C − V measurements

The capacitance is a measure for the charge stored within the device, studying the voltage and frequency dependence of the capacitance can therefore give information about the spatial and energetic distribution of localized charges such as defect levels within a semi-conductor. The capacitance can be derived from the phase shift of an applied alternative voltage bias and the current induced by it. The current response is determined by the complex conductivity of the device, which is called the admittance Y [79].

Y(ω) = G(ω) +iωC(ω), (2.4)

with G(ω) being the conductivity andC(ω) the capacitance, both in principle dependent on the frequency ω of the applied voltage. If no charging or discharging occurs, the admittance equals the conductivity and the phase shift becomes zero. If only charging and discharging occurs, the admittance equals the capacitance and the phase shift becomes 90°. By measuring the amplitude and phase shift of the current it is possible to calculate the capacitance, which is defined as following:

C(ω) = AdQ

dV , (2.5)

where dQ is the additional charge stored in the device by a small change of the voltage dV. A is the area of the device.

Space charge capacitance

When applying a small AC voltage (20 mV) oscillating around 0 V to a perfect p/n-junction, the conductivity G(ω) is zero due to the absence of shunts and the capacitance

originates from charging and discharging shallow defect states with free charge carriers at the edges of the space charge region. The capacitance can be then described similar to a parallel plate capacitor with an inserted dielectric [79].

C = ₀A

d_SCR, (2.6)

where₀ is the vacuum permittivity, the dielectric constant of the charge depleted semi-conductor and dSCR the width of the space charge region. This formula only holds for homo-junctions or for n +/p- or n/p+-hetero-junctions, where the space charge region width within one side of the junction can be neglected. The space charge width for such a junction can be described by the following equation [79]:

d_SCR= s

2₀(V_bi−V)

eN_CV , (2.7)

whereeis the elementary charge, V_bithe built-in voltage (defined by the difference of the Fermi levels in thepandnmaterial),V the applied DC voltage andN_CVthe shallow defect density at the edge of the space charge region. For homogeneously doped semi-conductors it is possible to deriveN_CVby combining Eq. 2.5 and Eq. 2.6. For non-homogeneous semi-conductors like CIGSe the doping profile has to be measured, which is shown in the next Section.

Profiling

In non-homogeneously doped semi-conductors, the change of the space charge region dd_SCR due to a small voltage bias dV depends on the local doping density N_CV(x) at the edge of the space charge region d_SCR. With the assumption that only shallow defects contribute to the capacitance, an equation for dd_SCR can be derived by extending Eq. 2.7 with dd_SCR/d_SCR and integrating over d_SCR [80]:

dd_SCR= ₀dV

edSCRNCV(dSCR). (2.8)

A change in dd_SCR leads to a change of the capacitance as defined in Eq. 2.5:

dC

dV =− A0

d_SCR² ddSCR

dV . (2.9)

Inserting Eq. 2.8 into Eq. 2.9 leads to the equation for the position dependent shallow defect density N_CV(x):

N_CV(d_SCR) = −C³

e₀A²dC/dV . (2.10)

By choosing small changes of the applied voltagedV, in a way that dCis linear todV, the expressiondC/dV can be exchanged with ∆C/∆V, which is more practical in application.

It has to kept in mind, that these equations are only valid for semi-conductors without deep defects, which is not necessarily the case for CIGSe. In the next section the influence of deep defects will be discussed.

Deep defect capacitance

The capacitance Cd of a homogeneously distributed deep defect whose charging and dis-charging is fast enough to follow the change of the applied voltage is described by the following equation [81]:

Cd = e²

kTNdf(Ed)(1−f(Ed)), (2.11)

whereN_dis the defect density,f(E_d) the Fermi function at the energy level of the defect,k the Boltzmann constant and T the temperature. Thus, the capacitance is highest, when f(Ed)=0.5, which is the case when the Fermi level crosses the defect level. Thus the position of the defect charging and discharging process does not occur at the edge of the space charge region, but somewhere within the space charge region. Eq. 2.10 is no longer valid, because of two effects. First, the deep defects increase the capacitance compared to the capacitance defined in Eq. 2.5. Secondly, they reduce d_SCR if they are charged, i.e.

if f(E_d) > 0. It is, however, possible to freeze out the influence of deep defects to the capacitance, since the average charging and discharging time depends on the temperature and on how deep the defect lies within the band gap. The cut-off frequency ω₀ defines the transition when the defect charging cannot follow the change of the applied AC voltage any more. It is given for a defect, which interacts with the valence band by [81]:

ω₀ = 2c_pN_Ve^Ea/kT, (2.12)

with c_p being the capture coefficient, N_V the density of states of the valence band and Ea the activation energy, which is the difference of the defect level and the valence band.

Thus for high activation energies or for low temperatures, defect charging can eventually not follow the change of the applied voltage and does not contribute any more to the capacitance. The study of the frequency dependence of the defect capacitance is called admittance spectroscopy.

Influence of series resistance

The influence of the solar cell series resistance R_s and parallel resistance 1/G on the measured capacitance C_m has to be kept in mind. The real space charge capacitance differs from the measured capacitance as follows [82]:

C_m= C

(1 +R_sG)²+ω²C²R²_s. (2.13)

For high Rs or high G = 1/Rp values the measured capacitance is lower than the real capacitance. Therefore these two conditions R_s<< R_p and R_s <<1/ω²C² should always be met. For R_p=1000 Ω, R_s=1 Ω ,C=500 nF, which are common values for the solar cells studied in this thesis, the measured capacitance isCm=0.1Catf=1 MHz andCm=0.99C atf=1 kHz. Therefore, all measurements shown in this work were done at f=1 kHz.

Experimental conditions: The C−V curves presented within this thesis were mea-sured at room temperature, with an Agilent 4284A precision LCR meter. The solar cells were contacted with metal probes, whose influence was corrected for prior to each mea-surement. The devices were relaxed in the dark for 5 minutes before each meamea-surement.