Page 315 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers
P. 315

Interacting Subsystems
                             Adding these two expressions together yields the Shockley ideal diode
                             equation
                                              J =  J ( x ) +  J ( – x )
                                                   P  n   N    p
                                         D      D          qV
                                                       
                                                               Applied
                                           P
                                                  N
                                        
                                     =  q ------- p n0  +  -------n  exp  --------------------- –  1  
                                                       
                                                     p0
                                          L  P  L  N         k T               (7.177)
                                                                B
                                                     qV  Applied  
                                            =  J  exp  --------------------- –  1
                                               0      k T       
                                                        B
                             where    the   reverse   saturation  current   density   is
                                   {
                             J =  qp D ⁄   L +  n D ⁄  L }  . Equation (7.177) is plotted in
                                     n0
                                                p0
                              0
                                            P
                                        P
                                                   N
                                                       N
                             Figure 7.21.
                                                                  (
                                                                 JV     )
                                                                    Applied
                Figure 7.21. The shape of the
                                                                 J
                                                                  0
                Shockley ideal diode equation,
                                                                                V
                (7.177), showing the reverse satu-                               Applied
                ration current density J  .
                                  0
                Junction     The space-charge region, with its separated positive and negative
                Capacitance  charges, represents a nonlinear capacitor, i.e., a capacitance dependent on
                             the applied voltage. In general, the nonlinear capacitance is defined as
                                    ⁄
                             C =  dQ dV  , i.e., the differential change in charge Q   that would result
                             for a differential change in voltage  V  . The total negative charge in the
                             space-charge region, per unit abrupt junction length, is:


                                                   +  -
                                                  N N q                     2k T
                                       -           d  a            −          B
                              Q =  qx N =   ( 2ε) ------------------------- ( V )  + V Applied  –  -------------  (7.178)
                                    P
                                       a
                                                            bi Abrupt
                                                       +
                                                   -
                                                ( N +  N )                   q
                                                   a   d
                             The derivative of the charge w.r.t. the applied voltage gives
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