Efficiency of thin film photocells
Efficiency of thin film photocells
D. Mozyrsky and I. Martin
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
(Dated: Printed November 4, 2018)
We propose a new concept for the design of high-efficiency photocells based on ultra-thin (sub-
micron) semiconductor films of controlled thickness. Using a microscopic model of a thin dielectric
layer interacting with incident electromagnetic radiation we evaluate the efficiency of conversion
of solar radiation into the electric power. We determine the optimal range of parameters which
maximize the efficiency of such photovoltaic element.
Improvement of efficiency of semiconductor photo-
voltaic elements (solar cells) has been an important tech-
nological challenge for several decades. The maximum
possible efficiency obtains when every incident photon
generates an electron-hole pair, which then separates into
electron flowing to cathode and hole flowing to anode[1].
The limitations that reduce the efficiency of the practi-
cal solar cells relative to the ideal are 1) light reflection
at the interfaces, 2) incomplete absorption of light en-
tering the device due to finite thickness, 3) electron-hole
relaxation inside the absorbing medium during diffusion
to the leads[2, 3]. Interplay between two latter mecha-
nisms leads to an existence of an optimal device thick-
ness, typically a few optical wavelengths. Here we show
that the interface reflection, commonly considered a com-
pletely independent loss mechanism, shows an interest-
ing interplay with absorption in ultra-thin film devices.
This opens a possibility for a new generation of ultra-thin
(sub-wavelength) photovoltaic elements with efficiencies
rivaling the best conventional devices.
A “working body” of a solar cell is typically a semicon-
ductor with relatively high absorption index at frequen-
cies corresponding to those of the sun quanta h̄ωsun ∼
kBTsun, Tsun ≃ 6000 K. Such semiconductors, however,
process a rather high refraction index n at these frequen-
cies. As a consequence, a fraction (n − 1)2/(n + 1)2 of
the incident light is reflected from the surface of the think
device. To reduce this loss, often anti-reflective coating
are applied to the surface of the device. On the other
hand, for sub-wavelength thin films, the reflection can
be significantly smaller (for a reason similar to why even
metallic films are transparent when thin enough). Thus
reducing the film thickness one should reach an optimum
where reflection is reduced but the absorption is still sig-
nificant.
Also, in such thin devices the carrier recombination
is naturally reduced. Electron-hole recombination which
prevents efficient charge separation in the photocell is a
major limiting factor in device operation. There are nu-
merous mechanisms which lead to the charge relaxation
in a bulk of a semiconductor. These mechanisms include
spontaneous emission as well as phonon or impurity in-
duced relaxation. While it is difficult to control these
processes in the bulk of a semiconductor, it is clear that
their contribution can be significantly reduced if diffu-
sion length of electrons and holes is large compared to
the width of the semiconducting layer.
radiation
(a) (b)
(d)(c)
FIG. 1: Insets (a) and (b): Schematics of the device. Insets
(c) and (d): Band structure of the device without and with
external load.
Diffusion length for most semiconductors used in pho-
tocells is of the order of a few microns. Since this distance
is comparable to a typical wavelength of the sunlight,
one could expect that the specific absorption (the ratio
of the absorbed power to the incident power of radiation)
in such a thin semiconducting layer is insufficient for any
practical use. In order to see whether this is the case,
it is instructive to look at the absorption of radiation
in a layer of thickness d, e.g. Fig. 1(a,b). For simplicity
we assume that the radiation is incident perpendicular to
the surface of the layer and is monochromatic with wave-
length λ. The layer has a dielectric constant whose real
and imaginary parts are ǫRe and ǫIm respectively. The
specific absorption of the dielectric layer can be easily
evaluated by solving the wave equation (n/c2)Ä = ∂2zA
(A is radiation field vector-potential, n = ǫ1/2 is the re-
fraction index, and c is the speed of light) in the three
regions, i.g., Fig. 1(b), and taking into account the conti-
nuity conditions at the boundaries of the dielectric layer,
A1 = A2, A2 = A3, ∂zA1 = ∂zA2, etc. After straight-
forward algebra one finds that the specific absorption is
http://arxiv.org/abs/0704.0651v1
Pabs = 1−|t|2−|r|2, where the amplitudes of transmitted
and reflected waves are
4n exp (id/λ)
(n+ 1)2 exp (−ind/λ)− (n− 1)2 exp (ind/λ)
, (1a)
(n2 − 1)[exp (ind/λ)− exp (−ind/λ)]
(n+ 1)2 exp (−ind/λ)− (n− 1)2 exp (ind/λ) .(1b)
The specific absorption evaluated according to the above
equations is presented in Fig. 2 for a GaAs slab as a
function of its thickness d. GaAs has a relatively nar-
row (∼ 1.4 eV) bandgap and therefore is widely used
in high-efficiency photocells. Since dielectric function
of GaAs is strongly frequency dependent [4], in Fig. 2
we plot the specific absorption for several energies typ-
ical to the quanta of solar radiation. The 2 eV curve
corresponds to the relatively low imaginary part of the
dielectric constant and thus saturates slowly exhibiting
several oscillations due to the interference between re-
flected and transmitted components. The 3 eV and 4 eV
curves correspond to much higher absorption (for exam-
ple ǫGaAsIm (3 eV ) ≃ 17) and saturate much faster. Prior
to saturation both curves exhibit a peak (again due to
the interference) at roughly d ≃ λ/|ǫ|. Remarkably the
value of the specific absorption at the peak (∼ 0.42 at
d ≃ λ/|ǫ| ≃ 20 nm for 3 eV curve) is nearly the same as
its saturation value (0.51 at d ≫ λ). Thus we conclude
that the solar radiation can be absorbed by a semicon-
ducting layer of submicron thickness almost as efficiently
as by an infinitely thick slab.
In this paper, following the simple above considera-
tions, we propose a new concept for the design of pho-
tovoltaic elements based on thin semiconductor films of
controlled thickness. To put our arguments on more rig-
orous footings in the following we consider a detailed
microscopic model of a dielectric layer interacting with
solar radiation. Effective Hamiltonian can be derived
from standard quantum-mechanical interaction between
matter and radiation, (e/mc)A(r)·p + (e2/2mc2)A2(r),
where here and in the following we assume Coulomb
gauge for the electromagnetic field. The radiation in-
duces transitions between valence and conduction band
of the semiconductor. We assume that the temperature
of the semiconductor is 0, and therefore these are the
only possible transitions in the system (the valence band
is completely full and the conduction band is empty). De-
noting the Bloch states for the valence(conduction) bands
(r) = exp (−ikr)uv(c)
(r), H0ψ
(H0 is a Hamiltonian of the crystal in the absence of
coupling to radiation field) one can rewrite the radiation-
matter interaction Hamiltonian in terms of single particle
states in the semiconductor as
Hint =
〈uc0|pα|uv0α〉 c
dk−q⊥αA
+H.c. . (2)
We make the following assumptions: (1) While the
electromagnetic field does not significantly vary with
distance inside the film, the electronic wave-functions
are effectively 3-dimensional - we assume that λsun ∼
2πh̄c/(kBTsun) ≫ d ≫ h̄/(m∗Eg)1/2, where d is the
thickness of the film, m∗ is exciton effective mass,
(m∗)−1 = m−1v +m
c , mv(c) are effective masses in va-
lence and conduction bands, and Eg is the band-gap (in
this paper we assume that bands have extremuma at zero
momentum). This assumption allows one to carry out an
analytic calculation with rather simple and transparent
results. We will discuss the validity of this approxima-
tion at the end of the paper; (2) The bands have different
symmetry, say s and p, so index α denotes angular mo-
mentum of an electron in p band. Since wave-vectors
of the incident radiation are nearly perpendicular to the
surface of the film and the film is assumed infinite in
x − y dimension, only in-plane components of the an-
gular momentum (α = x, y) are relevant in Eq. (2); (3)
The coupling in Eq. (2) is isotropic and the coupling con-
stant tα = 〈uc0|epα/(mc)|uv0α〉 ≃ Egp/(cS1/2), where p is
the effective dipole moment per unit cell of the film and
S is the surface area of the film. (4) Due to external
electric load the bands have effectively different chem-
ical potentials, µn and µp, e.g., Figs. 1(c) and 1(d).
That is, once electron is promoted from valence to con-
duction band, it immediately “rolls over” to the left lead,
which corresponds to the infinite transition rate between
the semiconductor and the metallic lead (an infinitely
thin Shotkey barrier). Clearly, were the rate comparable
or slower than the electron-hole relaxation rate, the effi-
ciency of the cell would have decreased. (5) The p − n
junction is prepared (doped) as shown in Fig. 1(c) - the
top of the valence band in the p-doped area of the junc-
tion lies just below the bottom of the of the conduction
band in the n-doped area. Therefore the maximum volt-
age the photo-cell can sustain is equal to Eg/e, which
corresponds to the assumption of maximum efficiency of
Shockley and Quasser[1], i.e., each electronic transition
from valence to conduction band generates energy Eg in
the circuit. The photocurrent is defined as the rate of
the charge transfer between the valence and conduction
band, Îph = [Hint,
ck]. To the lowest non-vanishing
order it can be expressed as
Iph =
(Eck − Evk−q⊥α, z = z
′ = 0) . (3)
The photocurrent of Eq. (3) is independent of the
voltage across the cell as far as it does not exceed
the bandgap of the semiconductor, e.g., Fig. 1(d).
For larger bias, within our assumptions, a reverse
current begins to flow. In Eq. (3) D<αβq⊥(ω, z, z
dtd2r⊥ exp i(ωt+ r⊥ · q⊥) 〈TKA−α (t, r⊥, z)A+β (0,0, z′)〉
is the “lesser” Green’s function of the electromagnetic
field defined along the Keldysh contour, where super-
scripts ± denote forward and return branches of the
contour [5]. The Green’s function is inhomogeneous
0 0.5 1 1.5
4 eV 
3 eV 
2 eV 
FIG. 2: Specific absorption of GaAs slab as a function of its
thickness. Different curves correspond to different frequencies
(energies) of incident radiation.
along z-direction, i.e., perpendicular to the film surface.
In order to incorporate effects of absorption and reflec-
tion from the film, it is necessary to include renormaliza-
tion of the photon Green’s function due to interactions
with the film according to Eq. (2). These effects can be
treated by means of Dyson equation, which, for the case
of two-dimensional film reads
D̂αβ(ω,q⊥, z, z
′) = D̂0αβ(ω,q⊥, z − z′)
+D̂0αγ(ω,q⊥, z)Σ̂γδ(ω,q⊥)D̂δβ(ω,q⊥, 0, z
′) , (4)
where hats denote the standard 2 × 2 matrix struc-
ture of the non-equilibrium Green’s functions. The
self-energy Σ̂γδ in Eq. (4) is quasi-two-dimensional.
Since l ≫ h̄/(m∗Eg)1/2, effects related to the finite
width of the semiconducting slab can be neglected and
Σ̂γδ(ω,q⊥, 0) =
Σ̂γδ(ω,q), where Σ̂γδ(ω,q) is the
self-energy defined for the bulk of the semiconductor.
Moreover, due to x− y symmetry Σ̂γδ reduces to a δγδΣ̂,
where Σ̂ depends only on |q⊥|. Therefore we obtain a
closed form equation for D̂αβq⊥(ω, z = z
′ = 0):
D̂αβ(0, 0) = D̂0αβ(0) + D̂0αγ(0)Σ̂D̂γβ(0, 0) , (5)
where components of the bare Green’s function D̂0αβ for
solar radiation are:
0αβ (ω,q⊥, 0) =
4π(δαβ − qαqβ/q2)
ω2 − ω2q ± iδ
, (6a)
D<0αβ(ω,q⊥, 0) =
(δαβ − qαqβ/q2)
δ(ω − ωq)ñq − δ(ω + ωq)(1 + ñ−q)
. (6b)
In Eq. (6) ωq = h̄c|q| and the ñq is the distribution
function of solar radiation. We assume that the incident
radiation wavevectors are uniformly distributed within a
cone with an opening angle 2φ (φ ≪ 1). Moreover, in
order for the incident power to be maximum, we assume
that the surface of the cell is perpendicular to the cone’s
axis. Then ñq = n
q θ(qz)θ(φqz −|q⊥|), where nBq is Bose
distribution function with temperature Tsun.
Furthermore due to the homogeneity of the Green’s
function D̂αβq⊥ in x− y plane one can seek for solution
of Eq. (5) in the form D̂αβ = D̂1δαβ + D̂2 qαqβ/q
2, where
D̂1(2) are 2×2 matrices in the Keldysh space, but depend
only on the absolute value of the wavevector q. Substi-
tution of this ansatz into Eq. (5) yields two independent
equations for D̂1 and for D̂3 = D̂1 + D̂2. After solving
those equations one finds
(DR01(3))
01(3)
(DA01(3))
−1 +Σ<
01(3)
)−1 − ΣR][(DA
01(3)
)−1 − ΣA]
, (7)
where D̂01 is the diagonal part of D̂0αβ , e.g., Eqs. (6),
and D̂03 = D̂01 + D̂02, where D̂02 is the transverse part
of D̂0αβ . Σ
R(A) and Σ< are retarded(advanced) and
“lesser” parts of photon self-energy. Also we find a stan-
dard expression for the retarded(advanced) Green’s func-
tions of the radiation:
01(3)
)−1 − ΣR(A)
. (8)
We can now evaluate the photo-current in Eq. (3) in
terms of the Green’s function of Eq. (7). The first contri-
bution is due to absorption of incident radiation accom-
panied by transfer of electrons from valence to conduction
band. It comes from the first term in the numerator in
the RHS of Eq. (7). This term can also lead to the reverse
current due to spontaneous and stimulated emission, i.e.,
transitions from conduction band to the valence band ac-
companied by creation of a real photon. It arises due to
the δ(ω+ωq) term in Eq. (6b). This process is, however,
not allowed while the energy gap Eg exceeds the applied
voltage µn − µp, e.g., Fig. 1(c,d). In this situation the
cell becomes a light-emitting diode, and, as was stated
above, we are not interested in such case in this paper.
The second contribution comes from the Σ< term in the
RHS of Eq. (7). It corresponds to the emission of vir-
tual quanta of radiation by one electron-hole pair and
their subsequent re-absorbtion by another pair, resulting
in an incoherent simultaneous transfer of two electrons
from conduction to valence band. This process gives a
reverse contribution to the current which, once again, is
non-zero only in the “light-emitting” regime (or at high
temperature).
The self-energies in Eqs. (7,8) can be evaluated in
terms of the electronic Green’s functions. The leading
contribution comes from the conventional polarization di-
agram, i.e., a convolution of two electronic Green’s func-
tions. For non-equilibrium situation one obtains
ΣR(ω, 0) =
GKvk(ω + ω
′)GAck(ω
+GRvk(ω + ω
′)GKck(ω
′) + (v ↔ c)
. (9)
and ΣA = (ΣR)∗. In Eq. (9) G
are retarded (ad-
vanced) Green’s functions of valence(conduction) elec-
trons and GK
is the Keldysh Green’s function. Note
that in Eq. (9) we evaluated the self-energy at zero
wavevector, since photon wavevectors are small com-
pared to those of electrons, and therefore the self-energy
Σ̂ is weakly dependent on q for direct bandgap materials.
The self-energies Σ(R)A can be easily evaluated for non-
interacting electrons. ThenG
v(c)k
(ω) = (ω−Ev(c)
±iδ)−1
and GKv(c)k(ω) = (1− 2n
n(p)k)δ(ω −E
), where nFn(p)k
are Fermi filling factors. The valence and conduction
electrons are assumed to have chemical potentials corre-
sponding to those of the two leads, µn and µp respec-
tively. For these Green’s functions the imaginary part of
the self-energy yields:
Im (ω, 0) = ±
Θ(ω − Eg)
ω − Eg , (10)
where we have introduced a dimensionless parameter
a = p2dE
∗)3/2/(21/2h̄4c). A similar calculation
for the real part of ΣR(A) yields an ultraviolet diver-
gence. This, however, is an artifact of our approxima-
tion of the infinitely thin absorbing layer (similar prob-
lem occurs in the quantum electrodynamics treatment
of electron-photon interaction). In a proper microscopic
theory this divergence in the real part of the self-energy
is exactly cancelled by the e2A2(r)/(2mc2) term in the
radiation-matter interaction Hamiltonian (the frequency
sum rule). Taking into account this cancellation is equiv-
alent to performing the Kramers-Krönig transformation
on the dielectric function [Re ǫ(ω) = 4πc2ReΣ(ω)/ω2],
rather than self-energy:
ΣRRe(ω, 0) =
ΣRIm(ω, 0)dω
ω′2(ω′ − ω)
|ω + Eg| −
|ω − Eg|) , (11)
Note that in Eqs. (10,11) we used the three-dimensional
expression for the self-energy and therefore we can re-
express parameter a in terms of the conventional zero
frequency dielectric constant ǫ0, a = 2lEg(ǫ0 − 1)/(h̄c).
From Eqs. (6,7,10,11) one can evaluate the photocur-
rent Iph given by Eq. (3). In this paper we are interested
in the maximum efficiency of the photocell, which can
be defined as ηmax = (IV )max/Pin, where (IV )max is the
power that is dissipated in the circuit assuming an opti-
mal load and Pin is the power of incident solar radiation.
0 2 4 6
FIG. 3: Dependence of photovoltaic efficiency on dimension-
less parameters a and b.
Since the photo-current is only weakly dependent on volt-
age for V < Eg and becomes negative due to spontaneous
and stimulated emission emission for V > Eg, we have
(IV )max ≃ IphEg/e. The incident power per unit area
Pin = cu, where energy density u = (2/v)
h̄ωqñq,
where v is the mode quantization volume and factor 2 is
due to two polarizations of the light waves. Carrying out
the calculation we obtain the following closed form ex-
pression for the maximum photovoltaic efficiency of the
cell:
ηmax =
30ab4
exp (bx)− 1
(x+ a
x− 1)2 + a2(2−
x+ 1−
x− 1)2
, (12)
where we introduced another dimensionless parameter
b = Eg/Tsun. Function ηmax(a, b) shown in Figure 3 has
a pronounced maximum at a ≃ 2.0 and b ≃ 2.4. This
maximum corresponds to the first maximum of the spe-
cific absorption curves in Fig. 2. Note that since our
microscopic theory is valid only for thin dialectic layers
(d ≪ λsun), it does not account for the saturation of
specific absorption at larger thickness, i.e., when d > λ.
However, since ηmax reaches maximum at d ∼ λsun/ǫ0,
our theory is fully self-consistent for semiconductors with
sufficiently high value of the dielectric constant.
The optimal value of parameter b corresponds to the
bandgap energy Eoptg ≃ 1.2 eV . Since this value is close
to the bandgap energy in GaAs (EGaAsg ≃ 1.4eV ), we
conclude that GaAs is a good candidate for the prac-
tical realization of thin film photocells. According to
Eq. (12) the optimal thickness of such GaAs layer is
dopt ≃ 1.2h̄c/[Eoptg (ǫGaAs0 − 1)] ≃ 15 nm, which is within
the fabrication capabilities of contemporary molecular
beam epitaxy technology. Another promising material
is amorphous Silicone which, unlike crystalline Si, has
large imaginary part of dielectric constant[6]. Moreover,
in ultra-thin devices considered here, the thermal equi-
libration of photo-generated carriers may occur on the
time scales longer than the charge separation timescale.
Thus the hot-carrier physics[7] may lead to further en-
hancement of the efficiency.
We thank D. Smith and A. Findikoglu for useful dis-
cussions. The work was supported by the US DOE.
[1] W. Shockley, H.J. Queisser, J. Appl. Phys. 32 (1961) 510-
[2] P. Wurfel, Physics of Solar Cells, Wiley-VCH, 2005.
[3] J. Nelson, The Physics of Solar Cells, Imperial College
Press, 2003.
[4] J.S. Blakemore, J. Appl. Phys. 53 (1982) 520-531.
[5] J. Rammer, H. Smith, Rev. Mod. Phys. 58 (1986) 323-359.
[6] K.C. Kao, R.D. McLeod, C.H. Leung, H.C. Card, H.
Watanabe, J. Phys. D: Appl. Phys. 16 (1983) 1801-1811.
[7] R.T. Ross, A.J. Nozik, J. Appl. Phys. 53 (1982) 3813-3818.
