. The Bell System technical journal . have KdE(x)/dx = dp. () 816 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Integrating this from the region where E is zero gives KdE(x, t) = / dp{x, t) () Equation () states that the dielectric displacement at x is equal tothe excess charge between the potential maximum and x. Evidentlyduring the transient following Fig. (a), the rate of change of thisextra charge is —8J(x, t) since the dc current is flowing in at the leftand an excess current 5J flows out at the right. Hence we have KdhE/dt - -bJ, = —{bpu + pbn).For the change in drift
. The Bell System technical journal . have KdE(x)/dx = dp. () 816 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 Integrating this from the region where E is zero gives KdE(x, t) = / dp{x, t) () Equation () states that the dielectric displacement at x is equal tothe excess charge between the potential maximum and x. Evidentlyduring the transient following Fig. (a), the rate of change of thisextra charge is —8J(x, t) since the dc current is flowing in at the leftand an excess current 5J flows out at the right. Hence we have KdhE/dt - -bJ, = —{bpu + pbn).For the change in drift velocity we may writehu = {du/dE) 8E = iM*dE. () () For high electric fields u increases less rapidly than linearly with E and/x* is less than the low-field mobility. For very high fields fx* is nearlyzero and there are theoretical reasons for thinking that there may be arange in which fi* is negative. We shall return to this point in the nextsection. In Fig. we show a diagrammatic representation of the transient so-. r (b) 0 f dp ft 0 . 3 Fig. — Graphical representation of the dependence of SE upon time. () NEGATIVE RESISTANCE IN SEMICONDUCTOR DIODES 817 lilt ion. Each of the dashed lines represents the decay of 8E as measured ina moving coordinate system: Thus wo consider 8E measiued at a positionx(so + 0; this is a position that moves with the dc velocity 7/. This BEis evidently expressed in terms of dE(x, t) by writing x = x(so -\- /): dE in moving system = 8E„Xso, t) = dE[x(so + t), t]. () The differential equation for SE^ is (d/dt) 8Em = (d8E/dt), + (d8E/dx)tdx/dt, = {d8E/dt), + {d8E/dx)tU, = - {u8p + p8u)/K + (8p/K)u, = -{pn*/K)8E = -v8E, where the quantity V = pn*/m () is an effective dielectric relaxation constant being the differential con-ductivity pn* divided by the permittivity K. E\idently j/ is a function of position x only and may be expressed asv(s) through the dependence of x upon s. Thus we may write (d/dt)8EUso, t) = -v(s
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