Eq. (2-17) suggests that the current output of a solar cell is basically determined by the recombination currents prevailing in the device. It is therefore necessary to have a closer look at the phenomenological description of the respective recombination proc-esses. The dependence of theses processes on the carrier densities n and p and hence on the terminal voltage of the device directly results in its current voltage characteris-tics. In the following, only the case of low injection conditions will be considered.
Reference [15] presents a more detailed view on the sources of recombination includ-ing the case of high-level injection particularly relevant for concentrator applications.
2.4.1 Recombination in diffused regions
As pointed out above, it is convenient to introduce an effective surface at the edge of the respective space charge region (SCR) just inside the neutral base. Assuming no recombination in the SCR (which is a good approximation for high efficiency solar cells), the recombination current due to the diffused region and the contact is equal to the recombination current flowing through this surface. Due to minority carrier injec-tion (e.g. injecinjec-tion of holes from the p-type base into a highly doped n-type region), the minority carrier concentration at the edge of the depletion region increases exponen-tially with voltage [12]. The net flow of holes into this n-region Jp (which is the part of the injected minority carriers that recombine) can be determined to be
,
where J0n is a temperature dependent quantity denominated the diffusion saturation current density. The pn product refers to the internal edge of the highly n-doped re-gion, but to a good approximation, the product can be evaluated at the edge of the SCR
Basic solar cell operation and surface passivation 11 region in the neutral base. The same relation holds for electrons injected into a p-type diffused region (pn-junction or high-low junction)
.
In combination with eq. (2-22) this yields
⎟⎟⎠
for the voltage dependence of the recombination current density in the diffused re-gions. In praxis the diffusion saturation currents J0nand J0p are often determined ex-perimentally. Since this is commonly done by using injection dependent lifetime measurements, the theory and practical issues concerning the determination of the emitter saturation current are outlined in chapter 3.2. Note that eq. (2-22) is based on the assumption of constant quasi-Fermi levels throughout the base (no voltage drops), which is equivalent to a constant pn product. This approach is often referred to as narrow base approximation.
2.4.2 Recombination in the base
There are three types of recombination processes present in the base, namely radia-tive recombination, Auger recombination and the recombination catalyzed by intrinsic (related to the crystal structure) and extrinsic (impurity related) material defects. Note that in the following section the phenomena are addressed in terms of recombination rates r. The respective recombination lifetimes τ can, however, be directly inferred by the definition
r , Δn
τ≡ (2-27)
where Δn=Δp is the excess carrier density in the absence of trapping. The experimental measurement of the overall effective recombination lifetime (corresponding to differ-ent simultaneously occurring recombination mechanisms in the material) and the means of distinguishing between the different contributions are outlined in chapter 3.2.
Radiative recombination – This recombination is the reverse of the absorption process and is proportional to the pn product. The radiative recombination rate rrad can be written as
(
i2)
.rad Bpn n
r = − (2-28)
12 Basic solar cell operation and surface passivation
B is the coefficient for radiative recombination and was determined experimentally to B=9.5×10-15cm3/s at 300°C [17]. Since this value is very small for indirect semicon-ductors such as silicon (phonon-mediated radiative process) this recombination path can be neglected for silicon solar cells for one-sun applications. However, it should be mentioned that the proportionality to the pn product results in the same voltage de-pendence as observed for the recombination current in diffused regions (eq. (2-25) and (2-26)) and therefore exhibits a diode ideality factor of one.
Auger recombination – Instead of emitting light, the excess energy released during the so-called Auger recombination process is given to a third particle, either an elec-tron or a hole. The suggestive description of the Auger recombination rate rAug is there-fore given by the n- and p-type Auger coefficients, respectively. Again, in silicon, these coefficients are small and the recombination current due to the Auger process may to a good ap-proximation be neglected for low injection application. However, this recombination path becomes dominant for higher injection levels as it quickly increases with rAug ~ (Δn)3. The empirical parameterization for the Auger recombination rate rAug used throughout this work was established by Kerr et al. [18]
(
1.810 610 00.65 31027 0.8)
.Defect mediated recombination – Carrier recombination through defects (often re-lated to as traps) in the band-gap is usually described following the phenomenological approach introduced by Shockley, Read [19] and Hall [20]. The recombination rate for a single defect level rSRH is given by
where τpo and τno are the fundamental hole and electron lifetimes which are related to the thermal velocity of charge carriers νth, the density of recombination defects Nt, and the capture cross-sections σp and σn for the specific defect by
t
Basic solar cell operation and surface passivation 13 where ψiagain refers to the potential referenced to the intrinsic level and Et is the energy level of the defect. Of course the total recombination rate due to defects in the bulk is the sum of the single defect rates.
Assuming a p-type base, the defect mediated recombination rate in the bulk (eq. (2-31)) for low level injection rSRH,li and deep defects (n1, p1 << p) can be written
In the narrow base approximation, the integration over the base yields a recombination current density JSRH,li of
no
with W the thickness of the base. Using again eq. (2-22), the voltage dependence of the low injection defect mediated current density results in
⎟⎟⎠
with NA the acceptor density of the p-type bulk.
2.4.3 Surface recombination
The charge carrier recombination at the surface or interface of the silicon device is dominated by a large number of intrinsic defects since the discontinuity of the crystal results in partially bonded silicon atoms (referred to as silicon dangling bonds). The phenomenological description of the recombination dynamics at the surface follows the model by Shockley, Read and Hall of the bulk defect recombination with the mere difference of introducing surface instead of volume related quantities. For a single defect at the surface, the rate of surface recombination rS is given by
,
where nS and pS are the concentrations of electrons and holes at the surface, and Sno and Spo are the fundamental surface recombination velocities of electrons and holes related to the density of surface states per unit area Nts and to the capture cross-sections σp and σn for the specific defect by
14 Basic solar cell operation and surface passivation
To account for the continuous character of the defect distribution throughout the band-gap, it is appropriate to refer to defect energy dependent capture cross-sections σp(E) and σn(E) as well as an energy dependent density of interface states Dit(E). The surface recombination rate is obtained by integration over the band-gap energies
( )
The quantity related to surface or interface related recombination is the (effective) surface or interface recombination velocity S. Its relation to the surface recombination rate is given by
S .
S S n
r ≡ Δ (2-40)
In general, S is an injection dependent quantity (or equivalently: rS does not increase linearly with ΔnS) which depends strongly on the surface condition. It is therefore particularly dependent on the surface passivation applied and, conversely, allows for the deduction of passivation specific quantities which in turn allow for conclusions on e.g. the passivation mechanism. This type of analysis is presented in chapter 5.4 for surface passivation by silicon rich a-Si1-xCx.
However, assuming S to be constant in a first approximation, it becomes evident that the description of defect mediated recombination in the silicon volume and at the surface is similar (eq. (2-34) and (2-40)) and the voltage dependence of the surface recombination current density in low injection considering a p-type surface can analo-gously be written as
Asurf refers to the fraction of surface where recombination takes place which, however, is not yet accounted for by the recombination currents in the diffused regions (Jn or Jp).
2.4.4 Recombination in the space charge region
The defect mediated bulk recombination rSRH (eq. (2-31)) is most effective for re-gions of equal hole and electron carrier densities n ≈ p when assuming deep defects (n1 ≈ p1) and similar cross sections for both types of carriers (σn ≈ σp). Hence, if SRH recombination in the space charge region (SCR) cannot be neglected, then the domi-nant contribution to SCR recombination stems from narrow regions of equal electron and hole density. Here eq. (2-22) results in
Basic solar cell operation and surface passivation 15
and the voltage dependence of the SRH recombination current density in the SCR can be approximated to
J02 is the temperature dependent dark saturation current of the SCR [21]. Given the outlined simplifications, note that the diode ideality factor referring to SCR recombina-tion is two.
2.4.5 Current voltage characteristics
Following the superposition principle, the overall current voltage characteristics of the device are given by eq. (2-17) when introducing the above derived voltage depend-ent recombination currdepend-ent densities for the various recombination paths. In low injec-tion, the voltage dependence of the output current density J =I/A is therefore
, ,
Note that the diode ideality factor of the space charge recombination generally deviates from 2 for real solar cells since the assumptions made in the preceding section over-simplify the conditions in the SCR. Series losses of the device may be accounted for by approximating
, ' V JRS
V≈ − (2-46)
where RS is the overall series resistance of the cell. Leakage currents in a real device may be considered as an additional recombination current density Jleak characterized by a parallel resistance Rp
. / ' p
leak V R
J ≈ (2-47)
The current voltage characteristics can then be written as
( ) ( )
with J01 referring to all saturation currents in the device except for the SCR and leak-age currents. For high-injection conditions (Δn >> n0, p0), Auger and radiative recom-bination in the base can no longer be neglected and the behavior of the recomrecom-bination
16 Basic solar cell operation and surface passivation
currents may change, which is particularly reflected by a change in the corresponding diode ideality factor (Table 2-1).
Table 2-1: Diode ideality factors for different recombination paths in low and high injection conditions [15].
diode ideality factor recombination path low injection high injection
diffused region 1 1
radiative 1 1
Auger 1 2/3
SRH (bulk, surface) 1 2
SCR 2 2