponding expressions.

\(^1\) In the present case this is connected with the dependence of \(p_2\) on \(p_1\) [4].

Frequency characteristics of the base transfer coefficient, emitter efficiency, and current gain

Let us consider the alternating component of the current (signal) \(\tilde{j}_p \ll j\), so that the alternating component of the field \(\tilde{E}\) may be neglected in comparison with the steady-state component (11). For a harmonic signal \(\tilde{j}_p=\tilde{j}_p(x)e^{i\omega t}\), \(\tilde{p}=\tilde{p}(x)e^{i\omega t}\), it follows from the continuity equation for holes [1] that

\[ \tilde{p}=-\frac{\tau_p}{1+i\omega\tau_p}\frac{d\tilde{j}_p}{dx} \tag{19} \]

(the dimensionless time is \(\bar{t}=\dfrac{D_p}{l_i^2}t\) and the frequency is \(\bar{\omega}=\dfrac{l_i^2}{D_p}\omega\)).

From (2) and (19) we obtain

\[ \frac{d^2\tilde{j}_p}{dy^2}+\frac{1}{y}\frac{d\tilde{j}_p}{dy} -\frac{1+i\omega\tau_p}{j^2\tau_p}\tilde{j}_p=0, \tag{20} \]

where, as above, \(y=C_1-jx\). Hence we find the signal amplitude under the boundary conditions \(\tilde{p}(0)=\tilde{p}_1\), \(\tilde{p}(w)=0\) [2, 3]:

\[ \tilde{j}_p(x)=i\tilde{p}_1 \sqrt{\frac{1+i\omega\tau_p}{\tau_p}}\, \frac{J_0(iz)Y_1(iz_w)-I_1(iz_w)Y_0(iz)} {I_1(iz_0)Y_1(iz_w)-I_1(iz_w)Y_1(iz_0)} . \tag{21} \]

Here \(I_\nu,\,Y_\nu\) are Bessel functions of the first and second kind,

\[ z=\frac{C_1-jx}{w}\sqrt{\frac{1+i\omega\tau_p}{\tau_p}}, \qquad z_0=z(0),\quad z_w=z(w). \]

The electron component of the current in the emitter can be calculated without taking the field into account [5] and, in dimensionless variables, has the form

\[ \tilde{j}_n(x)=bn_2 \sqrt{\frac{1+i\omega\tau_n}{b\tau_n}}\, e^{\,x\sqrt{\frac{1+i\omega\tau_n}{b\tau_n}}}. \tag{22} \]

From (21) and (22) we find the base transfer coefficient

\[ \beta=\frac{\tilde{j}_p(w)}{\tilde{j}_p(0)} = \frac{2}{i\pi z_w\left[I_1^wY_0^0-I_0^0Y_1^w\right]}, \tag{23} \]

emitter efficiency

\[ \gamma=\frac{\tilde{j}_{p}(0)}{\tilde{j}_{p}(0)+\tilde{j}_{n}(0)}= \]

\[ =\left[ 1-i b\,\frac{\hat{n}_{2}}{p_{1}} \sqrt{\frac{(1+i\omega\tau_{n})\tau_{p}}{(1+i\omega\tau_{p})b\tau_{n}}}\, \frac{I_{1}^{0}Y_{1}^{w}-I_{1}^{w}Y_{1}^{0}} {I_{0}^{0}Y_{1}^{w}-I_{1}^{w}Y_{0}^{0}} \right]^{-1} \tag{24} \]

and the current gain coefficient

\[ \alpha=\gamma\beta= \left\{ \frac{\pi}{2}\,i z_{w} \left[ I_{1}^{w}Y_{0}^{0}-I_{0}^{0}Y_{1}^{w} +\right. \right. \]

\[ \left. \left. {}-i b\,\frac{\hat{n}_{2}}{p_{1}} \sqrt{\frac{(1+i\omega\tau_{n})\tau_{p}}{(1+i\omega\tau_{p})b\tau_{n}}} \left(I_{1}^{0}Y_{1}^{w}-I_{1}^{w}Y_{1}^{0}\right) \right] \right\}^{-1}. \tag{25} \]

The superscript on the Bessel functions denotes the point at which the value of the argument is taken, \(I_{1}^{w}=I_{1}(i z_{w})\), etc. The boundary values of the concentrations are related to the displacement at the junction by the Boltzmann relations [1].

Let us note that, at low injection levels, \(n_{1}\simeq N\gg p_{1}\gg p_{2}\), \(C_{1}\simeq N+p_{1}\), \(C_{2}\simeq N-p_{2}\), and, as is not difficult to verify, all the expressions obtained above pass over into the expressions of Shockley’s small-signal theory [5 et al.].

This work was carried out under the supervision of Professor E. I. Adirovich.

References

1. Rittner E. S., “Extension of the Theory of the Junction Transistor,” Phys. Rev., vol. 94, No. 5, 1954.
2. Watson G. N., Theory of Bessel Functions, Part 1, 1951.
3. Smirnov V. I., Course of Higher Mathematics, Vol. 2, 1951.
4. Ryabinkin Yu. S., “Electric field in the base of planar transistors at low injection levels,” Fizika tverdogo tela, vol. I, p. 159, Academy of Sciences of the USSR, 1959.
5. Early J. M., “Effects of space-charge-layer Widening in Junction Transistors,” PIRE, vol. 40, No. 11, 1952.