Theory of the Ambipolar Model Applied to the Quantum Efficiency of Photocurrents in Semiconductors
Department of Physics, National Taiwan University, Taipei, Taiwan 107, R.O.C.
(Received June 27, 1997)
A theoretical analysis is made for a P-polarized electromagnetic surface wave propa-gating along a plane-parallel ambipolar semiconductor plate. Fourier transform analrrsis is used to derive a general formula for the determination of the dependence of the elec-tron and hole concentration distribution function, the photocurrents, and the quantum efficiency of the excess carrier generation function in the steady state. We give a com-plete two dimensional analysis of the problem, taking surface recombination and the vertical diffusion current into account. We present the analysis and discuss the ex-cess carrier photocurrent and quantum efficiency in the two cases for which the surface recombination velocity S, -+ 0, and S, ---f m.
PACS. 73.50.-h - Electronic transport phenomena in thin films PACS. 73.5O.P~ - Photoconduction and photovoltaic effects. PACS. 73.25.+i - Surface conductivity and carrier phenomena. I. Introduction
The phenomenon of the generation of excess carriers by optical excitation in semi-conductors has drawn wide interest, because of its basic physical aspects and its numerous applications. The ambipolar model is a widely used approach to explain the distribution and transport of excess carriers in semiconductors. This approach was first_ developed by Roosbroeck [l] for his analytical studies of the high level injection problem. Ishaque  developed the ambipolar model to describe the radiation induced photocurrent of a reverse biased p-n junction. Chatterjee  demonstrated mathematically the modulated photocur-rent in the presence of a bias monochromatic light and found the possibility of an appaphotocur-rent quantum efficiency greater than unity. (Q.E.> 1).
In this work, we have used a Fourier transform analysis  to solve the ambipolar transport equation and obtain general formulas for the excess carrier concentration, flux density, current, and the quantum efficiency. We give a complete two dimensional analysis of the problem , taking the surface recombination and vertical diffusion into account. II. Electromagnetic fields
In Fig. 1, the sample is in the form of a rectangular solid and the thickness ! is much less than the other dimensions z,y. We assume that the P-polarized electromagnetic w a v e
725 @ 1998 THE PHYSICAL SOCIETY
PIG. 1. The coordinate system and semiconductor film position for the analysis. The origin is chosen at the center of the sample. The dimensions of the sample in the 2 and y direc-tions are assumed to be very large compared to the thickness e. The incident P-polarized electromagnetic wave is propagating along the semiconductor surface in the 2 direction.
is incident along the semiconductor surface in the z direction. Since the incident P-polarized electromagnetic wave is propagating in the z direction, the y component of the electric field E, = 0.
The electromagnetic fields in medium 1 (air) and medium 2 (semiconductor) will be of the form 
Ej(r,t) = (EjZ,O, Ej*)exp i(kj,z + k+.z. - wt), Hj(r,t) = (0, H,,,O) exp i(kj,s + kj,t - wt).
In medium 1 (air), the subscript j = 0. In medium 2 (semiconductor), the subscript j = m.
In the general case, the wave vector k is complex, kmz -- k,l + ikz-2, k,, = k,l t ib.
In the semiconductor, the dielectric constant is &m -- El t i&2,
where ~1 is the static dielectric constant,
4xa 00 e2NS
-w ’ Cl= l-iuS;’ (To=-.me (1)
CT is the frequency dependent conductivity, 3 is the excess carrier lifetime, N is the number of excess carriers per unit volume, and m, is the mass of excess carrier.
Using Maxwell’s equations and the usual Maxwell boundary conditions, &r B &a, k,r B kz2, ~1 B ~2, and neglecting the second order terms, we
assuming have the
4&l + 2)
ac (1 + &1)3/2’
III. Steady state photoconductivity III-l. Excess carrier concentration
We calculate the steady state photoconductive response of a uniform semiconductor. Since in the steady state condition g = 0, the ambipolar transport equation takes the form
$P(x,z) = ~g.(x,*):
where P(z,z) is the excess carriers concentration, L, is the diffusion length (L, = &?i), D, is the diffusion coefficient, and gs(z,.z) is the generation rate of excess carriers. We define
g&, 4 =Fe-2q2e-2~2 (0Fe2Ve-2b
<_x < co)
(-oc < x 5 O),
where F, q and < are parameters. We will determine these parameters later. In order to solve the ambipolar transport equation, a Fourier transform used, i P+(x,z) =
JwP+(s,z)ees2ds, 0 e-27p - -som f(s)e-szds, P_(x,z) =
s0 e2vx = _-oo
f(4es%where f(s) = 6(s - 277).
(6)analysis is (7) (8) (9) We expand the excess carriers concentration Ph(s, z) in terms of a superposition of cosine functions.
P&, z) = 5 A,(s) cos /&z. (10)
Substituting equations (5), (7), (9) in to equation (4), using equation (10) and letting
M= ew2(’ cos ,&zdz, -e/2
we get P+(z,z).
From equation (6), (8), (9), using the same method, we get P-(2,~). They have the form
P+(x, z) = e-2qx 5 q&M cm t%Z, n=O
P-(X, z) = e2qx 5 q&M cosP,z, n=O where 4n = - F , D,l 4rj2 - p; - & P 2
M = 4t2 + p; 2[sinh[! . cos i,D,! + ,&cosh[! . sin ipn!
The surface boundary conditions are
-D, [“p~~-“)],=t = SpP+h4,+
where S, is the surface recombination velocity. In order to satisfy the surface boundary conditions (15), (16), we must have
g&e = y .
111-2. Excess carrier currents
The time averaged energy flux of an electromagnetic wave in a semiconductor is given by the real part of the complex poynting vector:
(S,) =ERe (E, x Hk).
Using the electromagnetic fields in the semiconductor and equation (18), we get
-V . (3,) = ERe [(--k,2Am2BX t k,2Am1Bk) e-2krZze~2kz2z] , (19) where -V - (s,) is the electromagnetic energy per unit volume per unit time lost in the semiconductor and A,l, Am2, B, are the amplitudes of E,, , E,, , H, respectively.
Since gs(5, z) is the generation rate of excess carriers per unit volume per unit time for every photon absorbed, one electron-hole pair is created in the semiconductor. We define
where cr is the absorption coefficient and ! is the thickness of the sample. From equations (19), (20), we get
77 = Re[b2], E = Re[h2],
F = &Re [-lc,2A,&IL + IC,2AmlB~] aeeae.
Applying Maxwell equations, and equations (l), (a), (3), (22), we get
where A01 is the amplitude of the incident electric field E,,, and w is the angular frequency of the incident electromagnetic wave, while wz = e is the plasma frequency.
The diffusion flux density of excess carriers can be calculated from the following relations:
Using equations (ll), (12) and the above relations, we get the diffusion flux density of excess carriers
J$(x, .z) = 2D,qe-2vx 2 &iW cosp,z, n=O
J;(z, z) = 2D,qe2v” F &M cosp,z, n=O
J$(x, z) = D,e-2q” E ,&&M sin&z,
J;(x,z) = -Dpe2vx F&&Msin,&z.
The excess carrier currents are defined by
JO” I, = elytz J,-(z,z)da: +
Jx hzPz ,+
JO” I, = etylz J,-(x,z)dz - +
Jz (WW 7
where e is the charge of the electron and hole, and & and e, are the width and thickness of the sample. Substituting equations (24), (25) in to equations (26) and (27), we obtain the currents at z + 4,
I, = 2eQ,f!,
I, = -~ee,Y,D, 5 ,&&M sin k,&&. (29)
Using equations (13), (14), (23), (28) and (29), we get the currents
111-3. Quantum efficiency
The time averaged energy flux of an electromagnetic wave in the spacer layer is given by the real part of the complex poynting vector
(go) =&Re [Eo x H,‘].
z component of ($0)
(so,) =ERe [Aol
where Aur and Bo are the amplitude of the electric field E,, and magnetic field Ho, respec-tively.
The total number of photons per unit time incident upon the surface of the semicon-ductor is
From equations (33), (34) and the boundary conditions, we find the total number of photons per unit time to be
4ntLw SW; IAor12.
The Quantum efficiency is defined by
Using equations (30), (31), (35) and (36), we get the Quantum efficiency in two cases. Case (l), when the surface recombination velocity is very small (S, -+ O), the current I, + 0, so the Quantum efficiency is
&E X = z P P 1 -ae,
Case (2). When the surface recombination velocity is very large (S, -+ oo), the current 1, -+ 0, so the Quantum efficiency is
2 c El
These are the ratio of the number of excess carriers per unit volume per unit time to the number of incident photons per unit volume per unit time.
IV. Discussion and conclusions
When an electromagnetic wave is incident on a semiconductor, the generated electron-hole pairs are excited in a thin surface layer and their subsequent decay is expected to be largely influenced by surface recombination and diffusion effects.
Now we apply the above theory to a few typical cases. There are two limiting cases that are of practical importance. We shall give numerical estimates for the currents z,I,, and the Quantum efficiency QEX, QEZ.
We take the thickness and width of the sample !J, = 5 x 10v2 cm, and & = 3 cm,
the diffusion length L, = 1 x 10m3 N 8 x 10m3 cm, the absorption constant cx = 250
l/cm, the incident electric field Eo = 15 stat volt/cm, the plasma frequency w,” = 1 x 1O25 I/ sec2, the excess carriers life time % = 6 x lo-’ set, the angular frequency of the incident
electromagnetic wave w = 1.7 x 1Or5 rad/sec.
Case (1). Low surface recombination velocity (S, + 0). From the surface boundary condition, if S, + 0, it gives E -+ 0, corresponding to the condition of no net diffusive flow to the surface. As S, + 0, from equation (17), 4,&e --+ 728, sin +,&J? -+ 0, therefore 1, --+ 0. Now from equation (30), (37) an using the above numerical value, we find thed excess carrier current to be 1, = 6 x lo-r5 Amp N 4 x lo-r3 Amp, and the Quantum
efficiency is QEX = 3.5 x lo-l5 N 2.25 x 10-13.
Case (2). High surface recombination velocity (S, + oo). From equation (17), if + 00, cot ‘p e + 0
?merical valui &d equatto; (31),2(38)
‘p e + ‘(2n + 1)7r, cos $p,e --+ 0, therefore 1, -+ 0, using the we find the excess carrier current I, = 0.25 Amp N 2.0 Amp and the Quantum efficiency is QEZ = 0.28 N 1.12. We plot the numerical value of I, and QEZ in Fig. 2, and Fig. 3.
When the surface recombination velocity S, + 00, if the diffusion length L, > 8x 10m3 cm, there is the possibility of an apparent Quantum efficiency greater than unity (QEZ > 1).
L (Amp) A 2.2 ' QEZ A 1.1 .I . 1.8 '. 0.9.. jj_i/_/< 1.4 " 1 . 0 " 0.5 0.7. ; . 0.3 / 0.1 ,f I > 0 1 0 30 50 7o L,(w) 0 10 30 50 70 -h&m)
FIG. 2. The excess carrier currents 1, function of diffusion length
L , . of excess carrier diffusion length L,.
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