Literature overview

→v_n = −µn

−

→F and −→v_p = µp

−

→F (4.13)

and the net drift current density flowing through a sample is

j_{drif t}(x) = eF [n(x)µ_n+ p(x)µ_p] = σF . (4.14) This equation expresses Ohm’s law, in which σ = e[n(x)µn+ p(x)µp] stands for electric conductivity of a material.

Diffusion current density jdif f can be derived from Fick’s law. Under the assump-tion of a steady state, i.e. when concentraassump-tion of charge carriers and hence cur-rent density do not change in time, one obtains

j_{dif f}(x) = eD_ndn

dx− eD_pdp

dx , (4.15)

where Dn and Dp denote electron and hole diffusion coefficients, that can be related to mobilities of these charge carriers by Einstein-Smoluchowski relation

D = µkT

e . (4.16)

Combination of Eqs. 4.14and 4.15gives the total bipolar, i.e. electron and hole, current density j flowing through the sample in a steady state

j = j_{drif t}(x) + j_{dif f}(x) = eF (nµ_n+ pµ_p) + e D_ndn

dx − D_pdp dx

!

. (4.17)

4.3 Single layer cells

4.3.1 Literature overview

Though efficiencies of single layer photovoltaic cells are rather low, studies car-ried out on simple metal/organic solid/metal (M/OS/M) structures are relevant

since they provide information on basic optoelectronic properties of investigated organic solids as well as on the nature of processes taking place in such struc-tures in the presence of electric field or illumination. Numerical approach to mod-eling of photoelectric properties of M/OS/M systems was taken by Signerski et al., who considered thermal and photogeneration of free charge carriers at both electrode/organic material interfaces as well as in the bulk of the sample [78].

Singlet excitons, neutral generation centers of exponential spatial distribution, dis-crete level of traps of finite concentration and unipolar charge transport were re-garded. Numerical calculations yielded current-voltage curves of illuminated sam-ples, short-circuit current vs. linear absorption coefficient and illumination inten-sity under different conditions. Symbatic or antybatic behavior of jsc dependent on trapping conditions and values of absorption coefficient along with a power-law dependence of jsc on light intensity

jsc ∼ I0n

, (4.18)

where n ranged from 0 to 1, that stands in a good agreement with experimental data, were obtained.

Numerical and analytical approach to model the performance of surface barrier solar cells of different architectures, in which band bending near the barrier-former/ semiconductor contact is observed, was presented by Fonash [38]. This model involved the presence of minority charge carriers, hence a bipolar cur-rent flow was considered. Moreover, no internal electric field within the bulk of the semiconductor beyond the barrier-region was assumed in this model and the results of numerical simulations shown that good photovoltaic performance, for which the energy conversion efficiency exceeds 12% is possible in this type of cells. Therefore, this model is rather inconsistent with the results of experiments conducted in the field of single layer organic photovoltaic cells.

Analytical dark current-voltage expression for a hole-only metal-insulator-metal (MIM) organic diode, in which the presence of a uniform electric field within the whole organic layer is assumed, was derived by Bruyn et al. [79]. Authors of this work considered a model of a diode with asymmetric contacts, i.e. one Schottky type contact and one Ohmic type contact, of different work functions. Dark current density under bias U lower than the built-in voltage derived in the absence of band

bending was

j_dark = eµ_pN_v(ϕ_b − U )[exp^eU_kT^− 1]

d[exp^eϕ_kT^b− exp^eU_kT^] , (4.19) while if band bending at the Ohmic contact, resulting from accumulation of free charge carriers diffusing from this electrode into an insulator or undoped semi-conductor, was considered the following expression was proposed

j_dark = eµ_pN_v(ϕb− b − U )[exp^eU_kT^− 1]

d[exp^eϕ_kT^b− exp^{e(U +b)}_kT ^] , (4.20) where Nv stands for the effective density of states, ϕb is the built-in voltage, d denotes the thickness of the organic layer, whereas b describes the magnitude of band-bending dependent on d, Nv and dielectric constant of the organic material.

Under bias greater than the built-in voltage space charge limited current (SCLC) following the Mott-Gurney equation in the trap free case, i.e.

j = 9

8εµ (U − ϕ_b)²

L³ (4.21)

was expected.

An overview regarding currents and photocurrents limited by some interface phe-nomena, like thermal, tunneling and photoinjection along with free charge carrier trapping taking place at the electrode/organic material junctions, as well as, at the organic material heterojunctions, was presented by Godlewski [80,81]. The open-circuit voltage developed in the one-dimensional system, in which both electrodes are active and inject the same carrier type, the rate of dark injection is constant, while the rate of photoinjection is proportional to the light intensity was described by Pope [82], according to whom the photovoltage step U₁ developed at the front electrode, for which x = 0, in a steady state at zero current flow is

U₁ = kT

Analogical expression can be written for the photovoltage step U2 developed at the back electrode, for which x = d and since these two voltages oppose each other, the resultant open-circuit voltage Uocgenerated across such cell is equal to

U_oc = U₁− U₂ = kT

Figure 4.2: Schematic representation of relative positions of energy levels of a single organic layer and electrodes before contact (A) and after contact (B). A simple case of no surface dipoles and no band bending is considered, d stands for the thickness of an organic layer, indexes f and b refer to the front and back electrodes respectively.

where indexes f and b refer to front and back electrodes respectively. In a simple case of identical electrodes the open-circuit voltage results from different con-centrations of free charge carriers at these contacts originating from their uneven illumination

U_oc= kT e lnn₀

nd

. (4.24)

If we assume that photoinjected current is proportional to the light intensity I0 and that a and b are the ratios of photoinjection efficiency to dark injection rate at front and rear electrodes respectively, then Uocbecomes

U_oc = kT e ln

1 + aI₀ 1 + bI0



. (4.25)

In case of only one active contact, e.g. the front illuminated electrode/organic layer interface, when illumination is sufficiently high we may simplify the equation given above to the following one

U_oc= kT

e ln (aI₀) , (4.26)

since in such case J_ph^f /J_d^f  1, while J_ph^b /J_d^b  1.