Journal of ELECTRONIC MATERIALS
DOI: 10.1007/s11664-017-5904-z 2017 The Minerals, Metals & Materials Society
Comment on I. Lashkevych, O. Yu. Titov, and Yu. G. Gurevich, ‘‘Ohm’s Law for a Bipolar Semiconductor: The Role of Carrier Concentration and Energy Nonequilibria’’ [J. Electron. Mater., 46, 585 (2017)] C.H. SWARTZ1,2 1.—Materials Science, Engineering, and Commercialization Program, Texas State University, 601 University Drive, San Marcos, TX 78666, USA. 2.—e-mail:
[email protected]
In a recent publication [J. Electron. Mater., 46, 585 (2017)], a number of formulae are presented for the effective conductivity of a bipolar semiconductor sandwiched between two metal contacts. However, the results are shown to be nonphysical, and the explanation is traced to errors appearing in previous literature on the subject. Key words: Bipolar semiconductor, effective electrical conductivity, recombination, nonequilibrium charge carriers, nonequilibrium temperature
In a recent publication,1 the conductivity of a semiconductor sandwiched between two metal contacts is calculated, taking into account the effects of temperature gradients, finite thickness (d), and finite surface recombination velocity (S) at the metal–semiconductor interfaces. However, the results are nonphysical, and are found to be based upon errors made in the cited references. The metal–p-type semiconductor–metal structure is modeled in several different settings. In the isothermal (or high thermal conductivity) case, the structure is found to have an effective electrical conductivity given by the manuscript’s Eq. (12), 1 1 rp0 1=x sinh x ¼ 1þ ; r r0 rn0 rn0 cosh x þ sS=LD sinh x
ð1Þ
where rn0 and rp0 are the electron and hole conductivities at equilibrium, r0 is their sum, s is the bulk excess carrier lifetime, LD is the diffusion length, and x ” d/LD. This formula makes the surprising prediction that the p-type structure will have zero conductivity when the equilibrium n-type conductivity rn0
(Received August 19, 2017; accepted October 23, 2017)
approaches zero. Small values of rn0 might result, for example, from stronger p-type doping, larger bandgaps, or lower temperatures. It is seemingly nonphysical for the total conductivity of a p-type structure to be eliminated entirely as rn0 becomes smaller. The reason for this nonphysical result is that the conductivity formulas were originally derived for the case of no interaction between the valence band and the metal contacts.2 Without the ability of metal electrons to transition to or from empty valenceband states (holes), this essentially becomes a metal–insulator–metal structure. While an Ohmic contact between a metal and a ptype semiconductor can indeed be difficult to form in many practical cases, the difficulty comes from the difference between the Fermi level of the p-type semiconductor (near its valence band) and the Fermi level of the metal; That is, a low transition rate of valence-band charge carriers to the metal implies an offset between the Fermi levels of the two materials.3 Two such materials placed in contact will soon reach thermal equilibrium, where the offset will be balanced by an electric field. This case is more realistically accounted for by the Schottky model for metal–semiconductor interfaces, as seen in Fig. 1.4
Swartz
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n0 þ p0 k0 ; L0 ¼ sT0 n0 p0 fEg
ð2Þ
where f is a function of the dominant mobility scattering mechanism for the charge carriers, n0 and p0 are the equilibrium electron and hole concentrations, T0 is the temperature, and Eg is the bandgap energy. Small disturbances to the temperature T0 + dT are said to obey the diffusion equation given by the manuscript’s Eq. (23) d4 dT d4 dT ¼ 0: ð3Þ dx4 dx2 However, it is apparent that L0 is zero in the limit of large Eg, as if thermal diffusion were to halt in highbandgap materials. There is no indication that the value of L0 applies only to low bandgaps, since Eg is described only as being much greater than the thermal energy. This nonphysical result was originally derived using the following expression for recombination5: 1 n0 p 0 dn dp Eg dT R¼ þ 3þ ; ð4Þ s n0 þ p 0 n0 p 0 T0 T0 L20
Fig. 1. Band diagram of a metal–p-type semiconductor contact in (a) the Schottky model and in (b) the model used by the manuscript’s authors, which only has charge carrier transport between the conduction band and the metal, as indicated by the arrow.
where dn and dp are the nonequilibrium excess carrier concentrations. This expression was in turn derived for the case of radiative recombination. However, the details of its derivation were unclear,6 and it appears to be at odds with standard expressions for spontaneous radiative recombination rates, which state that the rate R should go up, not down, when the temperature T0 increases.7,8 For these reasons, the versions of Ohm’s law given for bipolar semiconductors in the manuscript are not likely to be applicable to any system of physical interest. REFERENCES
The authors have assumed the charge neutrality approximation, with zero electric field throughout the structure, noting that the Debye length is less than LD and d. However, any depletion width made by the band offset must also be much less than LD in order for the effects of junction fields to be ignored. The formula for the adiabatic case, given in the manuscript’s Eq. (28), is similar to the isothermal case, with one key difference being the replacement of LD with the generalized diffusion length L0. The manuscript’s Eq. (30) gives L0 for the case of low thermal conductivity (k0) as
1. I. Lashkevych, O. Yu Titov, and Yu.G. Gurevich, J. Electron. Mater. 46, 585 (2017). 2. Y.G. Gurevich, G.N. Logvinov, G. Espejo, O.Y. Titov, and A. Meriuts, Semiconductors 34, 755 (2000). 3. B.G. Streetman and S.K. Banergee, Solid State Electronic Devices, 6th ed. (Upper Saddle River: Pearson Education, 2006), p. 227. 4. S.M. Sze and K.K. Ng, Physics of Semiconductor Devices, 3rd ed. (Hoboken: Wiley, 2007), p. 135. 5. Yu.G. Gurevich and I. Lashkevych, J. Electron. Mater. 44, 1456 (2015). 6. I.N. Volovichev, G.N. Logvinov, O.Y. Titov, and Y.G. Gurevich, J. Appl. Phys. 95, 4494 (2004). 7. Y.P. Varshni, Phys. Stat. Sol. 19, 469 (1967). 8. P. Wurfel, J. Phys. C Solid State Phys. 15, 3967 (1982).
