Measurement of the photocurrent as a function of the thickness of a light absorber has been shown herein both theoretically and experimentally to provide a method for determination of the minority-carrier diffusion length of a sample. To perform the measurement, an illuminated spot of photons with an energy well above the band gap of the material was scanned along the thickness gradient of a wedge-shaped, rear-illuminated semiconducting light absorber. Photogenerated majority carriers were collected through a back-side transparent ohmic contact, and a front-side liquid or Schottky junction collected the photogenerated minority carriers. Calculations showed that the diffusion length could be evaluated from the exponential variation in photocurrent as a function of the thickness of the sample. Good agreement was observed between experiment and theory for a solid-state silicon Schottky junction measured using this method. As an example for the application of the technique to semiconductor/liquid-junction photoelectrodes, the minority-carrier diffusion length was determined for graded thickness, sputtered tungsten trioxide and polished bismuth vanadate films under back-illumination in contact with an aqueous electrolyte. This wedge technique does not require knowledge of the spectral absorption coefficient, doping, or surface recombination velocity of the sample

Atwater, Harry A.

Leenheer, Andrew J.

Lewis, Nathan S.

Lichterman, Michael

Pala, Ragip A.

Caltech Authors - Main

Measurement of Minority-Carrier Diffusion Lengths Using Wedge-
Shaped Semiconductor Photoelectrodes
Supplementary Information
Calculation of photocurrent
Fig. S1 Geometry for one-dimensional photocurrent calculation.
The device model assumes abrupt, full depletion of minority carriers (holes in our n-type semiconductor 
example system) at a depletion-width boundary at a distance Wd from a barrier-type contact; a back 
contact with an arbitrary recombination velocity; low-level injection with a bulk minority-carrier lifetime 
τ; and simple exponential light absorption, with each absorbed photon generating one electron-hole pair. 
The geometry used to set up the following model is shown in Fig. S1, where light is incident on the left 
side of a semiconductor slab at x = 0 through a transparent contact (Contact 1), and any light that is not 
absorbed escapes through Contact 2 on the right side of the diagram. Photogenerated charge carriers are 
lost either by bulk recombination or by collection at either boundary with a collection (or recombination) 
velocity s. The solution to the diffusion equation for excess minority holes is then:10
, (S1)∆𝑝(𝑥) = 𝐴𝑒 ‒ 𝑥 𝐿𝑝 + 𝐵𝑒𝑥 𝐿𝑝 + 𝐶𝑒 ‒ 𝛼𝑥 
where the coefficients A, B, and C are given by
 ,
𝐴 = 𝐶((𝑆2 ‒ 𝛼𝐿𝑝)(1 ‒ 𝑆1)𝑒 ‒ 𝛼𝑊 + (𝑆1 + 𝛼𝐿𝑝)(1 + 𝑆2)𝑒𝑊/𝐿𝑝(1 ‒ 𝑆1)(1 ‒ 𝑆2)𝑒 ‒ 𝑊/𝐿𝑝 ‒ (1 + 𝑆1)(1 + 𝑆2)𝑒𝑊/𝐿𝑝 )
 ,
𝐵 = 𝐶((𝑆2 ‒ 𝛼𝐿𝑝)(1 + 𝑆1)𝑒 ‒ 𝛼𝑊 + (𝑆1 + 𝛼𝐿𝑝)(1 ‒ 𝑆2)𝑒 ‒ 𝑊/𝐿𝑝(1 ‒ 𝑆1)(1 ‒ 𝑆2)𝑒 ‒ 𝑊/𝐿𝑝 ‒ (1 + 𝑆1)(1 + 𝑆2)𝑒𝑊/𝐿𝑝 )
Electronic Supplementary Material (ESI) for Energy & Environmental Science.
This journal is © The Royal Society of Chemistry 2014
,
𝐶 =  (1 ‒ 𝑅) Φ 𝜏 𝛼1 ‒ 𝛼2𝐿2𝑝  
and the dimensionless boundary collection/recombination velocities are defined as  evaluated 𝑆 = 𝑠𝐿𝑝 𝐷,
at the edges of the quasi-neutral region of width W. If the junction is on the right (front) side of the 
device, the assumed complete minority carrier sweep-out implies . The photocurrent density 𝑆2→∞
resulting from full collection by drift in the depletion region, as well as partial diffusion collection by 
considering the continuity equation for carriers that diffuse to the depletion region boundary, is then:
, (S2)
|𝐽𝑝ℎ𝑜𝑡𝑜 𝑒𝑐| = 𝑊 + 𝑊𝑑∫
𝑊
𝐺(𝑥)𝑑𝑥 + 𝐷 𝑑∆𝑝𝑑𝑥 |𝑥 = 𝑊
where ec is the charge on an electron. Equation S2 was implemented in Mathematica, and is plotted under 
various conditions in Fig. 1 in the main text. 
Effect of porosity
Fig. S2 Pore geometry considered allowing non-1-D minority-carrier collection.
Though porosity and nanostructuring can be a critical part of efficient electrode design for 
photoelectrochemical devices, any deviations from a dense, approximately planar film are potentially 
confounding issues for electrochemical diffusion-length measurements. Porosity in the film causes the 1-
D approximation to break down because photogenerated carriers can be collected by electrolyte that has 
filled in any cracks or connected pores, and the effective film thickness for collection is therefore not the 
true film thickness. The simple 1-D model can be modified by replacing the film thickness with an 
effective film thickness, given by
   , (S3)
𝑊𝑒𝑓𝑓 = 𝑊𝑓𝑖𝑙𝑚sin [tan ‒ 1 (𝑊𝑔𝑟𝑎𝑖𝑛 2𝑊𝑓𝑖𝑙𝑚 )]
where the geometry under consideration is shown in Fig. S2. In the limit that , , while 𝑊𝑔𝑟𝑎𝑖𝑛→0 𝑊𝑒𝑓𝑓→0
in the limit that , . Though this is a very crude correction to the 1-D model, it 𝑊𝑔𝑟𝑎𝑖𝑛→∞ 𝑊𝑒𝑓𝑓→𝑊𝑓𝑖𝑙𝑚
reasonably reproduces the behavior expected when 3-D collection is active. The photocurrent will then 
tend to become constant as a function of apparent film thickness, as shown in Fig. S3 where a correction 
has been made to the dotted-line base case without pores. 
Fig. S3 Photocurrent behavior with pores. The black dotted line shows a base case with strongly absorbed light and 
high back-surface recombination without pores, and the red solid line is calculated with grains of width Wgrain = 
Lp/25.
WO3 Film Characterization
The tungsten oxide showed different film morphologies when films were grown at various substrate 
temperatures. XRD data for films deposited at different temperatures are shown in Fig. S4. All of the 
peaks could be matched to peaks ascribable to monoclinic WO3, but the texture and grain size varied with 
growth temperature.
Fig. S4 X-ray diffraction data for WO3 films deposited at various temperatures on F:SnO2-coated glass. The 
substrate peaks are shown as unlabelled red lines at the bottom, whereas the labelled lines are peak positions for 
monoclinic WO3. 
The linear thickness profile of the Si wedge was simply measured by cleaving and examining the cross-
sectional thickness of the sample, but determination of the thickness profile of the thin-film WO3 sample 
required more in-depth examination. An example of the optical interference fringes used to determine the 
thickness profile is shown in Fig. S4. To characterize the gradient in thickness, the film that was 
simultaneously deposited on the Si wafer was imaged in an optical microscope. The Si wafer provided a 
clear view of the optical reflectance fringes so that a reflectance image with 500 nm illumination allowed 
determination of the film thickness profile. An ellipsometric measurement for a known thickness WO3 
film was used to determine that the index at 500 nm was 2.34, so each maximum in the interference fringe 
profile corresponded to a thickness increase of λ/(2n)=107 nm. The thickness profile was fitted to a 
sigmoidal function. Additionally, the films deposited on FTO were often cleaved at the thick side to 
image the cross-sectional film morphology in a scanning-electron microscope (SEM), and the maximum 
thickness could be verified or used to adjust the optical data. The films grown on FTO could also be used 
to image the thickness fringes, but the lower index contrast at the substrate as well as the underlying FTO 
roughness gave less interference contrast. 
(a)
(b)
Fig. S5 (a) Image of WO3 wedge film on Si (top) and on FTO (bottom). Scale bar is 10 mm. (b) Top: Image of 
reflectance of WO3 wedge film on Si with illumination at 500-nm wavelength. Middle: Line profile of reflectance 
magnitude. Bottom: Thickness profile determined by positions of peak reflectance and known index.
An example of quiescent cyclic voltammetry under chopped illumination is shown in Fig. S6. The data 
were collected under chopped illumination so that the current both under illumination and in the dark was 
evident. The sweep was performed with the 1 mm focused illumination at an arbitrary point on the sample 
and under quiescent conditions, with some exposed FTO, so the absolute values were somewhat different 
than the standard case of fully filled front illumination on a well-sealed electrode. However, the general 
curve shape is as expected for WO3, with an onset potential of 0.2-0.3 V vs. E(AgCl/Ag) (0.4-0.5 V 
relative to the hydrogen potential). The photocurrent never fully saturated in this voltage range but began 
to level off at voltages of ~0.8-0.9 V vs. EAg/AgCl. For reference, the oxygen-evolution potential in this cell 
is at 1.03 V vs. E(AgCl/Ag).
Fig. S6 Cyclic voltammetry sweep of WO3 film under chopped 365-nm illumination.
Insulation of pores in porous WO3
After sputtering, the tungsten oxide was often too porous to obtain robust electrochemical data. To 
insulate the electrochemically active pores or grain sidewalls, the sample surface was coated with a 10 nm 
layer thick of aluminum oxide that was deposited by atomic layer deposition (ALD - Cambridge 
Nanotechnology). A 20 sec diffusion time in exposure mode was used to allow the precursor to fully 
diffuse inside the pores. The ALD-treated samples showed no electrochemical activity or photocurrent, 
verifying that the alumina was largely conformal and insulating, as desired. To expose the top surface, the 
samples were mounted on the tripod polisher, and were carefully levelled by use of a flat glass plate and 
by minimization of the number of Newton's rings visible between the plate and the sample. Gentle 
polishing was then performed using 50 nm colloidal diamond slurry (Buehler MetaDi Supreme 0.05um) 
on a porous polyurethane pad (Eminess Politex Reg) for 10-30 min on a polishing wheel rotated at ~50 
rpm. The best results and uniformity were obtained when the sample was rotated once or twice during 
polishing. For short polishing times, only the tops of the largest and highest-protruding grains were 
polished, as evidenced by the low SEM contrast and flat profile in an atomic-force microscope (AFM) 
scan. With somewhat longer polishing, most of the top grains were polished smooth, as evidenced by 
SEM images. As the sample was polished even further, the surface became extremely flat and was 
difficult to image in the SEM. One sample was also gently sputtered using the RF substrate bias in the 
AJA sputtering system. This process revealed the pores that had been insulated by aluminum oxide, 
because the tungsten oxide was more readily sputtered, leaving behind the aluminum oxide that had 
coated the sidewalls as protrusions. 
Large illumination spot scanned along a wedge with non-linear thickness variation
In the diffusion-length measurement analysis, a point spot illumination scanned over a film with a 
thickness gradient was assumed.  This procedure produced a photocurrent that exhibited an exponential 
decay with thickness, , where w is the thickness at the measurement spot and L is the 𝐽(𝑤) = 𝐽0 𝑒 ‒ 𝑤 𝐿
minority-carrier diffusion length. The experimental system however had a finite spot size, which 
produced a photocurrent value averaged over the illumination area with a gradient thickness. To evaluate 
the impact of the beam size on the diffusion length measurement, we assume a large beam spot incident 
on a wedge having a non-linear thickness variation (Fig. S7). 
Fig. S7 Schematic of a wedge with non-linear thickness variation illuminated by a large beam.
The average photocurrent at any point, i, can be calculated by integrating the photocurrent over the 
illuminated area:
 
?̅?𝑖 = 1∆𝑠∫
𝑖
𝐽(𝑤(𝑠)) 𝑑𝑠 ≈ 1
Δ𝑤𝑖
∫
𝑖
𝐽(𝑤) 𝑑𝑤
Here we assume the spot size, s, is small compared to wedge length, l, . The thickness gradient in ∆𝑠 ≪ 𝑙
each segment s is therefore assumed to be linear, i.e. .
𝑑𝑤
𝑑𝑠
= 𝑐𝑜𝑛𝑠𝑡
 
?̅?𝑖 = 𝐽0 1∆𝑤𝑖 
𝑤𝑖 + ∆𝑤𝑖 2
∫
𝑤𝑖 ‒ ∆𝑤𝑖 2𝑒
‒ 𝑤 𝐿 𝑑𝑤
         (S4)
?̅?𝑖 = 𝐽0𝑒 ‒ 𝑤𝑖 𝐿{ 𝐿∆𝑤𝑖 (𝑒∆𝑤𝑖 2𝐿 ‒ 𝑒 ‒ ∆𝑤𝑖 2𝐿)}
 
?̅?𝑖 = 𝐽0𝑒 ‒ 𝑤𝑖 𝐿 𝐿Δ𝑤𝑖 (1 + Δ𝑤𝑖2𝐿 + (Δ𝑤𝑖2𝐿 )212 + (Δ𝑤𝑖2𝐿 )316 + ⋯ ‒ 1 + Δ𝑤𝑖2𝐿 ‒ (Δ𝑤𝑖2𝐿 )212 + (Δ𝑤𝑖2𝐿 )316 ‒ ⋯)
 
?̅?𝑖 = 𝐽0𝑒 ‒ 𝑤𝑖 𝐿(1 + (∆𝑤𝑖𝐿 )2 124 + (∆𝑤𝑖𝐿 )4 11920 + ⋯)
for :  ∆𝑤𝑖 < 𝐿  ?̅?𝑖 ≈ 𝐽0𝑒 ‒ 𝑤𝑖 𝐿
The photocurrent is thus simply given by an exponential decay, provided that the thickness gradient is 
small, i.e. . ∆𝑤𝑖 < 𝐿
The value of   can be approximately given by , where  is the ratio of the film thickness to 
∆𝑤𝑖
𝐿
∆𝑤𝑖
𝐿
~ ∆𝑠
𝑙
𝑡
𝐿
𝑡
𝐿
the minority-carrier diffusion length. In our experiments, an illumination spot size of  was ∆𝑠 ≈ 1 𝑚𝑚
scanned over a wedge length of . Additionally, the film thickness was 2-3 times larger than the 𝑙 = 25 𝑚𝑚
minority-carrier diffusion length for each sample. Therefore  values were on the order of ~0.1 in the 
∆𝑤𝑖
𝐿
experiments.
Note that the requirement for small thickness gradient is less stringent when the wedge has a nearly 
uniform gradient. When the thickness gradient is constant, the photocurrent decay is given by the 
expression S4, which has the same exponential form but with an additional, constant, prefactor:
?̅?𝑖 = 𝐽0𝑒 ‒ 𝑤𝑖 𝐿{ 𝐿∆𝑤𝑖 (𝑒∆𝑤𝑖 2𝐿 ‒ 𝑒 ‒ ∆𝑤𝑖 2𝐿)}
?̅?𝑖 = 𝐽0'𝑒 ‒ 𝑤𝑖 𝐿
Diffusion length measurement on a wedge with non-uniform thickness variation
A non-uniform thickness can be a critical limitation in the determination of accurate minority-carrier 
diffusion lengths by the method developed herein. To evaluate the impact of non-uniform thickness 
variation, we assume that the films has a random thickness fluctuation characterized by a Gaussian 
distribution, as well as a large number of fluctuations within the illumination area. Assuming a thickness 
fluctuation with a standard deviation of σ, the average photocurrent over the illumination spot is given by:
?̅?𝑖 = ∞∫
‒ ∞
𝐹(𝑤) 𝐽0 𝑒 ‒ 𝑤 𝐿𝑑𝑤 = 1𝜎 2𝜋 𝐽0 ∞∫
‒ ∞
𝑒
‒
(𝑤 ‒ 𝑤0)22𝜎2  𝑒 ‒ 𝑤𝐿 𝑑𝑤
where w is a random thickness value in the distribution, w0 is the average thickness and L is the minority-
carrier diffusion length.
?̅?𝑖 = 1𝜎 2𝜋𝐽0 𝑒 ‒ 𝑤0𝐿 ∞∫
‒ ∞
𝑒
‒
𝑢22𝜎2 𝑒 ‒ 𝑢𝐿 𝑑𝑢
      (S5)?̅?𝑖 = 𝑒 𝜎
22𝐿2 𝐽0 𝑒 ‒
𝑤0
𝐿
?̅?𝑖 = 𝐽0' 𝑒 ‒ 𝑤0𝐿
The final expression indicates that that random fluctuations on the sample thickness will add a constant 
prefactor to the photocurrent expression, and thus will increase the absolute value of the measured 
photocurrent values while the exponential decay rate of the photocurrent with thickness remains the same. 
Furthermore, this prefactor can be ignored if the standard deviation is small compared to the minority-
carrier diffusion length, . Note that in this derivation, we assumed the standard deviation 𝑖.𝑒., 𝜎 ≪ 𝐿
remained the same along the thickness gradient. If thickness fluctuations were induced by film growth 
etc. then careful characterization of thickness variation would be required, especially if the fluctuations 
were comparable to the minority-carrier diffusion length. The samples used in our experiments were 
polished on top and the fluctuation in the thickness (~ 5 - 10 nm) was mainly caused by the substrate 
roughness (FTO). Hence the assumption of a constant standard deviation along the surface is justified for 
this experimental arrangement. 


Measurement of minority-carrier diffusion lengths using wedge-shaped semiconductor photoelectrodes

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Measurement of minority-carrier diffusion lengths using wedge-shaped semiconductor photoelectrodes

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