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Electrical Behavior of pn-Junctions To consider what happens electrically in a pn-junction, one can, again, consider a thought experiment. Suppose that two blocks of semiconductor of opposite extrinsic doping, i.e., one block is n-type and one is p-type, are initially separated widely. If these blocks are then joined, a net transfer of carriers from one block to the other must occur in order to establish equilibrium. Obviously, this situation is similar to the “simpler” cases of a metal-semiconductor contact and an MOS capacitor. Furthermore, just as in these previous cases, if the whole system is to be in equilibrium, then the Fermi level must be constant throughout the entire combined volume of the two blocks. It comes as no surprise that this requires that valence and conduction bands must become bent in the region of the junction. This situation is illustrated below: Evac EC E Fn E F Ei E Fp EV n-type p-type pn-junction Fig. 55: Band diagrams for an unbiased pn-junction The position of the junction is determined as the point that the intrinsic Fermi level and the actual Fermi level intersect. Of course, this defines the exact position at which the semiconductor changes from one type to the other, i.e., it is intrinsic at the junction. Moreover, even though this structure is hypothetically constructed by joining separate blocks of material, the physical mechanism used to form a pn-junction is immaterial to its electrical behavior. Obviously, in practice dopant diffusion processes can be used to form pn-junctions intentionally. An Unbiased pn-Junction A clear consequence of the existence of a junction is the existence of an internal or intrinsic electric field in the junction region. Returning to the previous thought experiment, one can easily see the cause of this behavior. Suppose that the two opposite type semiconductor blocks are brought together suddenly. At the instant of initial contact, mobile carriers are not in equilibrium since the Fermi levels do not coincide, i.e., carrier concentrations are constant right up to the interface between the two blocks. Furthermore, ignoring the effects of non-equilibrium generation-recombination processes, electron and hole fluxes, Fe and Fh, everywhere within the joined blocks of semiconductor (including the region near the junction) are governed by linear transport equations of the form: n Fe De neE x p Fh Dh ph E x Here, Dx and x are carrier diffusivity and mobility (x is either e or h, thus, denoting either electrons or holes), and E is electric field. Clearly, the first term on the right hand side of both of these expressions comes from Fick’s Law and describes diffusion of mobile carriers due to a concentration gradient. Similarly, the second term comes from Ohm’s Law and describes drift of mobile carriers under the influence of an electric field, E. Obviously, the change of sign of the drift terms is due to opposite charges of electrons and holes. In contrast, there is no sign change for diffusion terms since diffusion is independent of carrier electrical charge. (These equations illustrate coupling of different thermodynamic forces with the same transport flux.) Of course, corresponding current densities, je and jh, are trivially obtained from fluxes simply by multiplying by carrier charge: n je qDe qne E x p jh qDh qph E x Clearly, the ohmic terms have the expected form of electric field divided by resistivity. Within this context, it is clear that immediately when the two blocks touch, majority carriers from each side must diffuse across the junction since a large concentration gradient exists. (Of course, when majority carriers diffuse to the other side of the junction, they then become minority carriers since, by definition, the semiconductor type changes at the junction.) Initially, there is no drift since there is no intrinsic electric field. However, carrier mass action equilibrium requires that a large non-equilibrium excess concentration of minority carriers cannot be built up. Therefore, excess minority carriers rapidly recombine with majority carriers resulting in the depletion of majority carriers in both types of semiconductor in the vicinity of the junction. Obviously, depletion is greatest at the junction and decreases as distance from the junction increases. Furthermore, just as in the case of an MOS capacitor or metal-semiconductor contact, depletion also causes ionized impurity atoms to become uncovered. This implies the existence of a space charge within the depletion region around the junction. Naturally, an electric field must exist in this region of space charge. (One should note that the terms “space charge region” and “depletion region” are synonymous and describe different aspects of the same physical phenomenon.) Obviously, in the thought experiment the depletion region begins to form just as soon as the two blocks of semiconductor touch, however, this process cannot continue indefinitely. Returning to the transport equations, one observes that as the space charge increases, the strength of the associated internal electric field must also increase. Therefore, the magnitude of the drift component of net carrier current increases in response to the build up of an internal electric field. Furthermore, the drift current naturally opposes the diffusion current. Thus, the internal electric field can increase only until drift and diffusion currents become exactly equal and opposite, i.e., net carrier current vanishes. At this point, carrier equilibrium is established in the region of the junction as well as elsewhere throughout the volume of the semiconductor and no further net carrier transport occurs. The electrostatic characteristics of an unbiased pn-junction are illustrated in the following figure: E F r E V Fig. 56: Electrostatic characteristics of an unbiased pn-junction It is clear from the band diagram for an ideal abrupt pn-junction, that the space charge density, r, in the junction is dipolar and abruptly changes sign from positive on the n- type side to negative on the p-type side. Of course, Maxwell’s equation relating electric field, E, to charge density is: E r x Thus, the electric field is in the same direction (defined as positive to the right) on both sides of the junction, i.e., E “points” from n-type to p-type. Obviously, E reaches a maximum value exactly at the junction. Of course, the existence of the internal field implies a “built-in” or diffusion potential, V: V E x 2V r 2 x Clearly, V is more positive on the n-type side of the junction. It should be noted in passing, that in the preceding figure, net doping levels on each side of the junction are represented as approximately equal in magnitude. In actual practice, this is not usually the case. Typically, one doping concentration will be much larger that the other, but the total amount of uncovered positive and negative charges in the junction depletion region must be the same irrespective of doping. Hence, in such a case the junction must be asymmetric with the width of the depletion layer on the lightly doped side being much larger than on the heavily doped side. However, this does not substantially change the situation. Similarly, diffused junctions are not abrupt but, generally are “graded” (i.e., the net doping concentration decreases smoothly from some higher concentration far from the junction, to a lower concentration in the junction region). Again, this does not greatly change the behavior of a pn-junction (although graded junctions do exhibit some behavioral characteristics that are different than those of abrupt junctions). Naturally, Fermi potentials, Fn and Fp can be defined on each side of the junction, and far away from the junction region (i.e., far outside the depletion layer), one can write: 2 2 EF Ein kT Nn Nn 4ni Fn ln q q 2ni 2 2 Eip EF kT N p N p 4ni Fp ln q q 2ni Here, Nn and Np are net donor and net acceptor concentrations on each side of the junction and Ein and Eip are corresponding intrinsic Fermi energies. (Recall that the Fermi potential, F, is defined for an extrinsically doped semiconductor as |Ei EF|/q.) Clearly, the maximum value of the diffusion potential, Vpn, is just the sum of n-type and p-type Fermi potentials, hence: kT N N 2 4n2 kT N N 2 4n2 V ln n n i ln p p i pn Fn Fp q 2n q 2n i i Naturally, this can be simplified if extrinsic doping levels far exceed the intrinsic carrier concentration: kT Nn N p Vpn ln 2 q ni Obviously, Vpn is analogous to the contact potential defined in the case of a simple metal- metal junction (i.e., contact). Of course, it arises from the effective work function difference between extrinsic n-type and p-type semiconductor. However, since electric fields can penetrate semiconductors, in contrast to a metal-metal junction, a depletion layer of measurable thickness appears at a pn-junction and the potential difference is “stretched out” over a much larger volume of material. Depending on actual levels of extrinsic doping, for silicon Vpn is typically in the neighborhood of 0.6-0.7 volts. The Effect of Forward and Reverse Bias on a pn-Junction So far, only an unbiased pn-junction has been considered, however, if a voltage is applied across the junction, then one expects some current to flow. Within this context, the internal potential acts as a barrier to current flow from the p-type side to the n-type side of the junction. Hence, the existence of the diffusion potential indicates that a pn- junction can be expected to conduct electrical current through the junction more easily in one direction than the other.

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