621.382.8 : 621.385.1 The Institution of Electrical Engineers Paper No. 318OE Mar. 1960 A MODERN APPROACH TO SEMICONDUCTOR AND VACUUM DEVICE THEORY By R. D. MIDDLEBROOK, M.A., M.S., Ph.D. {The paper was presented at the INTERNATIONAL CONVENTION ON TRANSISTORS AND ASSOCIATED SEMICONDUCTOR DEVICES, 22nd May, 1959. The written version was received 28th July, 1959.) SUMMARY Vc — Collector voltage. An integrated approach to the understanding of charge-controlled Ve = Equivalent grid-plane voltage. electronic devices is presented. Although only vacuum triodes and Vg = Grid voltage. diffusion-type transistors are discussed in detail, the methods suggested Vj = Voltage across transition region. are also applicable to gas-filled and multi-electrode vacuum structures, V = (i) Anode voltage (Section 4). (ii) Voltage to surface-barrier and to drift-type transistors, and to space-charge- p limited solid-state devices. The treatment is tutorial in nature, and across charge-neutral /^-region (Section 6). begins with the development of general equations of current flow Vs = Diffusion potential (Fig. 8). applicable in any medium. The principles of charge-controlled devices Wb = Barrier height (Fig. 8). are then summarized, and a general functional relationship between Wg = Energy gap between valence and conduction the total charge in transit and the transit time is developed. These bands. results are then applied in turn to vacuum and semiconductor diodes Wf = Fermi level. and triodes to derive in a remarkably simple and consistent manner W, = kTje. the salient features of their operation. 'Ideal' vacuum triode and <f> = Work function of a metal. transistor structures are first discussed, and the voltage and current m amplification factors are then introduced as arbitrary parameters to </>s = Work function of a semiconductor. account for practical departures from ideality. Specific results Z = nQlpp, injection-level parameter. obtained are the d.c. characteristics and incremental equivalent dud2 = Distances from grid plane to cathode and plate. circuits for each device. The model established for the transistor is e = Magnitude of electronic charge. identical with the hybrid-77 circuit due to Giacoletto, and both low- and gm = Transconductance. high-level injection conditions are included. Finally, it is suggested i — Current density. that the transistor collector saturation current with open base is a /„ = Electron current density. more fundamental quantity than that with open emitter, and the k = Boltzmann's constant. temperature dependence of the base-emitter voltage is shown to be m — Mass of electron. linear at any injection level. Throughout, emphasis is on the principles involved and on the method of approach, and a particular effort is n = (i) Exponent in relation rt cc l/Q". (ii) Elec- made to present the development of the vacuum and the semiconductor tron density. devices in a completely analogous manner. /70 = Injected electron density. rij = Intrinsic carrier density. nn, np = Equilibrium electron density in /i-region and jp-region. LIST OF PRINCIPAL SYMBOLS p = Hole density. For simplicity in mathematics, the unit-area approach is used Po = Hole density at edge of transition region throughout; the few departures from this will be obvious. [eqn. (102)]. Ci, C = Input and output intrinsic capacitances. pn, pp = Equilibrium hole density in n-region and 2 /7-region. Ceg, Cep, Cec = Capacitances from equivalent grid plane to grid, anode and cathode (Fig. 6). v = Charge velocity. w = (i) Anode-cathode distance in vacuum diode C,(C,C, Cte) = Transition-region capacitance (at collector and emitter). (Section 4). (ii) Thickness of charge-neutral /7-region (Section 6). D(Dn, Dp) = Diffusion constant (of electrons and holes). E = Electric field. ws = Transition region thickness in equilibrium (Figs. 8 and 10). G|, G2 = Input and output intrinsic conductances. x = Transition-region thickness. I\, 12 = Input and output current densities. t j8 = Ratio of incremental collector and base Is = Saturation current density [eqn. (91), metal- semiconductor; eqn. (118), p-n junction]. currents. 8 — Geometry-dependent parameter defined in Na = Acceptor density. eqn. (78). Nd — Donor density. e(e0) = Permittivity (of free space). Q = Total charge in transit. = T — Absolute temperature. eS\Cep CJC2 = Ratio of incremental anode and grid V = Potential. voltages at constant current. r, = Average transit time. Vy, V2 = Input and output voltages. p = Charge density. Va = Vacuum-diode anode voltage. p = Fixed negative charge density. Vb = (i) Semiconductor diode voltage (Sections 5 a and 6.1). (ii) Base-emitter voltage (Sec- pd = Fixed positive charge density. tion 6.2). pe = Net charge density. pn = Mobile negative charge density. pp — Mobile positive charge density. Dr. Middlebrook is Associate Professor of Electrical Engineering at the California Institute of Technology, Pasadena, California. fxp) = Mobility (of electrons and holes). [887] MIDDLEBROOK: A MODERN APPROACH TO SEMICONDUCTOR AND VACUUM DEVICE THEORY (1) INTRODUCTION tions, is emphasized. General but simple results applicable to Whenever a new field of endeavour is opened up, much effort both low- and high-level injection are presented. The properties is expended in investigating all possible avenues of advance in of the practical semiconductor triode (transistor) are obtained the understanding and application of the new ideas. Inevitably very simply from those of an 'ideal' triode (whose base draws no some of these avenues are later recognized to be more funda- current and exerts sole control over the collector current) by mental or important than others: only the cumulative experience introducing the current amplification factor as an arbitrary of many workers can lead to a realization of the proper perspec- geometry-dependent parameter. A feature of the treatment is tive into which the many separate items of knowledge in the that the transistor is discussed throughout in terms of the field should be placed. common-emitter configuration, so avoiding the artificial and Such a crystallization of the salient features in the theory and unrealistic technique of first deriving all the parameters in the application of the ordinary vacuum tube has long since been common-base connection and then transforming them for the attained. This happy condition is at present evolving in the common-emitter one. Application of the results of Section 2 theory and application of transistors, although the process is leads to an incremental equivalent model identical with the far from complete, since new semiconductor devices continue to hybrid-7r model introduced by Giacoletto.3 Equations for the appear at a rapid rate. Upon contemplation of the basic model element values are given for both low- and high-level theories of the vacuum tube and the transistor, it soon becomes injection conditions. Finally, the temperature dependences of apparent that a further step in the evolution of each is desirable— the collector saturation current and base to emitter voltage are that of combining the two theories into one, in which the tube discussed, and it is suggested that the collector current with open and the transistor are examples of a general active device. base is a more fundamental quantity than that with open Although many workers have suggested similarities between emitter. tubes and transistors, the way to a really integrated treatment of Although only a few specific devices are discussed in the paper, both devices has been suggested only recently by Johnson and the general approach is applicable to all types of charge-con- Rose1 in the United States and by Sparkes and Beaufoy2 in trolled device, such as gas tubes and space-charge-limited England, who have pointed out that the tube and the transistor solid-state structures. Some comparisons between gaseous and are both fundamentally charge-controlled devices, and not, as semiconductor devices have already been drawn by Webster,4 had previously been believed, voltage-controlled and current- and the analogy between space-charge-limited current flow in controlled devices respectively. vacuo and in a semiconductor was first discussed by Shockley The paper represents an attempt to follow up the basic and Prim.5 It is felt that the approach here described is charge-control approach, to integrate the treatment of current capable of even further generalization in order to integrate the flow in vacuo and in a solid, and to derive in a remarkably theory of a large class of electronic devices into a complete simple manner the salient features of vacuum and semiconductor whole. diodes and triodes. In Section 2 the basic equations of current flow are established, applicable to charge motion in vacuo, in (2) BASIC EQUATIONS OF CURRENT FLOW solids, and in electrolytes. The Einstein relation between the In order to cause a current to flow, a potential gradient mobility and the diffusion constant of a carrier follows directly must exist. To determine the current density, /, at any from a discussion of the mechanisms of drift and diffusion flow. point, the charge density, p, and charge velocity, v, must be In Section 3 some results of Johnson and Rose on the basic known. There are thus, in general, four unknowns involved properties of charge-controlled devices are reviewed, and it is in the solution for the current in a given physical device, shown that the transit time of charged carriers across the active namely /, p, v, and V, some or all of which may be functions region is proportional to the nth power of the total charge in of positions and of time, and so four equations relating these transit, where n = 0 when the current is diffusion limited, 0 • 5 variables are required (magnetic fields are not considered). when it is space-charge limited in vacuo, and 1 when it is space- The simultaneous solution of these equations will be subject to charge limited in a solid. A simple incremental equivalent boundary conditions imposed by the particular physical device. circuit is presented which is independent of the type of charge- In a given structure the sources of charge density may be controlled device.