On the Theory of \(Pn\)-Junctions and \(Pin\)-Structures

Yu. S. Ryabinkin

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

\(pn\)-junction under reverse and small forward currents

In view of the mathematical complexity of the problem, in considering \(pn\)-junctions one usually neglects the resultant current in the junction, the concentration of electrons and holes in the space-charge region (the depleted-layer model), and the dependence of mobilities on the field. For semiconductors of the germanium type, theory and experiment show that the electron drift velocity is \(s_n=\mu_n E = 6\cdot 10^6\ \mathrm{cm/sec}\) at fields \(E>10^3\ \mathrm{V/cm}\) [1 and others]. Such values of \(E\) occur in alloyed \(pn\)-junctions under reverse and small forward biases. The indicated effects are taken into account in the present work both for \(pn\)-junctions and for \(pin\)-structures.

For simplicity of notation, introduce dimensionless variables (the notation is conventional):
\(\overline p=p/n_i,\ \overline n=n/n_i,\ \overline N=N/n_i,\ \overline x=x/l_i,\)
\(\overline\psi=q\Psi/kT,\ \overline E=(q l_i/kT)E,\ \overline j=j/q s_n n_i;\) (\(n_i\) is the intrinsic concentration;

\[ l_i=\sqrt{\frac{\varepsilon kT}{4\pi q^2 n_i}} \]

is the Debye length; \(N\) is the difference between the donor and acceptor concentrations). We shall neglect the difference between the drift velocities of electrons and holes \((s_p=s_n=s)\). Extrapolating Einstein’s relation to the strong-field region, we obtain \(D=\dfrac{kT}{q}\mu=\dfrac{kT}{q}S\dfrac{dx}{d\Psi}\). Then, in dimensionless variables, the current equations

\[ j_p=q\mu_p pE-qD_p\frac{dp}{dx},\qquad j_n=q\mu_n nE+qD_n\frac{dn}{dx} \]

and Poisson’s equation

\[ \frac{d^2\Psi}{dx^2}=-\frac{4\pi}{\varepsilon}q(p-n+N) \]

take the form (we omit the bar over the notation)

\[ j_p=p+\frac{dp}{d\psi}, \tag{1} \]

\[ -j_n=n-\frac{dn}{d\Psi}, \tag{2} \]

\[ \frac{d^2\Psi}{dx^2}=n-p-N. \tag{3} \]

Taking \(x=0\) at the alloying boundary, we have \(N=N_n=\mathrm{const}>0\) for \(x>0\), \(N=-N_p=\mathrm{const}<0\) for \(x<0\). We denote the boundary of the space-charge layer in the heavily doped \(p\)-type region by \(x_p\). The boundary in the \(n\)-region \(x=x_n\) is defined by the condition \(p-n+N_n=\frac{1}{2}N_n\) (in the depleted layer near \(x=0\), \(p-n+N\simeq N\); in the neutral bulk of the semiconductor, \(p-n+N\simeq0\)).

We solve equations (1), (2), and (3) separately in the \(p\)- and \(n\)-regions under the boundary conditions
\(\Psi(x_p)=\Psi_p,\ \Psi(x_n)=\Psi_n-\frac{1}{2}\),

\[ E(x_p)\simeq0,\quad E(x_n)\simeq4,\quad p(x_p)\simeq N_p,\quad n(x_n)\simeq \frac{1}{2}N_n,\quad p(+0)=p(-0); \]

\[ n(+0)=n(-0). \]

Here \(\Psi_p\) is the base and \(\Psi_n\) the top of the potential barrier of the \(pn\)-junction. The calculation of \(\Psi(x_n)\) and \(E(x_n)\) is based on [2]. The solution has the form:

\[ p=j_p+N_p e^{\Psi_p-\Psi}, \tag{4} \]

\[ n=j_n+\frac{1}{2}N_n e^{\Psi-\Psi_n+\frac{1}{2}}, \tag{5} \]

\[ \frac{d\Psi}{dx}= \sqrt{ 2\left[ \frac{1}{2}N_n e^{\Psi-\Psi_n+\frac{1}{2}} +N_p e^{\Psi_p-\Psi} +(N-j_p+j_n)\Psi+C \right]}, \tag{6} \]

\[ x=\frac{1}{\sqrt{2}}\int_{\Psi_0}^{\Psi} \frac{d\Psi}{ \sqrt{ \frac{1}{2}N_n e^{\Psi-\Psi_n+\frac{1}{2}} +N_p e^{\Psi_p-\Psi} +(N-j_p+j_n)\Psi+C } }. \tag{7} \]

The constant of integration is \(C=-N_p(1+\Psi_p)\) for \(x<0\) and \(C=N_n(\Psi_n-1)\) for \(x>0\). The value of the potential at the alloying boundary \(\Psi_0=\Psi(0)=\Psi_p+1\) follows from the condition \(\Psi(+0)=\Psi(-0)\), \(E(+0)=E(-0)\).

These expressions describe the distribution of holes, electrons, field, and potential in a \(pn\)-junction for given values of \(N_p\), \(N_n\), \(j_p\), \(j_n\), and for barrier height \(\Delta\Psi=\Psi_n-\Psi_p\). From their analysis it follows that \(p\simeq N_p\gg n\) for \(x<0\); \(p\gg N_n\gg n\) for \(0<x<x_1\), \(p\simeq n\ll N_n\) for \(x_1<x<x_2\), \(p\ll n\simeq N_n\) for \(x>x_2\), if \(x_1\) and \(x_2\) denote the boundaries of the depleted layer (then \(0<x<x_1\) corresponds to the inversion layer). The distribution \(p(x)\) for \(x<0\) changes appreciably only under the condition

\[ \Delta\Psi>\frac{N_p}{N_n}\left[1+\ln\frac{N_n}{10N_p}\right]. \]

In the broad sense of the word, by a \(pn\)-junction one usually means the entire space-charge layer in the vicinity of the contact of the \(p\)- and \(n\)-regions.

Within this layer one should distinguish the technological transition (the interface between regions with predominance of donor and acceptor impurities) and the physical transition (the surface of inversion of the type of the basic conductivity). From the condition \(p(\Psi_{pn})=n(\Psi_{pn})\), according to (4) and (5), we find the position of the physical transition \(\Psi=\Psi_{pn}\) (for \(j_p\gg j_n\)):

a) for
\[ \Delta\Psi \lesssim \ln \frac{N_pN_n}{100j_p^2} \]

\[ \Psi_{pn}=\frac{1}{2}\left(\frac{1}{2}\Delta\Psi+\ln \frac{2N_p}{N_n}-2\right) \]
and
\[ p(\Psi_{pn})=n(\Psi_{pn})=\sqrt{N_pN_n}\,e^{-\frac{\Delta\Psi}{2}}; \]

b) for
\[ \Delta\Psi \gtrsim \ln \frac{10N_pN_n}{j_p^2} \]
\[ \Psi_{pn}=\Delta\Psi+\ln \frac{j_p}{N} \]
and
\[ p(\Psi_{pn})=n(\Psi_{pn})=j_p, \]

i.e., with increasing barrier height the physical transition shifts toward the top of the barrier, while the values of \(p\) and \(n\) reach saturation.

pin structure under reverse bias

Completely analogous expressions are obtained when considering a pin structure under reverse bias (heavily doped \(p\)- and \(n\)-regions separated by a region of intrinsic conductivity). In this case the solution is added in the region \(0<x<w\) with intrinsic conductivity \((N=0)\), if \(x=0\) and \(x=w\) correspond to the diffusion boundaries. The boundary condition for this region is the matching condition at \(x=0\) and \(x=w\). There are now two inversion layers: of \(p\)-type near \(x=0\) (the \(pi\)-transition) and of \(n\)-type near \(x=w\) (the \(in\)-transition). For \(w\gg 10^{-3}\) cm, the influence of the inversion layers and of the current may be neglected, taking the field to be constant and equal to \(E=\Delta\Psi/w\), where \(\Delta\Psi\) is the external bias. This makes it possible to calculate the base transmission coefficient of a pin transistor, taking into account both the field and diffusion.

The dependence of the electron diffusion coefficient on the field \(D_n(E)\) can be found by averaging the known expression
\[ D_n=\frac{1}{3}v_n^2t_n \]
(in the present case \(t_n\) is the relaxation time for collisions of electrons with the optical branch of lattice vibrations). Using the electron energy distribution function in strong fields, derived in [1], we obtain (in the notation of [1])
\[ D_n= \frac{\Gamma\!\left(\frac{1}{3}\right)}{4\Gamma\!\left(\frac{2}{3}\right)} \frac{h\omega_0}{b_0c_0m^*} \left[ \frac{2a_0b_0c_0m^*}{3h\omega_0e^2} \left(1+\frac{C_{on}^2}{C_{ik}^2}\right) \right]^{1/6} E^{-1/3} \simeq 1.5\cdot 10^3\,(\mathrm{cgs}\ E) \tag{8} \]

For values \(E\gtrsim 10^3\ \mathrm{V/cm}\) and for a harmonic alternating signal, from the continuity and electron-current equations [3] we find

\[ \tilde n=\frac{1}{i\omega q}\frac{d\tilde j_n}{dx}, \tag{9} \]

\[ \frac{d^{2}\tilde{j}_{n}}{d x^{2}}-\frac{s}{D_{n}}\frac{d\tilde{j}_{n}}{d x}-\frac{i\omega}{D_{n}}\tilde{j}_{n}=0 \tag{10} \]

(without taking recombination into account; the units are dimensional).

Let \(x=0\) correspond to the plane of the virtual emitter, and \(x=w\) to the boundary of the \(i\)-type base with the low-resistance \(n\)-type collector. The solution under the boundary conditions \(\tilde{j}_{n}(0)=\tilde{j}_{1}\) has the form \((\tilde{n}(w)=0)\):

\[ \tilde{j}_{n}(x)=\tilde{j}_{1}e^{ax} \frac{a\,\operatorname{sh}\!\left[(w-x)\sqrt{b}\right]+\sqrt{b}\,\operatorname{ch}\!\left[(w-x)\sqrt{b}\right]} {a\,\operatorname{sh}\!\left(w\sqrt{b}\right)+\sqrt{b}\,\operatorname{ch}\!\left(w\sqrt{b}\right)}, \tag{11} \]

where

\[ a=\frac{s_{n}}{2D_{n}}, \qquad b=a^{2}+i\frac{\omega}{D_{n}}, \qquad s_{n}=6\cdot 10^{6}\ \text{cm/sec}, \]

\[ D_{n}=1.5\cdot 10^{3}E^{-1/3}\ (\operatorname{cgs} E). \]

By the definition of the transfer coefficient,

\[ \beta=\frac{j_{n}(w)}{j_{n}(0)} =\frac{\sqrt{b}\,e^{aw}} {a\,\operatorname{sh}\!\left(w\sqrt{b}\right)+\sqrt{b}\,\operatorname{ch}\!\left(w\sqrt{b}\right)} = \tag{12} \]

It follows from this that \(\beta\to 1\) as \(\omega\to 0\) and \(\beta\to 0\) as \(\omega\to\infty\), i.e., diffusion broadening of the signal in the base decreases the transfer coefficient. The dependence of the modulus and phase of \(\beta\) on frequency is determined in an obvious way from (12).

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

REFERENCES

1. Chuenkov V. A. On the behavior of valence semiconductors of the germanium type in a strong electric field, ZhTF, vol. XXVIII, no. 3, 1958.

2. Ryabinkin Yu. S. The electric field in the base of planar transistors at low injection levels, Fizika tverdogo tela, vol. 1, p. 159, Academy of Sciences of the USSR, 1959.

3. Rittner E. S. Extension of the Theory of the Junction Transistor, Phys. Rev., vol. 94, no. 5, 1954.