**4. Experimental PC Results and Fits**

tion recombination rates along the light propagation - i.e., at the surface and in the bulk re‐ gion - are different. At the surface, due to the increased density of recombination centers with respect to the bulk, the carrier lifetime is much shorter than in the bulk region. Specifi‐

s

é ù æ ö = -- - ê ú ç ÷ ë û è ø (29)

<sup>h</sup> (30)

a

<sup>h</sup> (33)

and assumed

b ( ) 1 1 exp( / ) *z z L* t

Where τs is the lifetime of the nonequilibrium carriers at the surface, τb is the lifetime of the nonequilibrium carriers in the bulk, and L is the length scale beyond which the recombina‐ tion rate is predominately ruled by τb. Thus we have a multi-time scale relaxation model.

0

 a

that diffusion length β-1 is a constant. We now substitute Eq. (30) into the general PC expres‐

 a

s

t

 th

w

= - =- - - - - - - <sup>+</sup> ò (32)

 a t

> a

 w

è ø

The first integral over the diffusion term is zero because of the zero flux boundary condition

( ) exp( )d 1 exp( ) (1 ) [1 exp( / )] <sup>1</sup>

*z L <sup>F</sup> zz d d dL*

 a

0 0 <sup>b</sup> ( )( )( ) *<sup>y</sup>*

m

Equations (31-33) basically correspond to the intuitively deduced PC dispersion Ref. [9]. We not that by expressing the PC in terms of the responsivity (R ) in [Ampere/Watt], i.e., for

*x <sup>l</sup> <sup>I</sup> I Ue l*

= D

 a

0 0

ò <sup>h</sup> (31)

 a

*L*

<sup>1</sup> exp( ) *d n <sup>I</sup> n z*

=+ -

Here, we have used the mean field approximation for the term Dτ≈D<τ>=1/β2

2

<sup>1</sup> ( )( ) ( ) exp( ) d

æ ö =D + - = ç ÷

 th

[Eq. (20)]. Thus, the dimensionless spectrum factor F takes the following form:

*ph Ph*

*<sup>l</sup> d n <sup>I</sup> I Ue <sup>z</sup> z z IF*

2 2

0

0 b

*Ph*

constant impinge light power, the proportionality R∝F holds.

b

*l dz* m

*d*

 w

t

cally, it was assumed that the spatial lifetime decay τ (z) takes the simple form:

b

2

2 2

b

*dz* th

 t

t

We rearrange Eq. (19), resulting in,

198 Optoelectronics - Advanced Materials and Devices

*y*

*x*

aa

where, the PC magnitude is given by,

sion Eq. (21), we find,

*d*

t

*b*

t The spectral PC dispersion was investigated by using the experimental arrangement in Fig. 1: The x, y plane of the CdS bulk material faces the incoming monochromatic light, while the z coordinate is parallel to the light propagation. The electric field of the light is perpendicu‐ larly oriented to the c-axis of the CdS sample, which was an industrially produced single crystal with d=1 mm. Two vacuum evaporated Al contacts with a gap of 0.5 mm between them were used for the electrical connection. The applied electric field driving the PC was 200 V/cm and the optical excitation was carried out with intensities typically in μW/cm2 range. The PC was recorded at room temperature with lock-in technique by chopping the impinging light at 25 Hz. The measured spectrum was corrected by a calibrated Si photo‐ diode in order to express the PC in terms of responsivity.

**Figure 1.** Experimental arrangement employed for the experiments. Note that d=lz.

#### **4.1. PC fit using Dutton's experimentally determined absorption coefficient**

The comparison of the experiment (symbols) with the fit (dotted line) using Eq. (31-32) is shown in Fig. 2(a). The best fit between the experimental data and theoretical formula was achieved with d=3.60×10-3 cm, L=8.65×10-5 cm, and τs/τb=0.068, while the used α(ωћ) values, which are shown in Fig. 2 (b), have been extracted from Dutton's paper [16]. The fit reveals that only an effective thickness of the sample, which does not necessarily correspond to the physical thickness, contribute to the formation of the PC signal. The corresponding conclu‐ sion was found by analyzing the photoluminescence of thin-film CdS [17].

s s b b <sup>1</sup> ln (1 ) ( ) ln (1 ) 12/ *L L <sup>d</sup>*

This shows that the maximum location (α \*d) in PC spectra is mainly controlled by two parameters τs/τb, L/d. Now we substitute the parameters d=3.60×10-3 cm, L=8.65×10-5 cm, and τs/τb=0.068 into α \* expression, we obtain α \*= 1108.54 cm-1. This corresponds to the energy E value between 2.41eV-2.42eV, and is exactly the energy region where the maxi‐

**4.2. PC fit employing the theoretical absorption coefficient based on the modified Urbach**

So far, we have used the experimental absorption edge data from Dutton directly, we may also fit the PC by modeling α(hω)with the density of states and the modified Urbach tail,

*α*(ℏ*ω*)= *A*<sup>0</sup> ℏ*ω* −*Eg*

<sup>0</sup> cr ( ) exp ( ) <sup>2</sup>

s

h h

*kT A E kT*

é ù <sup>=</sup> - ê ú ë û

where Eg is band gap energy, A0 is linked to the saturation value ofα(hω), kT is the thermal energy, and the crossover between Eqs. (36) and (37) takes place at Ecr=Eg+kT/(2σ ) [14], where σ defines the steepness of the Urbach tail [17]. Figure 3 shows the comparison of the measured PC [which is identical with the one in Fig. 2 (a)] with the fit using Esq. (32), (36), and (37), whereas the following fitting parameters were used: kT=26 meV, A0=3.26×10<sup>5</sup> cm-1 (eV)-1/2, d=3.69×10-3 cm, L=9.14×10-5 cm, σ=2.15, Eg=2.445 eV, and τs/τb=0.100. The fit parame‐ ters d and L are very close to the previous fit and we should stress that the σ value found (=2.15) is almost identical with the promoted CdS value (=2.17) of Dutton. Figure 4 shows the comparison of the calculated function *α*(ℏ*ω*) with Dutton's measurement and the good agreement of both curves proofs the suitability of the straightforward concept represented

s

w

*if*

*cr*

*E*

*if*

h

w

£

a w ℏ*ω* ≥ *Ecr*

\* æö æö =- - ç÷ ç÷ @- -

t

t

a

mum PC occurred.

which are expressed by [14],

by Esq. (35) and (36).

**rule**

and

*d Ld d*

 t

Theoretical Analysis of the Spectral Photocurrent Distribution of Semiconductors

http://dx.doi.org/10.5772/51412

201

 t

(36)

(37)

èø èø - (35)

**Figure 2.** a) The symbols represent the photocurrent measurements while the broken line was fitted with the Eq. (32). (b) The absorption coefficient for perpendicularly oriented CdS was used for the fit. The data were deduced from Dut‐ ton's paper (Ref. [16]).

Using Eq. (32) one can link the absorption coefficient to the location of the maximum of the

$$\begin{array}{cccc} \text{PC} & \text{spectra.} & \text{We} & \text{set} & \frac{dF}{d\hbar\omega} = \frac{dF}{da} \frac{da}{d\hbar\omega} = 0, & \text{and} & \text{ obtain,} \\\\ d\exp(-ad) - \left(1 - \frac{\tau\_s}{\tau\_b}\right) \mathbf{I} \frac{L}{\left(1 + aL\right)^2} (1 - \exp(-ad - d\left|L\right.)) + \frac{aL}{\left(1 + aL\right)} d\exp(-ad - d\left|L\right.) \mathbf{j}\mathbf{j} = 0 \end{array}$$

Since the term exp(-α d-d/L)<<1, we simplify the above condition to,

$$d \exp(-\alpha d) - \left(1 - \frac{\tau\_s}{\tau\_b}\right) \frac{L}{\left(1 + \alpha L\right)^2} = 0\tag{34}$$

resulting in −*αd* =ln((1− *τ*s *τ*b ) *L <sup>d</sup>* ) <sup>−</sup>2ln(1 <sup>+</sup> *<sup>α</sup><sup>L</sup>* )=0 and by using the approximation ln(1+ α L)≈ α L and L/d <<1, we finally find the α \* where the PC has the maximum value

$$\alpha^\* d = -\ln\left( (1 - \frac{\tau\_s}{\tau\_b}) \frac{L}{d} \right) (\frac{1}{1 - 2L/d}) \equiv -\ln\left( (1 - \frac{\tau\_s}{\tau\_b}) \frac{L}{d} \right) \tag{35}$$

This shows that the maximum location (α \*d) in PC spectra is mainly controlled by two parameters τs/τb, L/d. Now we substitute the parameters d=3.60×10-3 cm, L=8.65×10-5 cm, and τs/τb=0.068 into α \* expression, we obtain α \*= 1108.54 cm-1. This corresponds to the energy E value between 2.41eV-2.42eV, and is exactly the energy region where the maxi‐ mum PC occurred.

#### **4.2. PC fit employing the theoretical absorption coefficient based on the modified Urbach rule**

So far, we have used the experimental absorption edge data from Dutton directly, we may also fit the PC by modeling α(hω)with the density of states and the modified Urbach tail, which are expressed by [14],

$$\begin{aligned} \alpha(\hbar\omega) &= A\_0 \sqrt{\hbar\omega - E\_g} \\ \text{if} \\ \hbar\omega &\ge E\_{cr} \end{aligned} \tag{36}$$

and

**Figure 2.** a) The symbols represent the photocurrent measurements while the broken line was fitted with the Eq. (32). (b) The absorption coefficient for perpendicularly oriented CdS was used for the fit. The data were deduced from Dut‐

Using Eq. (32) one can link the absorption coefficient to the location of the maximum of the

<sup>2</sup> (1−exp(−*αd* −*d* / *L* )) +

*<sup>L</sup> d d*

α L and L/d <<1, we finally find the α \* where the PC has the maximum value

s

æ ö - -- = ç ÷

t

t

b exp( ) 1 0

Since the term exp(-α d-d/L)<<1, we simplify the above condition to,

a

*dF <sup>d</sup>*ℏ*<sup>ω</sup>* <sup>=</sup> *dF dα* *dα*

*αL*

2

*L*

(1 )

 a *<sup>d</sup>*ℏ*<sup>ω</sup>* =0, and obtain,

(1 <sup>+</sup> *<sup>α</sup><sup>L</sup>* ) *<sup>d</sup>*exp(−*α<sup>d</sup>* <sup>−</sup>*<sup>d</sup>* / *<sup>L</sup>* ) =0

è ø <sup>+</sup> (34)

*<sup>d</sup>* ) <sup>−</sup>2ln(1 <sup>+</sup> *<sup>α</sup><sup>L</sup>* )=0 and by using the approximation ln(1+ α L)≈

ton's paper (Ref. [16]).

*d*exp(−*αd*)−(1−

resulting in −*αd* =ln((1−

PC spectra. We set

) *<sup>L</sup>* (1 + *αL* )

> *τ*s *τ*b ) *L*

*τ*s *τ*b

200 Optoelectronics - Advanced Materials and Devices

$$\begin{aligned} \alpha(\hbar o) &= A\_0 \sqrt{\frac{kT}{2\sigma}} \exp\left[\frac{\sigma}{kT}(\hbar o - E\_{cr})\right] \\ \text{if} \\ \hbar o &\le E\_{cr} \end{aligned} \tag{37}$$

where Eg is band gap energy, A0 is linked to the saturation value ofα(hω), kT is the thermal energy, and the crossover between Eqs. (36) and (37) takes place at Ecr=Eg+kT/(2σ ) [14], where σ defines the steepness of the Urbach tail [17]. Figure 3 shows the comparison of the measured PC [which is identical with the one in Fig. 2 (a)] with the fit using Esq. (32), (36), and (37), whereas the following fitting parameters were used: kT=26 meV, A0=3.26×10<sup>5</sup> cm-1 (eV)-1/2, d=3.69×10-3 cm, L=9.14×10-5 cm, σ=2.15, Eg=2.445 eV, and τs/τb=0.100. The fit parame‐ ters d and L are very close to the previous fit and we should stress that the σ value found (=2.15) is almost identical with the promoted CdS value (=2.17) of Dutton. Figure 4 shows the comparison of the calculated function *α*(ℏ*ω*) with Dutton's measurement and the good agreement of both curves proofs the suitability of the straightforward concept represented by Esq. (35) and (36).

**5. Measurement of the lifetime at the surface (τs) and in the bulk (τb)**

the rather moderate intensity of about 0.6 W/cm2

the driving electric field was again 200 V/cm.

By measuring the temporal decay of the PC illuminating the sample with highly (*α*(ℏ*ω*)=105 cm-1) and less absorbed light (*α*(ℏ*ω*)<<102 cm-1) it should be possible the measure τs and τb. We made the temporal PC decay visible with a 500 MHz scope by employing chopped (18 Hz) continuous wave (cw) laser beams at 488.0 nm (2.54 eV) and 632.8 nm (1.96 eV), by us‐ ing an Ar-Kr and He-Ne laser, respectively. Both laser beams were unfocused resulting in

**Figure 5.** Photocurrent decay vs. time measured under the illumination of a laser emitting at (a) 488.0 nm and (b) 632.8 nm. The symbols represent the measurements, while solid and broken lines represent the fits done with Eq. (38).

Figure 5 shows the experimental results (symbols) and fits (solid and broken lines) of the decay. The fits were performed with the Kohlrausch function [19], which is an extension of the exponential function with one additional parameter γ that can range between 0 and 1,

= -

The following parameters resulted in the best fits for surface and bulk: τs=1.2 ms and γ=0.53, and τb=7.6 ms and γ=0.82, respectively, resulting in τs/τb=0.16. Despite the straightforward‐ ness of the experiment, which did not consider electric charging effects, the number is only approximately a factor 2.4 and 1.6 off from the predicted value using the data of Dutton and the modeled absorption edge, respectively. It is worthwhile to note that the Kohlrausch de‐

g

t

0 ph Ph s,b *It I t* ( ) exp[ ( / ) ]

cay can be expressed as linear superposition of simple exponential decays, i.e.,

and, as for the PC measurements above,

http://dx.doi.org/10.5772/51412

203

Theoretical Analysis of the Spectral Photocurrent Distribution of Semiconductors

(38)

**Figure 3.** Fit (broken line) of the measured photocurrent spectrum in Fig. 1 (symbols) using Eqs. (36) and (37). The fit hardly differs from the one in Fig. 2 (a).

**Figure 4.** Comparison of the absorption coefficient (a) after Dutton and (b) modeled with Eqs. (36) and (37).
