A model for photon-induced evolution of hydrogen at p-type semiconductor electrodes

0360-3199/86 $3.00 + 0.00 Pergamon Journals Ltd. © 1986 International Association for Hydrogen Energy.

Int. J. Hydrogen Energy, Vol. 11, No. 6, pp. 373--379,1986. Printed in Great Britain.

A M O D E L FOR P H O T O N - I N D U C E D EVOLUTION OF H Y D R O G E N A T p-TYPE S E M I C O N D U C T O R E L E C T R O D E S S. U. M. KHAN* and J. O'M. BOCKRIS'[ *Department of Chemistry, Duquesne University, Pittsburgh, PA 15282, U.S.A. and tDepartment of Chemistry, Texas A&M University, College Station, TX 77843, U.S.A.

(Received 29 October 1985) Abstract--An analytical expression for the hydrogen evolution rate in the form of photocurrent as a function of the physical properties of a semiconductor electrode and the neighboring electrolyte solution has been derived. The expression of photocurrent involves no implicit assumptions concerning the rate-determining step. The charge transfer to the proton across the double layer has been treated explicitly. The model used for the surface states includes their energy distribution and their concentration dependence on the electrode potential. Recombination of photoexcited carriers in the field-free and surface regions has been taken into account. Energies of acceptor ions in solution and their distribution are calculated. The predicted photocurrents for hydrogen evolution have been integrated over the solar spectrum. Numerical comparisons with experiment are made.

EvB

Eo v. v.

SCR X~

Eo W

R~ D~ kct ksr kbr

p~

so Lo Sth eo

NOMENCLATURE Fermi level energy in the semiconductor energy of band gap energy of bottom of conduction band energy of top of valence band ground state energy of surface state potential drop inside semiconductor potential drop in Helmholtz layer field-free region space charge region electron affinity of semiconductor ground state of acceptor ion width of depletion region recombination capture cross-section absorption coefficient of incident light intensity of incident photon reflection coefficient of photon at semiconductor surface diffusion coefficient of electron charge transfer rate constant surface recombination rate constant bulk recombination rate constant density of surface states drift velocity of electron diffusion length of electron thermal velocity of electron electronic charge

Recently, however, new results (such as the pinning of the Fermi level in the semiconductor [10, 11], the observance of Tafel linearity in the iphoto--V relation [2,12] and the correlation of the effect of submonolayer addition of metals on photocathodes with the exchange current densities for the dark metal/solution interface reactions [12] show that the interracial transfer of the photoelectron sometimes plays an important role in determining the photocurrent. It is therefore desirable to treat photo-stimulated interracial charge transfer processes in photoelectrochemical kinetics, especially for the hydrogen evolution reaction, in more detail, and to couple equations for interfacial control at the semiconductor/solution interface with those for photoexcitation and transport in the solid state.

M O D E L F O R A p-TYPE S E M I C O N D U C T O R / SOLUTION INTERFACE Inside the semiconductor, there is a depletion region of width, W, and a field-free region, which extends after the depletion region into the bulk. On the solution side, there is a region of specifically adsorbed ions, adsorbed water molecules and the acceptor ions in the outer Helmholtz plane (OHP) of the double layer (Fig. 1).

INTRODUCTION Photoelectrochemical production of hydrogen has become important in recent years [1-3]. It is becoming essential at this stage to have a detailed model of the hydrogen evolution reaction at a photocathode, e.g. at p-type semiconductor electrodes. Such model calculations will help to choose suitable photocathodes for the efficient production of hydrogen. Earlier treatments [4--9] of photoeffects at semiconductor/solution interfaces were primarily concerned with improved treatments of transport and recombination of carriers in the semiconductor. 373

SOLUTION OF TRANSPORT EQUATIONS FOR P H O T O E L E C T R O N DENSITY A T THE INTERFACE The basic transport equations in semiconductors given in textbooks [13] have been applied at an advanced level by several earlier workers [6, 8, 9]. In the present treatment, we have developed aspects of the treatments referred to, but in a form which enables equations for the concentration of electrons at the surface to be coupled in an explicit way with those for transfer across the interface.

374

S. U. M. KHAN AND J. O'M. BOCKRIS Vacuu T level =,

E¢,

2

1

FFR

SCR

~

. . . . . . . . . . . . . . . . . . . . . . .

,

I~

.... IlQ

~

Eo

I~ 11~

II

Inlerlacial ( ~ barrier =c /

t

aF.,dVac, scale)

~

::~--

[

I

oracceolor. H,O"

.................. 5:-\;5; r;:;;;;o.,.,.

.,-.... g) ! I j

t P'TYPE SEMICONDUCTOR

~lt~ adaorOed 1~ r ' ' w a t e r ~ dipoles OHP

'-'...

®

f~

SOLUTION

(~$pecilieally (~) G

adsorbed i°ns (~

Fig. 1. A schematic representation of the model of a p-type semiconductor-solution interface.

In the space charge region (SCR), i.e. Region 1 of Fig. 1, the transport equation can be expressed for the steady state (in the absence of recombination) as d2nt (x)

,ix 2

e0 , ~v(x)

dnl (x)

_~TV,,(x)na(x)

(1 - R~)I0 a v e - ~ =

De

(1)

where Iv is the intensity of the incident photons, R~ is the reflection coefficient of the photon at the semiconductor surface, and a~vis the photon absorption coefficient. It is assumed here that each absorbed photon gives rise to an electron if its energy, he ~> E. (the bandgap of the semmonductor). D e is the dlffuslon coefficmnt of the electron, nt(x) is the concentration of electrons at position x in the SCR and V'(x) is the field in this region. For the field-free region (FFR), i.e. Region 2 in Fig. 1, the transport equation can be expressed [13] as d2n2(x) (ix 2

n2(x) - L2

=

-

(1 - Rv)Ivolve-'~ D~

(2)

where -e0Jn~ (0) is the current across the semiconductor surface, kct is the heterogeneous charge transfer rate constant at the semiconductor/solution interface, and ksr is the surface recombination rate constant for electrons with surface traps (e.g. holes, surface states, impurity centers etc.). These rate constants may be potential dependent. The boundary condition (3) assumes that the thermal population of electrons is negligible compared to that of photoexcited electrons. The transport equation (1) can be solved in the following way. Integrating (1) twice with respect to x, one obtains: { A nl(X ) = e 0(x) C 1 G I ( X ) + - - G2(X ) 0¢v + nl(0)e-°s}

(7)

where the following substitutions have been used: A = (1 - R~)I~olJO e

(8)

e oV(x) 0(x)=

kT

(9)

where LD = (Der) 1/2 = diffusion length of electron. One can solve the transport equations (1) and (2) of regions 1 and 2 respectively, using the appropriate boundary conditions. The following four boundary conditions were used [8, 14]:

and V(x) is taken with respect to the bulk of the semiconductor.

n:(=) = 0

(3)

na(W) = n2(W)

(4)

where Vs is the potential at the surface of the semiconductor with respect to the bulk.

:.1 ( w ) = J . : ( w )

(5)

-e0Jnl(0) = e0(kct + ksr)nl(0)

(6)

and:

O(x = o) = Us =

Gl(x) = G2(x ) =

eoV(x) eoVs k---f- = k---f-

f:

(10)

e -°(x) dx

(11)

e-{a~x+ ex)} dx.

(12)

375

HYDROGEN EVOLUTION AT p-TYPE ELECTRODES Equation (2) for transport in the FFR can be solved by using the method of undetermined coefficients. The solution of (2) can thus be obtained as n2(x ) = C2e-X/CD + C3 ex/LD A 02 -- L ~ 2 e-ad:-

(13)

Equations (7) and (13) were solved with the use of boundary conditions (3)-(6) to find the concentration of photoelectrons in the surface region of the semiconductor, nl(0), as nl (0) =

(1 -

R,)lo

The potential dependence of the concentration of surface states, Nss(E°s, VH), which in turn depends on the degree of coverage, 0, by adsorbed anion or H atoms can be obtained using a suitable isotherm. Considering the Langmuir-type approximation for the coverage of anions, one can express the potentialdependent density of surface states due to adsorbed anions [16] as

X°s(0 = pss(E) -

1)

kT

e -(E°~-EI/kr 1 + e -eoVH/kr"

(20)

(kct + k~r + kbr) e-u~ W

×

dNss(E) Nss(E°, VH) e_(eo_e)/kr. (19) Pss(e) - d - - - - - ~ = kr

1 - (1 + ol~LD)[1 + L?)IGI(W)]

(14)

/

Now, using (20) in (18) and taking the limit of integration up to E°s, the ground state energy of the surface state, one gets after integration:

k~r = 6srSthNO~(O= 1)(1 + e-%VJkr) -I

LD + G I ( W ) J

× (1 -

(21)

e-(e°~s-Ev)/kr).

where the constants

De e -e°vs/kT kbr -

LD + G1(W)'

(15)

GI(W) =

e -~(~/dx - (zkT/4eoV~)a/2W (16)

GE(W) =

e - l ~ ÷°(xlIdx ~- GI(W) e -~w (17)

and

where in (14) and (15), kbr signifies the recombination rate constant in the bulk.

C H A R G E T R A N S F E R PROCESS At a p-type semiconductor, the charge transfer process at the semiconductor/solution interface is ratedetermining in the less negative potential ranges, while the photocurrent-potential curve is in its exponential growth region. The charge transfer process at the interface depends on the density of the photoelectrons at the semiconductor surface region, nl(0); the distribution of photoelectrons, f(E, hv); the velocity of photoelectrons, Se(E); the tunneling probability of the photoelectrons across the interracial barrier, P(E)(Fig. 1) and on the density of acceptor energy states of ions in the double layer, Da(E0, E).

R E C O M B I N A T I O N PROCESSES We have already taken into account recombination in the bulk in the second term of (2) and the bulk recombination rate constant, kbr, given in (15). Surface recombination [cf. (14)] is taken into account in terms of the surface recombination rate constant, ksr.

Surface recombination rate constant, ksr

Pss(E) d E

The rate constant due to charge transfer at the interface across the interfacial barrier (Fig. 1) to an acceptor ion in solution can be expressed within the framework of the quantum theory [17, 18] of electron transfer reactions as: kct = Se

The surface recombination rate constant can be expressed in a general form [5, 15] as ksr = •srSth

The charge transfer rate constant, ka

(18)

Ec where &st is the capture cross-section for the surface recombination, Sth is the thermal velocity of electron in the semiconductor, psi(E) is the density of surface states per unit energy and per unit area of the surface, Ev and Ec are energies corresponding to the valence band and conduction band levels at the surface of the semiconductor electrode respectively. The density of surface states per unit energy and per unit surface area can be expressed [16] as

P(E)f(E, hv)Oa(Eo, E) dE

(22)

Ec where Se is the drift velocity of outgoing electrons in the surface region of the semiconductor.

Tunneling probability of electron, P(E) Within the WKB approximation [19] the tunneling probability of an electron across the interracial barrier can be expressed along the x coordinate, perpendicular to the surface [20] as

P(E) = exp(- ~

{2m*[U(x)

-- El)l/2 d x )

(23)

376

S. U. M. KHAN AND J. O'M. BOCKRIS

where U(x) is the one-dimensional potential energy barrier at the interface (see Fig. 1),/~ is the width of the double layer and a is a distance of atomic dimension from the electrode surface.t

where Nt is the total number of sites for the reactants per unit area of the reaction plane.

The ground state energy of the acceptor ion, E0 The distribution of photoelectrons The distribution of photoelectrons, f(E, hv), in the conduction band used in (22) can be expressed in terms of a Fermi function, which is modified, to take into account the effect of illumination, having a photon energy, hv, ast

f(E'hv) = { l + e x p [ ( E - EF - hV)]}

(24)

where EF is the Fermi energy in the semiconductor.

To determine the density of acceptor states in ions in solution, D~(Eo, E), used in (22), one needs to know the distribution of electronic states in solution relevant to electron transfer. Such a distribution depends on the energy transfer process between the ion concerned and the solvent. One may consider a thermal equilibrium between the ion-solvent complex, e.g. H30 + and the solvent. The vibrational-librational levels of ion-solvent bonds will in effect be a continuum represented by the MaxweU-Boltzmann distribution law. Hence, one can express the distribution of acceptor states in ions in solution (including the potential drop in the Helmholtz layer, VH) [see 17, 21, 24] as N(E) = N~ e-~E°+eoVH-E)/kT (25) where Na is the number of acceptor ions per unit area of the reaction plane (OHP), E0 is the electronic energy of the acceptor ion corresponding to the ground vibration-libration energy of the ion-solvent bond in solution, fl is a symmetry factor, having the values, O~
1 dN(e)

E0 = Xsc - J - AHsol + AHads - Rp

AEo = Eo - Xsc

(28)

(29)

where sc(e) represents electrons in the semiconductor conduction hand, Xsc is the electron affinity of the semiconductor, J is the ionization energy of the H atom, AHsol is the solvation energy of the H 3 0 + ion, and AHads is the adsorption energy of the H atom on the semiconductor surface when it is in the stretched, sc--H activated state, and Rp is the repulsion energy of an H atom and the H 2 0 molecule. Thus, using (23), (24) and (26) in (21), one can express the transfer rate constant at the semiconductor/ electrolyte interface as N~ Se f ~Ec kct = kTN---~t

xexp(-lff {2mg[U(x)-E]}~/2dx) × [1 + elE-ev-h~)/kT]-I e-~Eo+eovH-E)/kr dE. (30) The potential drop in the Helmholtz double layer, VH, [to be used in (30)] can be expressed as VH ----"Vm . . . . . . d(NHE) + 4.5 V +/tg¢/e0 - Vs

Da(Eo, E) = ~ t ( ~ ]

E0- e0V, - E)/kT

where Eo becomes [1, 21] (with respect to the bottom of the conduction band at the surface, as in Fig. 1)

and in the vacuum scale:

Density of acceptor states in ions in solution

= - -fl- - eNa -IX kT N t

The ground state energy of the acceptor, E0, can be obtained by using a Born-Haber cycle. For H30 + as an acceptor ion in solution, one needs to use a cycle for the process E H3 O+ + sc(e)--~ s c - - H - - H 2 0 (27)

(31)

where /~_c is the chemical potential of the electron in the semiconductor. (26)

t Since the thermalization time (10 -12 S) is comparable to transition time (10-12s) for photoexcited electrons across the distance of 1 micron (104,~) where the absorption of photons is maximum, the modified Fermi function (24) is physically applicable for the photoexcited electrons in the semiconductor. The transition time t of photoexcited electrons is obtained using t = (m*~/2Ep)l/2dwhere me* is the effective mass of the photoexcited electron (which is much less than the free electron mass), Ep is the energy of the photoexcited electron (1-3 eV) and d is the distance of maximum absorption (of the order of 103--104/~ depending on absorption coefficient).

T H E EXPRESSION F O R T H E P H O T O C U R R E N T In expressing the photocurrent for H 2 evolution, no particular rate-determining step will be assumed, because the experimental evidence is that this changes with the rate of reaction. The photocurrent can be expressed in terms of the electronic charge, e0, charge transfer rate constant, kct and the density of photoexcited electrons at the surface of the semiconductor, nl(0 ) [cf. (14)] as

HYDROGEN EVOLUTION AT p-TYPE ELECTRODES

377

3.S

P r

,/

3.0

~

f '

~

o

o

~

o

Expertm@nfml/ '~ ~h.o,,,,¢,,

,"//

? 2.S E E

Z

2.C

c:

O.

1.5

1.(

0.~

ox 0,6

J, 0.4

0.2

0.0

,

-0.2

,

-0.4

I -0.6

L -08

-1!0

Potential, V(volI)/NHE

Fig. 2. The plot of photocurrent for H 2 evolution,/v (solar), as a function of electrode potential, V (V)/ (NHE).

ip(v) = e0k~nl(0) e0ka(1 - R~)Iv = (ka + ksr + kb,)

×

e -°tvw

f

/ 1 - (1 + a~vL51)(1 + GI(W)Lfi i')

o2(w) Ln ¥

(32)

where kbr, ks~ and kct are given, respectively, in (15), (21) and (30). G E N E R A L F E A T U R E S OF T H E P H O T O C U R R E N T EXPRESSION One may consider some qualitative aspects of (32) for the photocurrent. Thus, ip(V) should be zero when the absorption coefficient of light in the semiconducting electrode elv = 0. When ~v = 0, Gz(W) becomes equal to GI(W) [(16) and (17)] and the summation of the second and third terms in the third bracket of (32) becomes unity, so that the quantity in the third bracket of (32) becomes zero. Thus, the photocurrent becomes zero [cf. (32)]. Equation (32) indicated that the charge transfer becomes the rate-limiting step under the condition when k¢t< (ks, + kbr). In this situation, the photocurrent

depends on the charge transfer rate constant, kct. With increasing negative potential, the values of k,r (18) and (20) and also kbr (15) become small compared to kct (30). The number of surface states due to anion adsorption will fall and metalization of the interface will correspondingly be reduced, so that the rate will become controlled by transport inside the semiconductor. One notices from (32) that ip--~ ip (limiting) when W--~oo. The photocurrent ip--~0 when ksr--* oo or kbr --~ oo (32). These trends would be expected from a physical point of view. When kct ~ (ksr + kbr), charge transfer will not be rate-determining and the rate will again be controlled by the transport inside the semiconductor [cf. (32)]. P H O T O C U R R E N T IN SOLAR L I G H T To have information regarding the photocurrent at an electrode irradiated in solar light, one may integrate (32) over the solar spectrum. Thus, in solar light ip (solar) = f Vmax ip (v) d v

(33)

IIc

where vc is the critical frequency at which the major photoexcitation begins, e.g. vc = Es/h (neglecting the sub-bandgap excitation) and Vm~ is the maximum frequency available from the solar spectrum. E s is the bandgap of the semiconductor.

378

S.U.M. KHAN AND J. O'M. BOCKRIS

!

/

Theoretical

S c 0

eV~-..f6VoItlNHE I

"X~ ~N.

.

~

4O0O

3000

I

5OOO Wavelength.

A (A) --

Fig. 3. Plot of percent quantum efficiency wavelength of solar light.

RESULTS OF M O D E L C A L C U L A T I O N The model system p-type GaP electrode and HaO + ion as an acceptor for the hydrogen evolution reaction are used in the following calculations. The basic set of parameters used for p-GaP are the following. The doping concentration ND = 6.7 x 1017cm -s [2], Eg = 2.22eV [13], me* = 0.13m [13], LD = 2.8 x 10-4cm, for mobility/~ = 300 cm 2 s -1 V -1 [13], r, = 10 -l° s [22], dielectric constant of p-GaP es = 10, X~ = 4.3 eV, 6 = 2.7/~, e~ -- 6, eoVs = 0.4 eV, N°~(0 = 1) = 1015cm -2, Sth----- 107cms -1 [5] and ~sr = 10-16cm2, Nss(0= 1) = 1015cm -2. R~ was taken from Table VII, in [23], which ranges approximately from 0.25 to 0.5 within a wavelength range of 1.2--0.33 ~m in the solar spectrum. Ground state energy of acceptor, E0 = 1.2 eV [16].

The dependence of the photocurrent on the electrode potential The photocurrent as a function of the electrode potential has been computed, using (32) and (33). Figure 2 shows the photocurrent for hydrogen evolution, ip (solar), under solar light illumination, as a function of electrode potential. The shape of the photocurrentpotential relation corresponds to that observed [2] under xenon light illumination. Thus at the potential +0.5 to - 0 . 2 V w.r.t. NHE, surface recombination is fast, and the slow step is the interracial charge transfer (32).

The dependence of quantum efficiency on wavelength The quantum efficiency (i.e. the ratio of the number of reacting photoexcited electrons to the number of

< E

g m o

; VOIt/NHE co oea.

3.4

LOlL 10 !~1

I 10 t"

! 10 I I

| 10 t! Doping

I 10 t~' density,

I 10 I t

| 101l

N o ( c m "3)

Fig. 4. Plot of photocurrent as a function of doping density.

I 10 ~

HYDROGEN EVOLUTION AT p-TYPE ELECTRODES incident photons per unit time and area) as a function of wavelength is shown in Fig. 3. Figure 3 has taken into account the effect of R~ on ;L to explain the fall in quantum efficiency at lower wavelength, L The dependence concentration

of photocurrent

on

the doping

The photocurrent as a function of doping concentration, ND, for a range of electrode potentials is shown in Fig. 4. With the decrease of No, the width, W, of the SCR increases, and consequently photocurrent increases, as is observed in Fig. 4.

DISCUSSION The present communication contains the following specific features. The question of the rate-determining step has been left open, in this model, until some range of the current-potential relation on certain semiconductors has come under consideration. This model involves potential-dependent surface states and seems appropriate to situations in the semiconductor/solution interface, where there is evidence for influence of the variation of the Helmholtz part of the potential at the semiconductor/solution interface. One of the less discussed aspects of calculations at the semiconductor/ solution interface is the value of the so-called "potential" which should be introduced. This is often used as a parameter, and it is submitted that the method used here, in which the Galvani potential difference has been related to the well-founded potential on the vacuum scale, is an advance upon previous practice. The well-known relationship between quantum efficiency and wavelength, in which a maximum is observed, has been interpreted modelistically with quantitative evaluation (Fig. 3). Lastly, care has been taken to integrate the photocurrent calculated as a function of wavelength over the solar spectrum, so that, where relevant, results indicate the situation in terms of solar radiation, rather than monochromatic light. Acknowledgements--The authors gratefully acknowledge the

379

Robert Welch Foundation for the financial support of this work. REFERENCES 1. J. O'M. Bockris and K. Uosaki, J. Electrochem. Soc. 125, 223 (1978). 2. K. Uosaki and H. Kita, J. Eleetrochem. Soc. 128, 2154 (1981). 3. A. Heller, Acc. Chem. Res. 14, 154 (1981). 4. B. A. Butler, J. Appl. Phys. Lett. 48, 1941 (1977). 5. R. Wilson, J. appl. Phys. 48, 4297 (1977). 6. H. Reiss, J. Electrochem. Soc. 125,937 (1978). 7. J. Reichmann, Appl. Phys. Len. 35, 251 (1981). 8. W. J. Albery, P. N. Burtlett, A. Hamnett and M. P. D. Edwards, J. Electrochem. Soc. 128, 1492 (1981). 9. J. F. McCann and D. Haneman, J. Electrochem. Soc. 129, 1134 (1982). 10. A. J. Bard, A. B. Bocarsly, F. F. Fan, E. G. Walton and M. S. Wrighton, J. Am. Chem. Soc. 102, 3677 (1980). 11. J. N. Chazalviel and T. B. Truong, J. Electroanalyt. Chem. 114, 299 (1980). 12. J. O'M. Bockris and M. Szklarczyk, Appl. Phys. Comm. 2, 4; Appl. Phys. Lett. 42, 1035 (1983). 13. J. I. Pankov, Optical Processes in Semiconductors. Prentice-Hall, New Jersey (1971). 14. P. Lemasson, A. Etcheberry and J. Gautron, Electrochim. Acta 27, 607 (1982). 15. F. E. Guibaly and K. Colbow, J. appl. Phys, 53, 1737 (1982). 16. S. U. M. Khan and J. O'M. Bockris, J. Phys. Chem. 88, 2504 (1984). 17. R. W. Gurney, Proc. R. Soc. (Lond.) R134, 137 (1931). 18. J. J. Hopfield, Proc. natn. Acad. Sci. U.S.A. 71, 3640 (1974). 19. J. O'M. Bockris and R. K. Sen, Chem. Phys. Leas. 18, 166 (1973). 20. J. O'M. Bockris and S. U. M. Khan, Quantum Electrochemistry, p. 249. Plenum Press, New York (1979). 21. J. O'M. Bockris and D. B. Matthews, Proc. R. Soc. A292, 479; J. Chem. Phys. 48, 1989 (1968). 22. B. H. Bube. In R. K. Willardson and A. C. Beer (Eds.), Semiconductors and Semimetals, Vol. 3, p. 476. Academic Press, New York (1967). 23. B. O. Seraphin and H. E. Bennett. In K. Willardson and A. C. Beer (Eds), Semiconductors and Semimetals, Vol. 3, p. 512. Academic Press, New York (1967). 24. S. U. M. Khan and J. O'M. Bockris, J. appl. Phys. 52, 3640 (1981).