A calculation is presented for the direct conversion of a measured luminescence signal to the implied open circuit voltage. The effects of re-absorption, spectral dependence, and interference with front and back interfaces are all included, so long as the optical properties of the structure are known. The method is validated through a comparison of the terminal open-circuit voltage with the photoluminescence intensity of Cu(In,Ga)Se2 solar cells, each as a function of illumination intensity.

Luminescence imaging is a measurement technique used for development and rapid assessment of photovoltaic devices.1 Since the luminescence intensity has an exponential dependence on the quasi-Fermi level splitting between electrons and holes, ΔEF in Fig. 1(a), the implied open circuit voltage can be probed, even before the fabrication of an electrical contact structure.

FIG. 1.

(a) An illustration of electron and hole Fermi levels separated and within the gap of a semiconductor. (b) The absorption coefficient of CuIn1-0.5Ga0.5Se2 (EG = 1.3 eV) taken from Ref. 10, with the resulting normalized PL intensity calculated from Eq. (1).

FIG. 1.

(a) An illustration of electron and hole Fermi levels separated and within the gap of a semiconductor. (b) The absorption coefficient of CuIn1-0.5Ga0.5Se2 (EG = 1.3 eV) taken from Ref. 10, with the resulting normalized PL intensity calculated from Eq. (1).

Close modal

For a luminescence signal to take on a quantitative meaning, its relationship with the implied open circuit voltage must be calibrated. This can be done by determining photoluminescence (PL) or electroluminescence (EL) signals at several excitation (intensity) levels and solving for ΔEF within a multiple diode circuit model.2 Alternatively, the effective carrier lifetime may be independently estimated with some other measurements such as photoconductance and the corresponding ΔEF determined.3 However, uncertainty remains about the level of detail needed in a circuit model to give reliable results.4,5 Any such circuit analysis will also depend upon approximations such as excitation-independent photocarrier lifetimes. Scaling factors relating a measured PL signal to ΔEF are desirable for contactless characterization of solar cells and potentially other optoelectronic devices.

A number of authors have related ΔEF to absolute measurements of PL magnitude (in photons m−2 s−1) by way of

N(E)dE=a(E)Ω4π33c2E2exp(EΔEF)kBT1dE,
(1)

which is the “Lasher-Stern-Wurfel” (LSW) formula.6–9 This generalized form of both Planck's Law and the van Roosbroeck-Shockley (VRS) relation applies to band-to-band PL in a material with a non-equilibrium density of excited electrons and holes which have each reached a quasi-equilibrium within the conduction and valence bands, respectively. In Eq. (1), Ω is the solid angle through which detected photons of energy E=ω pass. The optical absorption coefficient α(E) of the material is contained within the absorptivity

a(E)=1exp[α(E)d],
(2)

where d is the thickness of the radiating layer. The other symbols in Eq. (1) have their usual meanings.

The spectrally resolved PL is related to α(E) by the LSW formula. Figure 1(b) shows the energy dependence for α(E)10 and the corresponding N(E) from Eq. (1). By restricting the analysis to PL photon energies somewhat above the bandgap (EG), the quantities α(E)d and exp[(EΔEF)/kBT] are both 1, the absorptivity becomes 1, and the energy-dependent PL expression for N(E) in Eq. (1) may be rewritten as

lndN(E)E2lnΩdE4π33c2(EΔEF)kBT,
(3)

which gives ΔEF as an offset to the PL magnitude versus energy relationship without reference to thickness or α(E). However, ambiguities and unrealistic results often follow from the subjective choice of high energy range for α(E). Possibly, this is because solving for ΔEF involves extrapolating Eq. (3) to the E0 limit, amplifying any variations from the chosen fit range or from non-uniform composition. Authors have attempted to instead fit the entire PL spectrum, although this relies on adopting an uncertain absorption tail model at low energies and is particularly problematic in multilayer structures.7,8

One alternative is to use the full, integrated PL intensity. Although this would invalidate the simplifying assumption of a(E)=1, the overall intensity should be fairly insensitive to uncertainties resulting from extrapolating the dependence in Eq. (3) to E0 and from the precise details of low energy tailing.

Since the total luminescence efficiency of a solar cell is directly related to its power generation efficiency,11,12 a number of calculations have already been put forward for the absolute emission rate of photons from a layered structure. Re-absorption and multiple front and back interface reflections should be considered13,14 as well as the angular and spectral dependence of the reflection and absorption.15 While interference is typically ignored with the use of ray optics, it can become significant with thin films, particularly those with a composition grading that changes over length scales less than the order of the wavelength.

Surprisingly, there does not appear to be any reported calculation relating ΔEF within layered structures to external emission rates, which quantitatively accounts for interference and re-absorption. One attempt was made to include all such effects, by finding the Poynting vector using the fluctuation-dissipation theorem.16 However, the resulting Poynting vector was not proportional to a(E) for thin layers, in contradiction to the LSW formula. Also, the fluctuation-dissipation theorem applies only in thermal equilibrium conditions, not simply to steady state conditions where thermal equilibrium has not been established.17 

Another calculation, taking into account interference in a graded-gap CuIn1-xGaxSe2 (CIGS) material stack,18 combined the LSW formula with an expression ultimately due to Lukosz.19,20 This approach gave the power of optical emission into an infinite medium (such as air) from an ensemble of radiating dipoles embedded in a slab of known dielectric properties, without considering re-absorption. However, inspection reveals that in the homogeneous case, when the slab has the same index as the infinite medium, this expression yields an unphysical result which continues to depend on the thickness of the slab.20 A reformulation of the Lukosz expression by Broeck had similar problems in the homogeneous case, with the s-polarized power increasing as the surrounding medium's index of refraction increased.21 Other closed-form expressions which have been reported for dipole radiation through interfaces apply to a dipole embedded in a semi-infinite medium, rather than within a film.22,23 A recently reported calculation included interference and re-absorption in relating radiative recombination rates, at a given depth and energy, to external emission rates,24 although the result was not in the closed-form and was computationally intensive.25 

This work presents a method of calculating the external emission of luminescence from an excited semiconductor within a layered structure of known optical properties. Sipe's dyadic Green's functions26 are used to find a closed-form expression for the impact of the structure on the external emission from a randomly oriented dipole radiation source at a given energy and location. The absolute emission at an energy E resulting from a value of ΔEF at a given location is taken from the LSW formula.

We wish to find the electric field in a medium with index of refraction n1, caused by a single dipole P0 depicted in Fig. 2 at position x=y=0,z=z0 and oscillating with frequency

ω=k0c=E
(4)

inside of a slab of thickness d and (complex valued) index of refraction n3. The polarization due to the dipole is

P=P0δ(x)δ(y)δ(zz0),
(5)

where the time dependence has been suppressed. While the electric field has not been explicitly given in the literature, the field for a thin sheet (at z=z0) of periodic polarization with a sinusoidally varying density has been solved.26 Therefore, we expand the delta functions δ(x)δ(y) in the polarization into a sinusoidal sheet representation

P=P0δ(zz0)(2π)2+exp(ikxx+ikyy)dkxdky=P0δ(zz0)(2π)202π0exp(ikcosφx+iksinφy)kdkdφ.
(6)
FIG. 2.

Geometry of a dipole embedded in a slab (3) of known dielectric properties, between two interfaces, which emits energy into a surrounding medium (1). (a) The xz plane is shown. (b) The xy plane, which is the surface of the slab (3), is shown from above.

FIG. 2.

Geometry of a dipole embedded in a slab (3) of known dielectric properties, between two interfaces, which emits energy into a surrounding medium (1). (a) The xz plane is shown. (b) The xy plane, which is the surface of the slab (3), is shown from above.

Close modal

For a given sheet of polarization

Psh=P0kdkdφ(2π)2exp(ikcosφx+iksinφy)δ(zz0),
(7)

the electric field was found to be26 

Es=As(k)dkdφγzγφsin2φγφsin(2φ)/20γφsin(2φ)/2γφcos2φ0000P0=Gs(k,φ)P0
(8)

for s-polarized radiation, where

As(k)=ikk02t31s(k)(βs+1)2πγ0w3μs
(9)

and

γz=exp(iw1z),γ0=exp(iw3z0),γd=exp(i2w3d),γφ=exp(ikxcosφ),βs=γ02γdr32s(k),μs=γdr31s(k)r32s(k)1.
(10)

The in-plane wavenumber has magnitude k and azimuthal direction φ, and it does not change from layer to layer. The z-component wavenumbers are w12=n1k02k2 and w32=n3k02k2. The reflection and transmission coefficients from the slab to the observer's medium are r31 and t31, and the back or bottom interface reflection coefficient is r32. The reflection and transmission coefficients are labeled with s or p for the differing values they will take from the Fresnel equations for incident s- and p-polarized light, at a transmitted angle of

sinθt=kk0n1
(11)

from Snell's law.

The classic Fresnel formulas for a single interface are

rijs=wiwjwi+wj,tijs=2wiwi+wj,rijp=winj2wjni2winj2+wjni2,tijp=2ninjwinj2+wjni2,
(12)

but the reflection and transmission coefficients are not limited to these since the back interface may be reflecting (e.g., a metallic contact) and the front interface in particular may have multiple layers. Multi-layered interfaces may be combined into a single pair of effective reflection and transmission coefficients using a coherent matrix formalism.27 Since dipole moments of x and y orientation are fully included, we have without loss of generality evaluated the fields only for an observer in the xz plane at a distance r2=x2+z2 and polar angle θ.

We integrate each sheet over dkdφ to obtain the resultant electric field from a single oscillating dipole. The magnitude of the Poynting vector, and hence power per area, is then found from

|S|=c8π|E|2.
(13)

However, note that the field cannot be squared until the integral over dkdφ is carried out; for the various planar wave components to combine to form a polarization delta function, they must be oscillating in-phase and thus able to mutually interfere. Only after the electric field due to a delta function source is found can they be viewed as separate sources emitting incoherently, with additive S values.

Taking the center element of Gs,

Gs22(k,φ)dφdk=0As(k)γz02πexp(ikxcosφ)cos2φdφdk,
(14)

the φ integral may be solved using Bessel functions. At this point, we take the fields at a macroscopically distant detector, so that the observer distance r2=x2+z2 is many wavelengths from the layered structure. This allows us to use the asymptotic form of the Bessel functions22 so that the right-hand side of Eq. (14) may be written as

=As(k)γzπ[J0(kx)J2(kx)]dkAs(k)γz2πkxcoskxπ4dk.
(15)

The remaining elements of Gs become zero after φ integration.

The macroscopic observer distance allows us to evaluate the integral over dk using the stationary phase method.28 Defining

g1=w1z+kxπ4,g2=w1zkx+π4,
(16)

we have

As(k)γz2πkxcoskxπ4dk=As(k)π2kx[exp(ig1)+exp(ig2)]dk.
(17)

At macroscopic distances, the products kx and w1z are very large. Therefore, As(k) is slowly varying, and the dominant range of the integral is in the neighborhood of the stationary points of the phases gi. Recalling that w1 is a function of k, the phase

g1(k)=zk02n12k2+kxπ4
(18)

has a stationary point at g1(kc)=0, which is solved by

kc=k0n1xr=k0n1sinθ.
(19)

Note from Eq. (11) that this value of kc corresponds to θt=θ or propagation in the direction of the observer angle. Since the phase g2 has no real positive stationary point, the integral is then solved by28 

As(k)π2kx[exp(ig1)]dkAs(kc)π2kcx2iπg1(kc)
(20)

for a total electric field of

Estot=πAs(kc)ikstxg1(kc)000010000P0=GstotP0.
(21)

For p-polarized light, the field for a sheet of oscillating polarization is26 

Ep=Ap(k)dkdφγz×w1w3(βp1)γφcos2φ0kw1(βp+1)γφcosφ000kw3(βp1)γφcosφ0k2(βp+1)γφP0=Gp(k,φ)P0,
(22)

where

Ap(k)=ikt31p(k)2πγ0n1n3w3μp.
(23)

For brevity, the elements that will integrate to zero over φ have already been dropped.

The integral over φ is carried out as before, this time leaving more than one nonzero element

Gp(k,φ)dφdkP0=Ap(k)γzw1w3(βp1)0ikw1(βp+1)000ikw3(βp1)0ik2(βp+1)×2πkxcoskxπ4dkP0.
(24)

For each element, the same phases g and stationary point kc still occur, for a total electric field of

Eptot=πAp(kc)ikcxg1(kc)×w1cw3c(βpc1)0ikcw1c(βpc+1)000ikcw3c(βpc1)0ikc2(βpc+1)P0,=GptotP0,
(25)

where the subscript c indicates that kc is substituted for k within that term. This solves for the electric field due to both polarizations, providing the dyadic Green's function relating a dipole moment to the electric field at a macroscopically distant observer

Etot=(Gstot+Gptot)P0=GtotP0.
(26)

The total Poynting vector magnitude (S) is found by squaring and summing the electric field amplitudes for each dipole orientation, according to Eq. (13). For an ensemble of randomly oriented, incoherent dipoles of magnitude P0, the power from the three possible dipole orientations is averaged. Since an individual dipole could be oriented, e.g., between the x and z axes, it might seem that the x and z components of P0 could produce mutually interfering electric fields and that the field should be found for each possible orientation and integrated before solving for S. However, for a random ensemble, the constructive/destructive effect of the dipole x and z components interfering with each other is always matched by a destructive/constructive effect from some complementary dipole with opposite x or z components. Thus, once the planar polarization densities are integrated into δ function densities, then the fields may be squared, and all components and dipole orientations may be summed and divided by 3.19 

For s-polarized radiation, the result is

Ss=|t31s(kc)|2ck04n13cos2θ|βsc+1|224r2πσ|μsc|2P02,
(27)

and for p-polarized,

Sp=|t31p(kc)|2ck04n1cos2θ(σ|βpc1|2+n12sin2θ|βpc+1|2)24πr2σ|n3|2|μpc|2P02,
(28)

where

σ=|n32n12sin2θ|.
(29)

To find the power passing through a detector aperture subtending half-angle θobj, we integrate

P(E)=02π0θobj(Ss+Sp)r2sinθdθdφ=2πr20θobjStotsinθdθ,
(30)

where P(E) is a function of energy because ck0=E/, as in Eq. (4).

Let us first examine the homogeneous case, that of a bare dipole outside of any layered structure, where n3=n1 and all reflectivity coefficients are zero,

Pbare(E)=2π0θobjPo2ck04n112πr2r2sinθdθ=Po2ck04n1(1cosθobj)/6.
(31)

For a randomly oriented radiative recombination event, the ratio P/Pbare gives the average fraction of power that radiates into a detector aperture, as a portion of the power which would reach the aperture in the case of no interfaces. That is, given a rate of photons per second which would reach the detector in the homogeneous case, multiplying by P/Pbare gives the expected rate of photons per second that would successfully reach the detector in the case of absorption and reflections. If the light-emitting area is sufficiently small, such as that illuminated by a focused laser beam, then the observer position (r,θ) may be considered constant for every part of that area, and P/Pbare is the same for all of the region of interest.

For an area A in the homogeneous case, the LSW formula already tells us the rate R of photons passing through an aperture with solid angle Ω per second, R=AN(E)dE. Multiplying this by P/Pbare, we have the total rate R reaching the detector, which has solid angle Ω=2π(1cosθobj),

R=P(E)APbare(E)a(E)(1cosθobj)2π23c2E2exp(EΔEF)kBT1dE.
(32)

R is numerically integrated over the energy range of expected luminescence, near the bandgap energy Eg. Note that if ΔEF becomes large enough to enter the energy range of luminescence, the LSW formula breaks down unless the change in absorption α(E) at high excitation ΔEF is also accounted for.

If the luminescing portion of a film has a significant thickness, R is also integrated over z0, with each thin layer dz0 having an absorptivity of a(E)=α(E)dz0, from Eq. (2). Recalling that the values represented by the μ and β symbols remain functions of kc and hence of θ, the integral over θ in Eq. (30) cannot generally be solved in the closed form, but we will see that a first order approximation in θ is reasonable for typical apertures. We make the common assumption that ΔEF is flat throughout the absorber, and that any reduction in splitting at the absorber's edge, near the back Ohmic contact, has a negligible effect on total luminescence.29,30

To relate the number of photons to a raw detector signal, a standard source of known absolute radiant flux is needed. Small, circular light-emitting diodes (LED) have been suggested,31 though we found it difficult to keep the LED in the linear output regime while radiating an intensity comparable to an easily measured PL signal. Instead, we used the excitation laser spot on a Spectralon reflectance standard, which has a reflectance of nearly unity with a highly diffuse character. When illuminated with a laser spot of intensity I, the surface of the reflectance standard forms a Lambertian source with radiance I/π. The rate of photons passing through the aperture in this case would be32 

Rref=A02π0θobjcosθIπωLsinθdθdφ=AIωLsin2θobj,
(33)

where ωL is the photon energy of the laser light. The power AI can be measured with a power meter, yielding a known Rref for that aperture and laser intensity. A conversion factor C is then found by setting the detector signal (in V or A) equal to CRref, to account for detector responsivity and any optical losses between the sample surface and the detector. C is also multiplied by the ratio of the detector responsivity at the PL wavelength to the detector responsivity at wavelength 660 nm for the laser. When measuring PL, this allows us to equate the detector signal to CR, obtaining ΔEF by Eq. (32).

The approach described is validated using solar cells based on CIGS absorbers. The cells were in the substrate configuration, where a 2.5 micron layer of CIGS was deposited on a molybdenum back contact followed by a 70 nm window layer of CdS. The back contact reflectance was taken as 0.3.33,34 For electrical measurements, devices were completed with a 210 nm ZnO bi-layer and Ni/Al grids. Power conversion efficiencies greater than 15% were obtained when absorbers were graded, with a back contact composition of x = 0.50, a front composition of x = 0.35, and a minimum of x = 0.25, where x = Ga/(Ga + In).35 Other absorbers were non-graded and fixed at one of two compositions x = 0.50 or 0.35.

For VOC measurements, illumination was provided by a spectrally uniform laser-driven light source filtered from 400 to 850 nm. For PL-I measurement, illumination was provided by a 660 nm diode laser focused by a 6.5× objective lens. A was found from the spot diameter, or full width at half maximum (FWHM), of 150 μm which was obtained using the standard knife edge method.36 The power was measured with a NIST-traceable power meter.

The resulting PL was collected through the same objective used to focus the excitation laser, with θobj=10° and aperture NA = 0.17, and passed through filters to remove laser light. The PL intensity was recorded by an InGaAs photodiode connected to a lock-in amplifier. The laser intensity was changed over six orders of magnitude by a series of calibrated neutral density filters.

For photon energies in the range of PL, about 1.1 to 1.3 eV, the complex index of refraction for CIGS at varying alloy composition (x) was used,10 and the CdS window was assumed to have a real index of refraction of 2.3. Resulting calculations are shown in Fig. 3(a) for x = 0.50, using either a full numerical integration over θ or by approximating the integrand in Eq. (30) to first order in θ, as f(θ)f(0)+θf(0). For most values of the aperture size θobj, corresponding to NA0.5 in air, the first order approximation appears to be adequate. In Fig. 3(b), an example calculation is shown. For convenience as a reference, it is given under the simplified conditions of a CuIn0.5Ga0.5Se2 absorber layer with no CdS coating in air and a back reflectance of unity. The simplified conditions give a 15% larger signal for a fixed ΔEf. Photons are passing through an objective with θobj = 10° at room temperature.

FIG. 3.

(a) Calculated detector signal as a function of the aperture opening, comparing full integration over the polar angle with taking only the first order. (b) Example of calculated photons per second entering into objective with aperture NA = 0.17.

FIG. 3.

(a) Calculated detector signal as a function of the aperture opening, comparing full integration over the polar angle with taking only the first order. (b) Example of calculated photons per second entering into objective with aperture NA = 0.17.

Close modal

The measured PL intensity is equal to CR. Since C is established by the calibration, the value of quasi-Fermi level splitting ΔEF is found which corresponds to the correct value for R in Eq. (32). This ΔEF is shown in Fig. 4, where it is compared to the terminal open-circuit voltage VOC, each found at a range of excitation intensities. The ranges generally do not quite overlap, since it is difficult to maintain intensities greater than 10 equivalent suns over a wide area for VOC measurements, while PL is difficult to detect at lower excitation intensities. Even with the same excitation, the terminal VOC is not necessarily the same as ΔEF, but the departure is often small in CIGS.29 

FIG. 4.

Comparison of the measured open circuit voltage to the quasi-Fermi level splitting extracted directly from the measured PL intensity without free parameters. An intensity of 1.7 × 1017 photons cm−2 s−1 is labeled as an equivalent sun.

FIG. 4.

Comparison of the measured open circuit voltage to the quasi-Fermi level splitting extracted directly from the measured PL intensity without free parameters. An intensity of 1.7 × 1017 photons cm−2 s−1 is labeled as an equivalent sun.

Close modal

The increase in implied ΔEF as x increases from 0.35 to 0.5 can be seen either with VOC or with optical measurements. For the graded sample, ΔEF is the largest, and the PL can be detected at the lowest I. It appears that at high intensities, ΔEF for this sample falls below that of the 0.5 sample. This is at least partly due to the majority of the PL coming from a position z0 near the minimum energy gap within the graded material. This allows the relatively high ΔEF to quickly reach a level where it would overlap the relatively low energy of the emitted PL spectrum at the bandgap minimum. This means that the use of a fixed α(E) in the LSW formula breaks down for the highest excitations I in the graded gap absorber, where the excitation is strong enough to change α(E) from its literature value.10 

The luminescence from a source within a slab of known optical constants is found for the first time in a way which includes re-absorption and interference with front and back interfaces. This leads immediately to a quantitative relationship between ΔEF and radiance. Using a radiance standard, an absolute relationship between the detector signal and ΔEF can also be obtained. The calculation contains a double numerical integration over energy and depth, but a triple integration to include the polar coordinate should not be necessary. No calibration cells with independently known ΔEF values are needed, nor are any assumptions concerning band offsets or electrical properties, so long as the optical properties are known and ΔEF is approximately flat through most of the absorber region. The PL method is useful for as-grown material stacks and does not require additional device processing. Furthermore, it is applicable to other photonic devices such as light-emitting diodes where the evaluation of the quasi-Fermi level splitting is important.

This work was supported by the U.S. Department of Energy's (DOE) Office of Energy Efficiency and Renewable Energy (EERE) under Solar Energy Technologies Office (SETO) Agreement No. DE-EE0007541. This work was authored in part by the National Renewable Energy Laboratory, operated by the Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.

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