a mathematical model for the influence of deep-level...

118 J. Electrochem. Soc., Vol. 139, No. 1, January 1992 �9 The Electrochemical Society, Inc.

photocatalytic efficiency of different TiO2 powders used in photoassisted oxidation processes may have its origin in their different distribution of surface traps, where electrons are immobilized and from which electrons are trans- ferred to oxygen. Materials with high densities of shallow (-0.1-0.3 eV) traps are likely to be most effective. However, catalysis of 02 reduction by group VIII metals should in all cases substantially increase the quantum efficiency of the photoassisted oxidation of organics by molecular oxygen.

Acknowledgment This work is supported by the Division of Advanced

Energy Projects, Office of Basic Energy Sciences, U.S. De- partment of Energy under Grant DE-FG05-90ER12101- A0002.

Manuscript submitted June 10, 1991; revised manuscript received July 31, 1991.

The University of Texas at Austin assisted in meeting the publication costs of this article.

REFERENCES 1. H. Gerischer and A. Heller, J. Phys. Chem., 95, 5261

(1991). 2. D. M. Eagles, J. Phys. Chem. Solids, 25, 1243 (1964). 3. D. C. Cronemeyer, Phys. Rev., 87, 876 (1952). 4. P. Salvador, J. Appl. Phys., 55, 2977 (1984). 5. "Handbook of Chemistry and Physics," 62nd Ed.,

p. D-229, RCR Press, Boca Raton, FL (1981).

6. M. W. Ribarsky, in "Handbook of Optical Constants of Solids," p. 795, Academic Press, Inc., New York (1985).

7. C. F. Bohren and D. R. Huffman, "Absorption and Scattering of Light by Small Particles," p. 166, John Wiley & Sons, Inc., New York (1983).

8. Ibid., p. 130. 9. H. Gerischer, J. Phys. Chem., 95, 1356 (1991).

10. S. C. Abrahams and J. L. Bernstein, J. Chem. Phys., 55, 3206 (1971).

11. B. Parkinson, F. Decker, J. F. Juliao, M. Abramovich, and H. C. Chagas, Electrochim. Acta, 25, 521 (1980).

12. D. Tafalla and P. Salvador, Ber. Bunsenges. Phys. Chem., 91, 475 (1987).

13. M. Gr~tzel, in "Photocatalysis," N. Serpone and E. Pelizzetti, Editors, p. 151, Wiley, New York (1989).

14. R. A. Marcus, J. Chem. Phys., 24, 1966 (1956). 15. N. Sutin, Progr. Inorgan. Chem., 30, 441 (1983). 16. "Standard Potentials in Aqueous Solution," A.J .

Bard, R. Parsons, and J. Jordan, Editors, Marcel Dekker, Inc., New York (1985).

17. J. Lind, X. Shen, G. Merenyi, and B. P. Jonsson, J. Am. Chem. Soc., 111, 7654 (1989).

18. M. O. Finklea, in "Semiconductor Electrodes," M. O. Finklea, Editor, p. 71, Elsevier Sci. Pub., Amsterdam (1988).

19. I. Izumi, W. W. Dunn, K. O. Wilbourn, F. F. Fan, and A. J. Bard, J. Phys. Chem., 84, 3207 (1980).

20. D. W. Bahnemann, J. MSnig, and R. Chapman, ibid., 91, 3782 (1987).

A Mathematical Model for the Influence of Deep-Level Electronic States on Photoelectrochemical Impedance

Spectroscopy I. Theoretical Development

D. Bivings Bonham *'1 and Mark E. Orazem **'2

Department of Chemical Engineering, University of Virginia, Charlottesville, Virginia 22901

ABSTRACT

A mathematical model is developed to calculate the impedance response of a semiconductor electrode to a sinusoidal current perturbation under subbandgap monochromatic illumination. The model accounts explicitly for electron and hole transport as well as generation and recombination through band-to-band mechanisms and through bulk deep-level electron acceptors of specified energy. The model results are compared to experiment and illustrate how the impedance response obtained under monochromatic subbandgap illumination can be used to identify the energy, density, distribution, and recombinat ion rate constants associated with deep-level electronic states. This may have application to in situ characterization of semiconductor-electrolyte interfaces and to characterization of solid-state materials and devices.

Electrochemical impedance spectroscopy, coupled with monochromatic subbandgap illumination, may provide a room temperature approach for characterizing deep-level and interfacial electronic states in large bandgap semiconductors. The electronic states of interest lie within the bandgap of the semiconductor. The goal is to determine that these states exist as well as to determine their concentration distribution and the associated rate constants for electronic transitions. Knowledge of these parameters is essential for the engineering of many electronic devices. For example, deep-level states are undesirable when they facilitate electronic transitions which reduce the efficiency of photovoltaic cells. In other cases, the added reaction pathways for electrons result in desired effects. Electrolu- minescent panels, for example, rely on electronic transitions that result in emission of photons. The energy level of the states caused by introduction of dopants determines

* Electrochemical Society Student Member. ** Electrochemical Society Active Member. 1Present address: Department of Chemical Engineering, The

Johns Hopkins University, Baltimore, Maryland 21218. 2 Present address: Department of Chemical Engineering, Univer-

sity of Florida, Gainesville, Florida 32611.

the color of the emitted light. Electroluminescent panels represent one approach for development of thin-film color television screens. Interfacial states play a key role in elec- troluminescence, and commercial development of this technology will hinge upon understanding the relationship between fabrication techniques and the formation of deep-level states.

The impact of deep-level states can be significant, even in concentrations that are very low by normal chemical standards. Several states can be associated with a chemical species, and such states may also appear as a result of vacancies or other crystalline defects. Traditional chemical means of detection, therefore, do not provide complete identification of deep-level electronic states. The techniques commonly employed to detect deep-level states [such as application of Mott-Schottky theory, see e.g. (1, 2), deep-level transient spectroscopy (DLTS) (3), or photocapacitance spectroscopy (4-9)] tend to be electrical in nature since it is through their electronic behavior that these states influence device performance. These techniques rely on determining the change of space charge capacitance associated with ionization of deep-level states. The capacitance is usually determined from a single, high-fre-

J. Electrochem. Soc., Vol. 139, No. 1, January 1992 �9 The Electrochemical Society, Inc. 119

quency measurement. Since the charge held in the states may be very small as compared to that associated with doping species, the sensitivity of capacitance-based techniques requires precise measurement of the imaginary part of the impedance response. The real part of the impedance response has been shown to be more sensitive to ionization of deep-level states, particularly at low frequencies. Nicollian and Goetzberger (10), DeClerck et al. (11), and Nagasubramanian et aI. (12) have attributed variations observed in the real component of the impedance with potential to midbandgap surface states.

This paper addresses a light-enhanced form of electrochemical impedance spectroscopy. In this technique, the effect of photonic excitation of electronic transitions by light at selected wavelengths is detected by impedance spectroscopy applied over a broad frequency range. The photonic energy of the light used is less than the bandgap energy; therefore, any changes in the impedance spectrum with i l lumination can be attributed to transitions involving energy levels within the bandgap. This method differs from the more commonly used DLTS in that the wavelength of monochromatic subbandgap light is varied to ex- cite electronic transitions at a fixed temperature (e.g., room temperature): whereas, in DLTS, temperature is varied to change the occupancy of the states. A broad range of frequency (with emphasis on lower frequencies) for the impedance measurements is used instead of measuring an effective capacity at a single high frequency. The use of a broad frequency range is the essential distinction between this approach and photocapacitance spectroscopy.

In principle, physical parameters such as the energy at which deep-level states exist, the concentrations of these states, the rate constants for transitions involving these states, and the distribution of these states could be ex- tracted from data obtained using light-enhanced impedance spectroscopy. Such detailed interpretation of impedance data required an extension to currently available models for this system. The overall goal of this work, therefore, was to develop a comprehensive mathematical model which treats the physical phenomena governing the response of the system without use of overly restrictive assumptions. This work was motivated by two goals: (i) to support the current application of Mott-Schottky theory and photocapacitance spectroscopy for identification of deep-level states and (ii) to provide a framework for identification of deep-level states through interpretation of both the real and the imaginary parts of the impedance spectrum. The theoretical development and validation of the model by qualitative and quantitative comparison of model results to experimental data is presented in this paper. The application of the model to assess Mott- Schottky methods of analysis is presented elsewhere (13).

Modeling Overview The usual approach in developing a general mathemati-

cal model for the impedance response of a given system is sequential. The (usually nonlinear) system of equations is first solved under the assumption of a steady state. The impedance response is obtained by linearization of the sinusoidal steady-state equations about the steady-state solution. The work presented here provides an extension to a series of steady-state and impedance models that are re- viewed in this section.

Steady-state models . - -Both analytic and numerical models have been developed for the steady-state behavior of semiconductor systems (see, e.g., Ref. (14) for a review of both analytic and numerical modeling work). Most of these models are extensions to a model described by G~rtner (15) for a reverse-biased p-n semiconductor junction. This model allows calculation of current density based on a drift component for carriers generated inside the depletion region and a diffusion component due to minority carriers generated outside this region. Major constraints are that recombination of carriers must be negligible in the space-charge region and that the electrochemical potential (quasi-Fermi level) must be constant throughout the phase.

Recent models have improved upon the Gartner model for surface state analysis by using kinetic expressions for steady-state trapping and by calculating changes in the Helmholtz layer [e.g., Kelly and Memming (16), Li and Peter (17), and Kobayashi et al. (18)]. The same restrictions of the G~rtner expression still apply, however, and these restrictions prohibit t reatment of bulk midbandgap electronic states. McCann and Haneman (19) have developed analytic expressions to circumvent these restrictions, but a depletion layer approximation is still used.

One-dimensional steady-state numerical studies have been presented by Orazem and Newman (20, 21, 14), Orazem (22), and Schwartz et al. (23). These models circumvent the restrictions necessary for the analytic solutions described above. Bulk and surface recombination reactions were included in all cases, but allowances were not made for changes in potential distribution due to deep- level state charging. The work by Schwartz et al. was in- tended for solid-state applications and is distinguished by the use of Auger recombination terms for very high light intensities. The influence of limitations to electrolytic mass transfer was examined by Orazem (22) for charge transfer between the electrolyte and valence band, conduction band, or surface states.

Laser and Bard (24, 25) present the results of steady- and unsteady-state numerical models for the semiconductor electrode. Their work illustrates the difficulty encountered in treating the coupling between the space charge and electrically neutral regions of the semiconductor in that convergence was obtained only for thicknesses equal to the space-charge region thickness.

Impedance models . - -The simplest and most common method for analyzing the impedance response of semicon- ducting electrodes and other electrochemical systems uses electrical circuit analogues. Electrochemical systems often have the same impedance response as that expected for electrical circuits composed of capacitors, resistors, and inductors. A specific circuit is deemed to provide a good model if the component values, obtained by nonlinear regression of the data, are "independent of frequency.

Interpretation of circuit component values in terms of physical parameters such as diffusivities and rate constants generally requires development of a model that in- corporates the physics and chemistry assumed to be rele- vant to the system. Dare-Edwards et al. (26), Kobayashi et al. (18), Li and Peter (17), and Allongue and Cachet (27) have developed and applied analytic impedance techniques based on perturbations of steady-state expressions. Dare-Edwards et al. (26) obtained electrical circuit component values in terms of interfacial and bulk semiconductor parameters by comparing solutions obtained from transport and kinetic expressions appropriate for an equil- ibrated, ideally-polarizable semiconductor containing surface states with those for a suggested equivalent circuit (see also Kobayashi et al. (18)). Analytic treatment of the effect of i l lumination or bulk states required more restrictive assumptions. The need for restrictive assumptions is lessened b y numerical solution of the governing equations. Transport-based numerical techniques have been employed by Macdonald (28, 29) and Bonham and Orazem (30) to describe the impedance behavior for ideally polarizable semiconductor electrodes under dark conditions. The system modeled by Bonham and Orazem (30) was an ideally polarizable semiconductor electrode containing bulk electronic states with a single energy level and an arbitrary spatial distribution. This model extended the work of Macdonald (28, 29) by treating the transport of both majority and minority carriers with different values of diffusion coefficients and by treating generation and recombination through deep-level electronic states as well as through band-to-band mechanisms. The work presented here provides an additional extension by including the effects of subbandgap illumination.

Theoretical Development The equations which govern the impedance response

under subbandgap i l lumination of an ideally polarized semiconductor with a distribution of deep-level states at


fixed energy levels relative to the valence band energy are Ec presented in this section. The equations apply to both steady and unsteady conditions and are based on estab- lished expressions for single-crystal semiconductors. The relationships resemble those employed for dilute electrolytic solutions in that electrons and holes are treated as Et chemical species to which mass transport and mass action principles can be applied.

Mass t ranspor t . - -Conserva t ion of species i is given by

c3Ci C3Ny i - + Gi [1] Iv

Ot Oy

where G~ represents the net rate of generation of species i and the flux Ni of species i is given by

dq dO Ni = - D i ~ y - uiziFci dy [2]

In a manner consistent with the use of dilute solution theory, the diffusivities D~ and mobilities u~ were assumed to be related through the Nernst-Einstein equation

Di = RTui [3]

The use of these equations is consistent with the assumption that the semiconductor is not degenerate. This assumption can be relaxed by including expressions for electron and hole activity coefficients [see, e.g., Res (31) and (32)].

Poisson's equation

02ap F = - - - [ p - n + Nd -- Na] [4]

Oy 2 esc

was used to relate the electrostatic potential to the charge held within the semiconductor. The scaling length for this system, found by making the governing equations dimensionless, is given by the Debye length

= [ .1 ~ LF2(Nd - N,) J [5] and

The term (Nd - N,) was taken here to include the charge associated with partially ionized midbandgap acceptors (which may be a function of applied potential) as well as the completely ionized dopant species (which may have an arbitrary distribution, but is usually assumed to be independent of operating conditions).

Electronic transi t ions . - -Calculat ion of a rate expression for G~ requires the choice of a kinetic framework. In this work, electrons are allowed to pass between the conduction band (with energy Ec), the valence band (with energy E0, and the deep-level species (with energy Et). A general scheme for the various electron transitions associated with this approach are shown in Fig. 1. The rates for electronic transitions between the various energy levels can be described by applying mass action principles [see, e.g.,Grove (33)] to give

rl = klc~ [6]

r 2 = k 2 ( c ~ - c ~ ) p [7]

r 3 = k ~ ( c ~ - c ~ ) [8]

r4 = k4c~n [9]

r s = k 5 [10]

and

r6 = k6np [11]

where ki is the rate constant of reaction i, c~ is the concentration of positively charged, deep-level donor species, c~ is the total concentration of deep-level donors, n is the electron concentration, and p is the hole concentration.

@ C

Fig. 1. Generalized reaction scheme showing electronic transitions between the conduction bandedge Ec, the valence bandedge Ev, and energy Et associated with deep-level states.

Under an assumption of a steady state, the net rate of pro- duction for electrons (and holes) is given by

Ge_ = k~ [ 1 + k2k4c~ 1 k6(~ + tc2p) J (n~ - up)

B2]

This expression shows that under equilibrium conditions np = n~.

An introduction of Eq. [6] through [11] requires selection of six rate constants associated with these expressions. This apparently arbitrary selection can be approached by deriving equil ibrium expressions to relate the rate constants for the reversible, homogeneous reaction pairs explicitly in terms of the energy differences between the valence band, deep-level species, and the conduction band, i.e.

k, E,2 = -~2 = Nvg exp L R T [13]

k3 Nc [ F(Et - Ec) ] E~4 - - exp / / [14]

k4 g L R-T -J

k5 [ F ( E v - Ec) ] E56 = ~ = N3krv exp

L R T [15]

where Eij is the equil ibrium constant for reaction pair i-j, g is the degeneracy associated with the deep-level state, Nc is the conduction band density of states, and N. is the valence band density of states. These expressions were de- rived by assuming thermal equil ibrium and substituting standard statistical expressions for electron, hole, and deep-level state concentration in terms of energy level. The numerical value for g is determined by the electronic character of the state, e.g., g = 4 for electron acceptors, and g = 2 for electron donors (34). Parameter variation studies were simplified by the assumption that the rate constants were also interrelated such that, given energy levels for the electronic states, all rate constants could be obtained from a single-rate constant. For example, the relationship

(E12~ '~2 k4 = k2 \E34/ [16]

was obtained by assuming that changes in the free energy of reaction associated with varying the energy of an electronic state are distributed equally between the activation energies for the forward and reverse directions. This is similar to the standard approach used to separate the free energy of an electrochemical reaction into chemical and electrical terms. The symmetry factor in this application is assumed to have a value of one-half (see e.g., Newman (35)). Similar expressions were developed for band-to- band recombination, i.e.

(E56~ 1/2 k2 = k8 \ E 1 J [17]

J. Electrochem. Soc., Vol. 139, No. 1, January 1992 �9 The Electrochemical Society, Inc. 121

The use of Eq. [17] to relate the homogeneous, band-to- band rate constant k6 to the corresponding deep-level constants k2 (and k0 is equivalent to assuming that the reaction cross section is the same for recombination through deep-level states as it is for direct band-to-band recombination. This, of course, is a major assumption which could easily be relaxed to account for enhanced rates of recombination through trap sites. In fact, the sensitivity of the model to the values for these rate constants provides an opportunity for experimental determination of these im- portant parameters through comparison between model calculations and experiment.

Treatment of optical excitation by light with photon energies smaller than the bandgap required expressions for the effective absorption coefficient. An expression was found from the literature (36) for the absorption coefficient m for the transition from deep-level acceptor states to the conduction band, i.e.

mel 3/2 m = 128"~ca - a* - - A ( h , - Eg + Ev) 1/2 [18]

\mr~ where

3/2 3/2 mr = rn~h + m~ 2 [19]

1 1 1 - + - - [ 2 0 ]

mrh mhh m e

and

1 1 1 - + - - [ 2 1 ]

mrl m~ m e

4 ~ e h 2 a* - [22]

m * e 2

(see the List of Symbols section for definitions of the constants). This expression was assumed to apply as well to deep-level donors and for transitions from the valence band to the deep-level state. This assumption, however, is not critical in this work since illumination has only been used to determine the position of the species within the bandgap. The combination of semiconductor thickness, deep-level species density, and absorption coefficients used in this work results in negligible absorption of subbandgap illumination, allowing the effects of subbandgap il lumination to be included as a modification of the rate constants in the expressions for r, and r3.

Numerical Method The dependent variables were separated into com-

ponents which represent their steady-state values and con- tributions from the sinusoidal steady state (28). Thus, for a current density given by

i = ~ + ~r exp (jcot) [23]

the concentration of electrons is given by

n = n + (~ + jfij) exp (jcot) [24]

Similar expressions were used for potential and the concentrations of holes, ionized electron acceptors, and ionized electron donors. In the above equations, an overbar represents the steady-state value, and a tilde represents the perturbation value. The actual concentration or potential at a given point in t ime and space is given by the real part of the expressions given above. This approach is similar to that taken to model the impedance response of elec- t rochemical systems [see, e.g., Ref. (37-40)].

In order to solve the governing equations, the above expressions were substituted and solved sequentially for the steady-state and the sinusoidal steady-state portions, respectively. The impedance

Z = Z~ + j Z j [25]

was calculated from real and imaginary components given by

Z~ - _. [26] ~r

and

zj - _ [273

respectively.

S t e a d y s t a t e . - - T h e governing equations were initially solved under the steady-state condi t ion , subject to the boundary conditions Np = O, d O / d y = 0, and i = 0 at the semiconductor-current collector interface (ohmic contact) and Nn = 0, �9 = 0, and d ~ / d y = -qsc/e~c at the semiconductor-electrolyte interface (ideally polarized contact). These conditions are appropriate for a semiconductor- mercury contact or for a semiconductor-electrolyte contact where the electrolyte is chosen such that no chemical reaction occurs. These equations were then linearized, written in finite-difference form and solved using the BAND algorithm developed by Newman (41) coupled with Newton-Raphson iteration. No linearization approxima- tions or assumptions involving depletion or neutral zones were required using this method for solution of the governing equations. The Debye length was used to scale the distances in the semiconductor crystal.

S i n u s o i d a l s t e a d y s t a t e . - - T h e t ime-dependent equations were solved for the response to a superimposed sinusoidal current by introducing expressions for the dependent variables (such as Eq. [24]) into the governing Eq. [1] through [11] and linearizing around the steady-state solution obtained in the previous step. The resulting set of linear ordi- nary differential equations was also solved with the BAND algorithm, and an iterative technique was used to mini- mize round-off errors. The boundarycondi t ions for the impedance calculations were given by Npj = Np.~ = 0, pj = p~ = 0, and n~ = n~ = 0,~ at the semiconductor-current-collector interface, and by Nnj =/~/.,r = 0, ~j = (Pr = O, d ~ j / d y = ~/e~c(O, and dCpr/dy = 0, at the semiconductor-electrolyte interface. Again, these conditions are consistent with an ideally polarized electrode where the superimposed current acts only as a charging current.

Results and Discussion The object of this work was to develop a mathematical

framework that would describe the impedance response of a semiconductor without requiring the use of electrical circuit analogues. This type of model can play a predictive role by suggesting experimental designs that may prove more sensitive to deep-level states. It can also play an edu- cational role by exploring the validity of less rigorous models for the capacitive behavior of semiconductors. These roles are explored in Ref. (13) and (45). This type of model can also be used for direct regression of experimental data, an approach being studied in our laboratory.

A key question that must be resolved is whether the physical phenomena incorporated within the model are sufficient to account for experimentally observed behavior. In this section, the results of the model are compared qualitatively to experimental results taken from the literature. The results of the model are shown to be consistent with an equivalent electrical circuit proposed for a semiconductor with surface states, with observation of frequency dispersion and shifts in Mott-Schottky plots attributed to surface states, and with observation of capacity shifts associated wi th photocapacitance spectroscopy. All numerical results discussed in this section are for an n-type, ideally polarized, GaAs semiconductor electrode. The fixed parameters associated with this system are given in Table I. Other parameters such as deep-level state concentration distribution and energy are given where appropriate.

I m p e d a n c e - p l a n e p l o t s . - - W h i l e the mathematical development presented in this work did not employ electrical

122

Table I. Values for parameters used in the calculations.

J. Electrochem. Soc., Vol. 139, No. 1, Janua ry 1992 �9 The Electrochemical Society, Inc.

T Temperature 300 K Eg Bandgap (34) 1.43 eV

Nd - N~ Doping level 1.0 • 10 TM cm -3 D~ Electron diffusivity (34) 155 cmZ/s Dh+ Hole diffusivity (34) 8.02 cm2/s e~ Permittivity (34) 9.65 x 10 13 farad/cm k4 Recombination rate 1.74 X 1014 cm3/equiv./s

constant (42) N~ Conduction band density 4.7 x 1017 cm 3

of states (34) Nv Valence band density 7.0 x 10 TM cm -3

of states (34)

c i r cu i t analogues , such circui ts are c o m m o n l y used as a conven i ence in in te rpre t ing expe r imen ta l data. Further , a n u m b e r of workers have p re sen ted their work on deep- level states in t e rms of equ iva len t circuits. In order to pro- v ide compa r i son to the i r work, a n u m b e r of equ iva len t circui ts were p roposed and tes ted against va lues calcula ted by the t ranspor t -based model . The circui t shown in Fig. 2 was found to p rov ide the bes t corre la t ion wi th the mode l results. The i m p e d a n c e re sponse of this c i rcui t is g iven by

C~R1 Z r = R 2 -I- [28]

(C1 + C~) 2 + ~2(C1C2Rl)2

C1 + C2 + ~2C~C2R~ Zj = [29]

o ) ( C 1 4- C 2 ) 2 q- o~3(C1C2R1) 2

and

E x a m i n a t i o n of Eq. [29] indicates that the imaginary com- ponen t Zj is insens i t ive to deep- level states i f C1 (the capacity associa ted wi th deep- leve l states) is negl ig ib le w h e n c o m p a r e d to C2 (the space-charge capaci ty associated wi th the comple t e ly ion ized dopan t species). U n d e r these con- dit ions, the imaginary c o m p o n e n t is g iven by Zj -~ -lkoC2, the same resul t as ob ta ined f rom a s imple R - C series circuit. U n d e r ident ica l c i rcumstances , the real part of the impedance Z~ at tow f requenc ies is g iven by Zr ~ RI/ [I+(~C2Rt)2], ind ica t ing that Zr is sens i t ive to deep- level states t h rough the pa rame te r R~ (the res is tance associa ted wi th deep- leve l states). The paramete rs R1, R2, C1, and C2 in Eq. [28] and [29] were de t e rmined by fitt ing these expres- s ions to the data genera ted by the t ranspor t -based com- pu te r m o d e l us ing the I M S L nonl inear least squares al- go r i thm Z X S S Q .

The inf luence of the concen t ra t ion of deep- level species can be seen in Fig. 3 whe re impedance -p l ane plots are pre- sen ted for an n- type s e m i c o n d u c t o r wi th a dop ing dens i ty o f 1016 cm -3 as a func t ion of deep- level species concentra- t ion (Et = 0.8 eV). The symbols r ep resen t the resul ts of the t ranspor t -based c o m p u t e r mode] and the curves are the resul t of the non l inea r least squares cu rve fit. F igure 3 illus- t rates the increase in Zr at low f requenc ies (top por t ion of curves) associa ted wi th an increase in deep- level species concentra t ion. In the absence of deep- level states, the impedance -p lane plot is cons is ten t wi th a s imple R - C series equ iva len t circuit .

The inf luence of appl ied potent ia l can be seen in Fig. 4 where impedance -p l ane plots are p resen ted as a func t ion

Jt C 2

C 1 R l

Fig. 2. An equivalent circuit representation which gives the same re- suits as does the numerical calculations which account for the effect of deep-level electronic states. Values of circuit components are independent of frequency but are functions of potential.

1 0 I~

10 ~

•EE 10 6 (3

10 4

I

102

10 o

a

/

/

/

/

I I I

t m

I 0 -6 1 0 -4 1 0 -2 1 0 ~

Z r (0hm-cm 2)

02

Fig. 3. Impedance-plane plots for n-type GaAs as a function of deep- level species concentration. The doping concentration was 1016 cm -3, the deep-level donor energy was 0.8 eV, and the applied potential was 0.6 V, referenced to the flatband potential. The frequency range was from 10 s to 10 -2 Hz. Deep-level concentration: curve a, 0 cm 3; curve b, l 0 s cm-3; curve c, 1010 cm-3; curve d, 1012 cm-3; curve e, 1014 cm -3. Symbols represent calculated results. Lines represent the nonlinear least squares curve fit.

of potent ia l for an n- type s e m i c o n d u c t o r wi th a dop ing dens i ty of t 0 l e c m -3 and a trap dens i ty of 10~4cm -3 (Et = 0.8 eV) . The low f r equency va lue of Zr (top por t ion of curves) increases as the deep- level species b e c o m e ionized. A t low potent ia ls , these species are near ly filled, and the cu rve approaches that observed in the absence of deep- leve l species (see Fig. 3).

The inf luence of subbandgap i l lumina t ion on the impedance-p lane behav ior of this sys tem was exp lo red in a simi-

I O ~~

10 e

10 6 o

~D ~ 1(14

i

10 2

10 o

0 -6 ] 0 -4 1 0 -2 1 0 0 ] 0 2

Z r ( 0 h m - cm 2 )

Fig. 4. Impedance-plane plots for a uniform distribution of deep-level states as a function of potential. The dopant concentration was 1016 cm -3, the deep-level donor concentration was 1014 cm 3 and the deep-level donor energy was 0.8 eV. The frequency range was from 10 s to 10 -2 Hz. Potentials: a, 0 V; b, 0.05 V; c, 0.2 V; d, 0.4 V; e, 0.6 V. Symbols represent calculated results. Lines represent the nonlinear least squares curve fit.

J. Electrochem. Soc., V o l . 1 3 9 , N o . 1, J a n u a r y 1 9 9 2 �9 T h e E l e c t r o c h e m i c a l S o c i e t y , Inc. 1 2 3

I 0 lO

10 ~

10 6

10 4

N

10 2

10 0

r r i T - -

i 0 -6 1 0 "4 1 0 -2 10 ~ 1 0 2

Z r tOhm-cm 2)

0 4

Fig. 5. Impedance-plane plots for a uniform distribution of deep-level states with subbandgap illumination energy as a parameter. The dopant concentration was 1016 cm -3. The deep-level donor concentration was 1014 cm -3, and the deep-level donor energy was 0.8 eV. The frequency range was from 105 to 10 -2 Hz. Optical energy: curve a, no illumination; curve b, 0.63 eV; curve c, 0.73 eV; curve d, 0,83 eV; curve e, 0.93 eV. Symbols represent calculated results. Lines represent the nonlinear least squares curve fit.

lar fashion and is presented in Fig. 5, which includes the impedance data for an n-type semiconductor with a doping density of 10 l~ cm -3 and a deep-level state density of 1014 cm -3 (Et = 0.8 eV). The effect of illumination is to charge the deep-level species and increase the magnitude of the real impedance at lower frequencies. This is essen- tially the same effect as obtained by increasing the applied potential in the previous example.

Experimental results generated in this laboratory indi- cate that the numerical model and the equivalent circuit shown in Fig. 2 can be useful for system characterization. The exper imental data presented in Fig. 6 (43) were obtained at the flatband potential for semi-insulating GaAs with high concentrations of deep-level states. These data were simulated using the transport-based model proposed in this work. The symbols represent the experimental data, and the curve represents the frequency response of an equivalent circuit that has been matched to the values generated by the computer model. The downturn and rebound observed at low frequencies in Fig. 6 is predicted by the model when the concentration of deep-level states becomes much larger than the doping level. While this

i 0 8 o

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2:

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/ - ~ m

03 ] 0 4 ]@5 ] 0 6 l O z

R e s C o m p o n e n L ( o h m )

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Fig. 6. Experimental impedance-plane plots for an intrinsic GaAs semiconductor with high concentrations of deep-level states at fiat- band. Symbols represent experimental data: the line represents results calculated from the mathematical model.

28. b

20.(~

o N x 'rJ

12.0

4.0

f l , I , , , I I 0.0 + 0.4 § 0.8

V vs .SCE

Fig. 7. Mott-Schottky plots for n-MoSe2 in 0.2M K4Fe(CN)6 taken from Ref. (44). a, 500 Hz; b, 1 kHz; c, 5 kHz.

analysis was conducted through a trial and error approach (45), it should be possible to apply nonlinear regression techniques directly to the model.

The electrical circuit described above was also used by Dare-Edwards et aL (26) to represent their model results for the effect of surface states. Quantitative agreement with their calculations was obtained when the effect of surface states was modeled in the absence of distributed bulk deep-level species. Comparison with the results of Dare-Edwards et al. (26) shows that the same equivalent circuit can be used to characterize the effect of either bulk or surface deep-level species.

Mott-Schottky plots.--In accordance with Mott-Schottky theory, a plot of 1/C~ as a function of applied potential for an evenly doped semiconductor under reverse bias should provide a straight line, where the flatband potential and the doping level could be obtained from the intercept and slope of the line, respectively. Experimental deviations from Mott-Schottky behavior, however, are often observed and attributed to the presence of surface states. For example, f requency-dependent Mott-Schottky slopes, such as those shown in Fig. 7 (44), have been observed for a number of semiconductor systems (7, 8, 13-16) and attributed to the presence of surface or bulk deep-level states. The results of the mathematical model were consistent with this interpretation. For example, an n-type semiconductor with a doping level of 10 TM cm -3, a deep-level donor concentration of 10 TM cm -3 (Et = 0.8 eV), and an illumination energy of 1.1 eV was found to exhibit f requency-dependent Mott- Schottky curves at lower frequencies (see Fig. 8). The capacitance in this example was calculated using an R - C series equivalent circuit, appropriate for an ideally polarizable semiconductor in the absence of deep-level states. Such frequency dispersion was not observed when the space charge capacity was calculated using the equivalent circuit shown in Fig. 2.

Shifts in Mott-Schottky plots with illumination or potential have also been attributed to deep-level states (16, 27, 46, 47). The appearance of two distinct Mott-Schottky slopes, shown in Fig. 9 for a p-type semiconductor, was at- tr ibuted to the influence of the filling of surface states by electron transfer from an ionic species (Fe 2§ (48). To inves- tigate the effect of near-surface distribution of deep-level states on Mott-Schottky plots, deep-level donors were modeled with a distribution near the semiconductor surface and the resulting Mott-Schottky curves plotted in Fig. 10 for an n-type semiconductor with a uniform doping level of 10 TM cm -3. In the absence of deep-level states (curve a), typical Mott-Schottky behavior is observed. Curve b was obtained for an arbitrary near-surface distribution given by c~ = c~ exp ( -my ) where c~ is the surface concentration, m = 3.638, and y is the distance from the surface in units of Debye lengths. The energy level of the deep-level states was 1.1 eV, referenced to the valence band. In curve


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Fig. 8. Mott-Schottky plots for o uniform distribution of deep-level donors (Et = 0.8 eV) under 1 mW illumination (hv = 1.1 eV) with frequency as a parameter. The dopant level was 1016 cm -3 and the deep- level state concentration was 1016 cm -3 with energy 0.8 eV. Curve a, 10 -2 Hz; curve b, 10 -1 Hz; curve c, 10~ s Hz. Symbols represent calculated results, and lines were drawn for clarity.

b, the influence of the near-surface states is reflected by a decrease in the Mott-Schottky slope in the potential range of 0.2 to 0.4 V. At higher potentials, however, the curve re- turns to the slope associated with dopant species (e.g., curve a). This observation indicates that shifts in Mott- Schottky curves can indeed be attributed to the charging and/or discharging of surface states. This result is discussed further in a subsequent paper (13).

Photocapacitance spectroscopy.--Photocapacitance spectroscopy has been shown to be a viable illumination- based technique for detection of deep-level states (4-9). This technique is based on relating changes in the measured space-charge capacitance as a function of subbandgap illumination energy to changes in the occupation of deep-level states.

The results of these experiments are usually presented in plots of capacitance (or change in capacitance) as functions of optical wavelength or photon energy. The capacitance is usually determined at a frequency where an R - C series equivalent circuit can be used to describe the system. A typical plot is shown in Fig. 11 (8) where sudden changes in the slope of the experimental data represent electron transitions due to the presence of deep-level species at a specific energy. These data give the energy of the deep-level state directly, since a transition from the valence band to the deep-level state results in a decrease in capacitance, while a transition from the deep-level state to the conduction band results in an increase (8).

While the results of the model are fully consistent with the observations of photocapacitance spectroscopy, the calculations also point out some experimental difficulties. Since the capacity used for photocapacitance techniques is obtained from an R - C series circuit, the concentration

-1D -0.~ 0 *0.5 V, V(vs. SSE)

Fig. 9. Mott-Schottky plots taken from Ref. (48) for p-GaP in contact with 0 .25 real/din s K2S04 + 1 mol/dm 3 FeS04 + H2S04 (pH = 1) measured at a frequency of 10 kHz.

Fig. 10. Mott-Schottky plots for an n-type semiconductor at equilibrium with varying distributions of deep-level donors. The dopant level was 1016 cm -3 and the deep-level state energy was 1.1 eV referenced to the valence band for both curves. In curve a, the deep-level concentration was 1014 cm -3 throughout the semiconductor; in curve b, the deep-level concentration was 1016 cm -3 at the surface, decaying expo- nentially to I % of this value at 500/~. Symbols represent calculated results.

of deep-level state sites must be a significant fraction of the doping concentration in order to be observed. This difficulty is demonstrated in Fig. 12 where model-generated R - C series capacitance is plotted on a linear scale as a function of il lumination energy for an n-type semiconductor with a doping level of 10 TM cm 3, a deep-level donor concentration of 10 I~ cm -3 (Et = 0.76 eV), and a deep-level acceptor concentration of 1012 cm -3 (Et = 0.82 eV). Note that the scale for the capacitance has been expanded significantly to observe the changes in capacity. These results confirm the previous results of Allongue and Cachet (8) who report that an increase in capacitance corresponds to an electron transition from the deep-level state to the conduction band, while a decrease in capacitance corresponds to an electron transition from the valence band to the deep-level species. This result is independent of the charge of the deep-level species.

The need for extremely accurate capacitance measurements associated with photocapacitance spectroscopy was

I I I I

20 / _ E-130

10 Ev+ 0.75 3 -

t_) - <3

0 _~

0.6 0.8 1.0 1.2 I I I t

ENERGY (eV) Fig. 11. Photocapacitance spectrum for n-GaAs in 1M KOH at

0 .95 V referenced to flatband taken from Ref. (8).

J, Etectrochem. Soc., Vol. 139, No. 1, January 1992 �9 The Electrochemical Society, Inc. 125

6 7 . 9 I I I I

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o

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ENERGY (eV]

Fig. 12. Calculated R - C capacitance as a function of photon energy for an n-type semiconductor at a reverse bias of 0.2 V. The dopant concentration was 1014cm-3, the uniform deep-level electron donor concentration was 10 u cm -3, the uniform deep-level electron ac- captor concentration was 10 u cm -3, and the deep-level energies were 0.76 and 0.82 eV, respectively. Symbols represent calculated results with lines drawn for clarity.

addressed by development of exceptionally precise experimental technique (6, 8). The capacitance is usually determined at a frequency greater than 1000 Hz. The calculated results presented in Fig. 4, 5, and 6 suggest, however (i) that the impedance response may be much more sensitive to deep-level states at lower frequencies and (ii) that interpretation of both real and imaginary components may be useful. These implications of the modeling effort are discussed in a subsequent paper (45).

Conclusions The objective of this work was to develop a mathemati-

cal framework that would describe the impedance response of a semiconductor without requiring the use of electrical circuit analogues. The mathematical development is presented in this paper. The treatment of the physical phenomena incorporated within the model is shown to be sufficient to account for a wide variety of experimentally observed behavior. The mathematical model presented here shows quantitative and qualitative agreement with published experimental data. The results of the model are shown to be consistent with an equivalent electrical circuit proposed for a semiconductor with surface states, with observation of frequency dispersion and shifts in Mott-Schottky plots attributed to surface states, and with observation of capacity shifts associated with photocapacitance spectroscopy.

In a separate paper, this model was used to explore the interpretation of Mott-Schottky plots in terms of nonuni- form distributions of dopants and both uniform and non- uniform distributions of deep-level states (13). This model was also used to guide development of analytic asymptotic solutions to the governing equations to develop a means of interpreting the photoelectrochemical impedance response of semiconductors in terms of the energy, average concentration, distribution, and rate constants for discrete single-energy deep-level states (45). These applications illustrate the value of developing models for impedance response that are based on solving the differential equations governing the physical problem.

A c k n o w l e d g m e n t s This material is based upon work supported by the Na-

tional Science Foundat ion under Grant No. EET-8617057. During a portion of this work, D.B. Bonham was supported by a fellowship from Texaco. This work was conducted at the University of Virginia and written in final form at the University of Florida.

Manuscript submitted July 31, 1989; revised manuscript received May 3, 1991.

LIST OF SYMBOLS A a constant in Eq. [18], 2.6 x 105 m -1 eV -1/2 ci concentration of species i, cm -~ C space-charge capacitance calculated from an R - C

series circuit, F/cm ~ AC change in C from a chosen reference level, F/cm 2 Ct space-charge capacitance associated with deep-level

states, defined by the equivalent circuit shown in Fig. 2, F/cm a

C2 space-charge capacitance associated with ionized dopant species, defined by the equivalent circuit shown in Fig. 2, F/cm 2

Di diffusivity of species i, cm2/s e charge of proton, 1.602 • 10 -'" C Ea deep-level acceptor energy, eV Ec conduction band energy, eV Ed deep-level donor energy, eV Ef Fermi energy, eV Eg bandgap energy, Er - Ev, eV E~k equil ibrium constant for reversible reactions j and k

energy of generalized deep-level trap species, eV Ev valence bandedge energy, eV J~ activity coefficient for generalized species i F Faraday's constant, 96,487 C/equiv. ~i degeneracy of deep-level species

volumetric rate of generation of species i i current density, mA j k Boltzmann's constant, 1.381 • 10 -23 J/K kj rate constant for species j m absorption coefficient, cm -1 me electron mass, g m* electron density of states effective mass, g m~ hole density of states effective mass, g mhh heavy-hole mass, g m~h light-hole mass, g mr reduced mass, g mrh reduced hole mass, g m~ reduced electron mass, g n electron concentration, cm -3 ni intrinsic carrier concentration, cm -3 Nc effective density of conduction band states, cm 3 Nd doping concentration, cm -3 N~ effective density of valence band states, cm -~ Nyi molar flux of species i, moYm 2. s ri rate of reaction of species i, mol/cm 3. s R universal gas constant, 8.314 J/mol - K R1 resistance associated with deep-level states, defined

by the equivalent circuit shown in Fig. 2, F/cm 2 R~ resistance associated with ionized dopant species,

defined by the equivalent circuit shown in Fig. 2, F/cm 2

t time, s T absolute temperature, K ui mobility of species i, m2/V �9 s V applied potential, referenced to flatband, V 5: steady-state symbol for variable x 5z~ real component of the perturbation in variable x 5:j imaginary component of the perturbation in variable

x y distance from interface, cm zj charge number for species, i Z complex impedance, ~-cm 2 e permittivity, Farad/cm ~1~ dimensionless deviation of energy band E~ from the

Fermi energy 0 phase angle, rad x Debye length, cm ~ electrochemical potential of species i, J/tool ~o reference electrochemical potential of species i,

J/mol 4) electrostatic potential, V ~ change in the real or imaginary portion of the poten-

tial across the semiconductor sample, V co frequency, s i

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a mathematical model for the influence of deep-level...

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