ECE 475 - Lecture 21
March 27, 2025
1 The photovoltaic diode
• For optical to electrical energy conversion, photodiodes are operated in the photovoltaic regime,
i.e. are attached to a passive external load, not an external battery
– In this regime, the pn junction is operated under forward bias (as opposed to the photodiode
case)
• To understand the photovoltaic mode of operation, consider:
• Open-circuit mode (I ≡ 0):
– Photogenerated EHPs are separated by the internal field, so that there is a net transfer of
positive charge from n − p
∗ This produces a photovoltage that opposes, i.e. reduces the built-in junction voltage
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– The light effectively forward biases the junction (see Vp1 and Vp2 in Fig 18.3-3)
• Short-circuit mode (V ≡ 0)
– As above, photogenerated EHPs are separated by the internal field
– However, these e− and h+ simply transit around the circuit and recombine on the opposite
side
– There is no change in built-in junction field (see ip1 and ip2 in Fig 18.3-4)
– We can then write:
Isc = −Iph = −k · Iˆ
where Iˆ is the optical irradiance [W/m2 ] and k is a device dependent constant [Am2 /W]
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2 Photovoltaic device principles
• Most photovoltaic cells are based on crystalline silicon
– Typically a n+ p junction is used, and the device is illuminated on the n+ side
• The depletion region (and built-in field) is almost entirely on the p-side of the junction
• Recall that the penetration depth δ is the average distance a photon ”survives” prior to absorption
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δ=
α
4π
α= κ
λ0
where α is the absorption coefficient and κ is the extinction coefficient
• For crystalline Si:
– The absorption coefficient for λ0 ≤ 1.11 µm increases rapidly with decreasing wavelength
∗ Short wavelength photons are absorbed much nearer to the surface
– Photons with hν < Eg , i.e. λ0 > λg = 1.11 µm are not absorbed. The silicon solar cell
”wastes” the portion of the sun’s spectrum in the 1.11 ≤ λ0 ≤ 2 µm range
∗ 25% of the total energy is in this range
– Photons with relatively long wavelength λ0 ∼ 1.1 µm are absorbed, but possibly ”deep”
beneath the surface
• Note: Some EHPs that are generated beneath the surface are not effectively separated by the
built-in junction field
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– They tend to be lost to recombination and do not contribute to the photovoltaic device
• To first order, only EHPs generated inside the depletion region or within a minority carrier diffusion length
away from the depletion region can contribute
• Recall:
p
Le = De τe [m]
where Le is the mean e− diffusion length prior to recombination inside the p-region, De is the
diffusion coefficient [m2 /s] and τe is the minority carrier recombination lifetime
Notes
• Typically, Le ≫ Lh in crystalline Si, so that more long λ0 photons contribute to the photovoltaic
effect when the p-region is ”underneath” the junction, i.e. illumination from the n-side
– To maximize absorption of long λ0 photons, the p-side is typically thick (∼ 200 − 500 µm),
even though Le is not typically this long
– Keep in mind that Le is a statistical parameter, some EHP ”outside” Le do get collected
• High energy (short λ0 ) photons tend to be absorbed very near to the surface
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– To maximize the collection of associated EHPs, the n-region is typically very thin (≤ 0.2 µm
– However, a high percentage of these EHPs are lost due to recombination associated with
surface defects
– Surface recombination can reduce the efficiency of the Si PV cell by ≃ 40%
3 Photovoltaic I-V characteristics
• In practice, we want to use the photovoltaic cell to drive an external load, R, which is neither a
short nor an open circuit
• As above, the light produces a positive voltage across the diode, resulting in a net circuit current
corresponding to a negative diode current
Iload = −Ipv
• Since the diode is forward biased, there must be a forward diode current Id , which opposes Iph
• We can write the total current as:
Ipv = −Iph + Id
eV
Ipv = −Iph + I0 [e ηkB T − 1]
where I0 is the diode scale current, η is the ideality factor (1 < η < 2), and V is the photovoltage
• The ”dark” I-V characteristic of the diode is effectively shifted downward under illumination, by
an amount ≈ Iph , which increases approximately linearly with light intensity. The open-circuit
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voltage, Voc , follows by setting I = 0:
ηkB T Iph + I0
Voc = ln
e I
0
ηkB T Iph
≃ ln
e I0
Notes
• The exact value of Voc for a given cell thus depends on light level, i.e. Iph = k Iˆ
• However, due to the steepness of the diode I-V curve above cut-in, Voc is typically similar to, but
less than, the cut-in voltage. For example, a Si diode has a cut-in voltage Vcut-in ≈ 0.6 V
• For a given load, R, and given light level, the operating point of the current can be found using:
−V
Ipv = I = = −Iload
R
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– Note: Positive directions of I and V are defined in terms of the conventional directions for
the diode here. The current through R is the opposite to direction indicated by I in Figure
5.42
• The operating point can be solved numerically or graphically, and corresponds to current I ′ and
V ′ . The power is then:
Pout = I ′ V ′
• For a given level of illumination, there is a specific value of load resistance that maximizes Pout
• A very important parameter to characterize photovoltaic cells is the fill factor
Im Vm
FF =
Isc Voc
where Im /Vm are the current and voltage, i.e. operating point, corresponding to Pout,max
Notes
• For a given cell and light level, Pout,max is obtained by connecting an optimal load (Roptimal ), the
value of which is unique to each case
• A fill factor as cose to unity as possible is desireable
• For crystalline Si solar cells, typically 0.7 ≤ F F ≤ 0.85, depending on the design/structure of the
cell
4 Energy from the Sun
• The Sun’s radiation resembles a blackbody @ T = 6000 K
• We can write
Z ∞
Iˆtot = Iˆλ dλ
0
where Iˆtot is the total irradiance [W/m2 ] and Iˆλ is the spectral irradiance [W/m3 ], i.e. Iˆλ dλ is the
power per unit area in a wavelength interval of λ to λ + dλ
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• The total (integrated) intensity of the Earth’s atmosphere is IAM 0 = 1.353 kW/m2
– At Earth’s surface, the power is lessened due to absorption IAM 1.5 = 1 kW/m2 = 100 mW/cm2
• Due to increased absorption and scattering by the atmosphere, we define:
AMm = AMsec θ
where AMm is ”air-mass” m, and
h
m = sec θ =
h0
where h is the path length through the atmosphere, h0 is the minimum, i.e. normal incidence,
path length, and θ is the ”solar angle,” ie. the angle of direct (non-scattered) photons arriving
from the Sun relative to the Earth’s surface normal
– Any amount of cloudiness or increased path length through the atmosphere reduces the avail-
able power, relative to these values
Notes
• The AMm values correspond to the spectral irradiance for a surface oriented normal to the (non-
scattered or direct) Sun’s rays
– If the surface, e.g. the solar cell, is oriented parallel to the Earth’s surface, the irradiance is
reduced by a factor cos θ, as we will show
– For this reason, some advanced solar cells ”track” the sun by adjusting their tilt throughout
the day
• Due to scattering by molecules in the atmosphere, some of the Sun’s photons arrive at the Earth’s
surface from random directions. This is the diffuse constant
– Diffuse component increases with θ and with cloudiness
– Can be ∼ 20% of the total
– Since short wavelength photons are scattered more efficiently, the diffuse component is blue-shifted,
has its peak power at shorter λ0 , relative to the direct component
• Significant energy is available in the range 0.3 µm ≤ λ0 ≤ 2 µm, i.e. throughout the visible and
near-IR
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– An ideal PV cell would absorb all of these with 100% efficiency into electrical energy
• Consider a particular ”solar angle” θ For Sun irradiance Iˆ1 = IˆAM sec θ
– When the cell is facing the Sun, it receives
P1
Iˆ1 = IˆAM sec θ =
A1
where P1 is the power received by the cell and A1 is the area of the cell
– For the cell parallel to the Earth’s surface, it receives the same power P1 , when its area is
larger,i.e.
A1
A2 = = A1 sec θ
cos θ
– Stated another way, the irradiance received by the cell parallel to the Earth is reduced:
Iˆ2 = Iˆ1 cos θ
