Space-charge-limited currents for organic solar cells optimisation

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Solar Energy Materials & Solar Cells 87 (2005) 235–250 www.elsevier.com/locate/solmat

Space-charge-limited currents for organic solar cells optimisation Franz Schauer Polymer Centre, Faculty of Technology, Tomas Bata University in Zlin, T.G. Masaryk Sq. 275, CZ -762 72 Zlin, Czech Republic Received 15 May 2004; received in revised form 20 July 2004; accepted 23 July 2004 Available online 23 November 2004

Abstract Basic suppositions and microphysical origin of the occurrence of the space-charge-limited currents (SCLC) are presented in general and for the temperature-modulated space-chargelimited currents (TM-SCLC) in particular. The criteria are given for the spectroscopical method TM-SCLC to be used for localized electron states elucidation in organic semiconducting materials for organic solar cells optimization and modelling. The ‘‘visibility ‘‘of the localized states by SCLC method, i.e. the power of the SCLC method to distinguish the localized states, is tested by the modelling, varying the temperature, energy position of localized states and their concentration. Generally, it was determined that the SCLC measurements results are more reliable with the increased energy of the states, with their increased concentration and with decreased temperature. The correlation (or its absence) between the measured current and activation energy on applied voltage, expressed by the dependence of preexponential factor of conductivity on activation energy (Meyer–Neldel rule), gives the possibility to determine the energy range where the reconstruction of density of localized states function is reliable. r 2004 Elsevier B.V. All rights reserved. Keywords: Space-charge limited currents; Organic solar cells; Density of states; Spectroscopy

E-mail address: [email protected] (F. Schauer). 0927-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.solmat.2004.07.020

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1. Introduction The organic solar cells have undergone a major development and started to attract considerable scientific efforts during past 3 years, when 5% white light efficiencies have been realized [1]. To increase the efficiencies of the organic solar cells with the distributed heterojunction concept formed by small molecules and polymers, the modelling with a detailed knowledge of the microphysical properties of the organic semiconductors as input parameters for simulations and thus optimization is needed (see the first models of electrical transport in organic solar cells [2,3]). One of the input parameters for detailed simulations is the density of electron states (DOS) of organic semiconducting materials. The general deficiency of spectroscopical methods bringing DOS in organic semiconductors renewed the interest in the method of space-charge-limited currents (SCLC). The SCLC method is extraordinarely suited for the examination and optimization of thin film organic solar cells with both types of carriers as the DOS of trapping states may be studied in one system after slight modifications. This method was originally devised for the spectroscopy of the energetically discrete states [4] and later for disordered substances like e.g. amorphous hydrogenated silicon (a-Si:H) with an arbitrary energy continuum of electron localized states with the need for a differential method based on the numerical differentiation of the measured data [5]. The method gained popularity due to its apparent experimental simplicity, though the theoretical background turned out to be more complicated. Later, the introduction of the concept of activation energy of SCLC was introduced, its meaning with respect to the DOS spectroscopy was presented and the importance of the Meyer–Neldel rule for the application of the studying the transport properties and electron structure was stressed [6] introducing the new method of temperature-modulated space- chargelimited currents (TM-SCLC). Later, the possibilities and limitations of the method TM-SCLC has been presented in Ref. [7]. The present paper intends to present a unifying look at the SCLC in organic materials, and discuss the potential possibilities in their studies. The layout of the paper is as follows: first we introduce the limitations stemming from the variablerange hopping transport pertinent to organic materials, then we bring a short description of the physical background of the existence of the space-charge in a solid thin film and the impact of relevant transport properties, then we introduce the space-charge-limited transport and the physical background of the TM-SCLC spectroscopical method with emphasis on its potential strength and applicability to organic materials on one hand and its limitations on the other.

2. Organic semiconductors—charge transport and injection In contradiction to the inorganic semiconductors, where delocalized charge carrier transport prevails, it is widely accepted that the transport in disordered organic semiconductors and polymers is by variable-range hopping in a positionally and energetically disordered system of localized states [8]. This fact introduces serious

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modifications in the framework of space-charge-oriented methods especially those based on the transport, as e.g. is the case of the method of SCLC. In Ref. [6] Eq. (1) we used for the building of the framework of SCLC the electrical conductivity s in the band of delocalized states with the effective concentration of states Nb and the mobility edge situated at Ec. This energy, supposed temperature and injection independent, was crucial in a spectroscopical method of SCLC, as it constituted the energy with the prevailing transport of charge carriers and also the reference point in the DOS energy positioning. The movement of the carriers was characterized by field independent and to the first approximation temperatureindependent microscopic mobility m0 within the delocalized standard transport model [9], s ¼ em0 N b eðE c E F Þ=kT ;

(1)

where T is the temperature, k the Boltzmann constant and EF the Fermi energy position. For the sake of applicability of the spectroscopical SCLC method based on the existence of space-charge and charge carrier transport, we want to check these assumptions taken in our previous papers [6,7] and show the applicability of SCLC for the study of organic semiconductors. 2.1. Transport path concept Exact analytic treatment of carrier hopping in arbitrary energetically and positionally disordered electron states is notoriously difficult problem. Among the wide spread approximations those, using the effective transport paths [10,11], turned out to be the most efficient for solving the problems of transient and steady-state transport and mobility evaluation. In Ref. [12] it was shown that introducing the so-called effective transport energy Etr, and the effective density of transport states Ntr, where the majority of charge carrier transport takes place, the problem was essentially reduced to the trap-controlled transport with the transport energy playing the role of the mobility edge Ec. Subsequent papers solved the problem how the deep traps [13] and doping [14] influence the position of transport energy Etr in an exponentially energetically distributed tail states. If we summarize these results, we claim that the position of the transport energy Etr for the reasonable tail exponential slopes (Ttr4T) and low concentration of deep trapping states (Nto1018 cm3 for EtX0.6 eV) is only slightly temperature dependent, and so, taking into consideration the temperature range used generally for SCLC T 2 150; 350 K, we can neglect the temperature dependence of the transport energy Etr. Another limitation may stem from the dependence of the transport energy Etr on the injection level. According to the modelling in Ref. [14] we can suppose injectionindependent transport energy Etr for the tail states filling by injection up to 102Nt, where Nt is the total concentration of the Gaussian distribution of tail states, which condition is always fulfilled in SCLC measurements. Then we can claim, that the transport energy Etr is a good approximation of the mobility edge Ec and may be considered as a good energy reference point.

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2.2. Mobility field dependence In the variable-range hopping model the carriers occupying hopping sites around Etr mostly jump down to deeper states and, therefore, their mobility at the transport path only weakly depends upon both the field and temperature [15]. So the concept of the microscopic mobility m0 in Eq. (1) can be replaced by the field and temperature-independent hopping mobility at the transport path mðE tr Þ; that will be further denoted by m: And, finally, as the concept of (quasi-) Fermi level for the injection of the charge carriers into the bulk of a material is justified even under the variable-range hopping model, we can rewrite Eq. (1) for the electrical conductivity of the organic semiconductor with variable-range hopping transport s ¼ sM eðE tr E F Þ=kT ;

(2)

where we put for the preexponential factor sM ¼ emN t : 2.3. Injection into organic semiconductors The injection contact and the charge carrier reservoir occurrence are stringent conditions for the existence of SCLC. In inorganic substances the charge carrier injection was either by heavy doping of the surface of the semiconducting material (n+ for the injection of electrons and p+ for injection of holes), by heterostructural contacts, or by metal contacts with a proper work function. In general, the injection into an organic material is basically influenced by the hopping transport. As recently shown [15], the injection from the metal contact is by a two-step process, where carriers jump from the Fermi level of a metal contact into localized states close to the interface, creating the image force and lowering the barrier height. Then, in the next step the carriers may experience either neutralization at the contact, or crossing the barrier by the diffusion process similar to the Onsager model. But, as the barrier thickness is in the range of several nanometers and the semiconductor thickness is usually 4100 nm, we can consider the injection as independent from the bulk processes and provided the barrier height is low (o0.4 eV) and at moderately low temperatures (o300 K), we can consider the contact as invisible and neglect its influence on the transport.

3. Space-charge and steady-state transport 3.1. Simple approach To explain SCLC we have to start with the explanation of basic terms. Spacecharge is the net charge existing in a material, breaking the condition of the charge neutrality. For its description we will further use the quantity of space-charge density

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rsc ; defined as the net charge in the unit volume, i.e. rSC ðxÞ ¼ e½ns ðxÞ  ns0 ;

(3)

where the ns and ns0 are the total concentrations of charge carriers, forming the space charge at a point x formed by concentrations of free nf and trapped nt charge carriers, ns0 is the charge carrier concentration at the neutrality. For the explanation, how the space charge can arise in a material by injection and take part in the charge carrier transport in the semiconductor let us introduce the conditions for its existence first. For this purpose let us suppose a plane-parallel homogeneous sample of a material in thermodynamical equilibrium with the well-established Fermi energy and electrical neutrality with mobile charge carriers, provided with the charge injecting contact that creates reservoir of mobile charges at x ¼ 0 and the extracting contact at x ¼ L: Then we can introduce two typical time scales that represent the relaxation towards thermodynamical equilibrium after space-charge injection (in the form of small perturbation) and its drift in the applied electric field. The Maxwell relaxation time tM is the measure of the time needed for the space-charge neutralization by mobile carriers, defined 0 ; (4) tM ¼ s where s ¼ enf m is the electrical conductivity given by the mobile charge carriers of the concentration nf and their mobility m; and 0 is the permittivity. The transit time tt is the time needed for the crossing of the sample by moving space-charge under the electric field created by the external voltage U tt ¼

L2 ; Umd

(5)

where we introduced a somehow arbitrary quantity of the drift mobility, md ; fulfilling the condition nf md ¼ m ¼ mY: (6) nf þ nt Depending on the relation of the Maxwell relaxation time tM and the transit time tt, three basically different regimes of transport may be observed: For tM ott ;

(7a)

tM 4tt

(7b)

tM ¼ tt :

(7c)

and If the Maxwell relaxation time is less then the corresponding transit time (7a), the charge carriers forming the space-charge perturbation do not manage to cross the sample as the movement of the mobile carriers neutralizes their charge. In this case, the low or high electric field transport is observed and no space-charge exists for time

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t43 tM (due to the factor exp (3) 0), the carriers forming space-charge cannot reach the extracting contact as they are neutralized. In the opposite case (7b) the uncompensated charge carriers manage to cross the sample, space-charge exists in nonsteady state for times t tt, the carriers forming space-charge reach the extracting contact, crossing the sample before the neutralization. From this transit time tt the quantity of the drift mobility md may be derived, what is the basis of time of flight (TOF) method. It is proper to stress that drift mobility measured by TOF is quite a different quantity compared to that derived from SCLC (or other steady-state method) as the concentration of deep trapping states that are in equilibrium with transport states is dependent on the time scale of the method used. The movement of the carriers is a complex process, resulting from both the trapping in all the available states and retrapping from those, being in quasi-equilibrium with the transport states [16]. So, the transit time of the carriers across the sample tt is not a straightforward quantity, but for the purpose of the present explanation we consider it as a measurable and meaningful observable. The crucial case for the purpose of the present discussion is the case, where the condition given by Eq. (7c) tM ¼ tt is fulfilled. The injected carriers experience the full transit across the sample, so that after their extraction at the extracting contact an exactly equal number of carriers is injected from the injection contact. This situation exhibits features of the self-controlled steady-state process, which may be kept for infinite time with the space-charge existing in the whole volume of the sample in question. On increasing the voltage, thus decreasing the transit time tt, the conductivity of the sample has to rise correspondingly in order to decrease the Maxwell relaxation time tM as well. The increased concentration of the space-charge in the sample formed by mobile and trapped charge carriers and the super linear increase of the current result. This is the basic scenario for the SCLC occurrence. Let us examine what the fulfilment of these conditions means in microphysical terms in a typical material with energetically distributed localized electron states. Combining definitions (4) and (5) with the condition of the SCLC existence (7c) we obtain the condition for the voltage necessary for the maintaining of SCLC U¼

enf L2 0 Y

(8)

and the corresponding current density of SCLC j¼

0 mYU 2 : L3

(9)

Eq. (9) is the simplified version of the SCLC master equation, more precise expressions are to be found e.g. in Ref. [4]. At this moment it should be pointed out that the spectroscopical character of the SCLC is contained in the voltage-dependent distribution function YðUÞ; carrying information on the space-charge in traps. Besides, information on the steady-state drift mobility md ¼ mY is also available from the measurements of SCLC.

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We can rewrite the space-charge volume density rsc ðxÞ in Eq. (3) using the energetically distributed density of states (DOS) function hðEÞ Z rSC ðxÞ ¼  e½ns ðxÞ  ns0  ¼ e hðEÞ E

½f ðE F ðxÞ; EÞ  f ðE F0 ; EÞ dE;

ð10Þ

where EF and EF0 are the quasi- and thermodynamical Fermi levels, respectively. The main idea of the SCLC spectroscopy is the shifting of the Fermi level position EF (by changing the applied voltage U) and thus scanning the DOS of traps by increasing the voltage on the sample and the space-charge density rSC : Further, we show how and under what conditions it is possible to extract information on DOS hðEÞ distribution using SCLC from the experimentally available spacecharge rSC :

4. Space-charge-limited currents 4.1. SCLC without traps In case of the absence of the deep localized states in the semiconductor, then the coefficient Y defined in Eq. (6) is Y ¼ 1 and we can observe SCLC current density 9 0 mU 2 : (11) 8 L3 (Eq. (11) is a more precise form of the Eq. (9), see e.g. Ref. [4]). From Eq. (11) the mobility m may be easily extracted. This approach was taken by many authors, recently in Ref. [17] studying PPV derivative by SCLC and in Ref. [18] elucidating temperature dependence of mobility in tetracene single crystals. This approach may be used even in case the mobility is field dependent [19]. j¼

4.2. SCLC with traps When the deep localized traps are present in the organic semiconductor, then equation 9 0 mYðUÞU 2 (12) 8 L3 should be applied. In past, many procedures were devised for the extracting of the DOS of traps, after the first recognition of the spectroscopical character of the SCLC method by Sto¨ckmann [20]. Many authors (cf. for example Ref. [21] and references quoted therein) presented model calculations of the most frequent typical DOS distributions—uniform, exponential, Gaussian and bell shaped and tried to compare their data with the model calculations by sophisticated fitting techniques. The insight into the SCLC methods was deepened with the development of the differential methods, based on the first or higher derivatives of the experimental j(U) SCLC j¼

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dependences where no a priori assumptions had to be taken, but for the analysis the knowledge of the mobility and Fermi level position was necessary [22]. This approach was recently used e.g. by Santoshi Mizuo et al. [23] for Alq3 DOS analysis. Recently, the procedure for studying both the field-dependent mobility and traps distributed in energy has been introduced, using the combined measurements of the differential SCLC and capacitance–voltage method [24]. 4.3. Temperature modulated SCLC—possibilities and limitations In order to make the analysis of SCLC more straightforward the method of temperature–modulated SCLC (TM-SCLC) was introduced in a series of papers [6,7] and applied on some inorganic semiconductors. The name temperature-modulated stems from the temperature ‘‘modulation’’ during TM-SCLC measurement during which the temperature is not fixed constant at the preselected value T, but is swept in a predefined manner in a small temperature interval, usually, DT ¼ 5 K round the preselected value T DT: During every temperature run, the current is sampled and its the activation energy is calculated by standard procedures. The new differential method uses two pieces of independent SCLC information, current–voltage j(U) and activation energy–voltage Ea(U) dependences to reconstruct DOS without any assumptions. The reason for introducing two independent measurements is that for the reconstruction of DOS we have to find two quantities for each value of the applied voltage, the DOS value h(Ei) and the corresponding energy position Ei. These two quantities, (i.e. h(Ei) and Ei) may be obtained from experimental SCLC data, i.e. current–voltage characteristic, j(U) and the dependence of the activation energy of the current Ea(U) on the applied voltage, using the expressions derived in Ref. [6] Z dnsL 1 ¼ hðEÞf ðE; E F Þ½1  f ðE; E F Þ dE F kT E 1 0 2m  1 ; ð13Þ dE ¼ kT eL2 m2 and E ¼ E a þ kT

3  4m n; mð2m  1Þðm  1Þ

(14)

where nsL is the total concentration of space-charge at the extracting contact x ¼ L; U is the applied voltage across the sample, 0 is the electrical permittivity, e is the elementary charge and L is the sample thickness; m is the first derivative of the j(U) characteristic with respect to the voltage, m ¼ dðln jÞ=dðln UÞ; and n is the corresponding first derivative of the Ea(U) characteristic with respect to the voltage n ¼ dðE a =kTÞ=d ln U: The activation energy Ea of the SCLC (experimentally available) is defined as Ea ¼ 

dðln jÞ : dð1=kT Þ

(15)

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The couple of Eqs. (13) and (14) represents the framework for DOS reconstruction by TM-SCLC method (the exact procedure is given in Ref. [6]). Next we want to show how the measured TM-SCLC characteristics, i.e. current–voltage j(U) and its temperature dependence expressed by activation energy–voltage Ea(U) dependence, give the possibility for the TM-SCLC validity check for DOS reconstruction. The combined information of TM-SCLC i.e. the current j, the voltage U and geometrical dimensions give the possibility to calculate the average SCLC conductivity s: Then, using the value of the measured activation energy Ea we can write s ¼ s0 eE a =kT :

(16)

Comparing the expressions in Eqs. (1) and (16) we can see that both the preexponential factors and exponents are different. In Refs. [6,7] we have shown that this difference may be explained by the statistical temperature shift of the Fermi level EF(T) [9] and also how it may be used for the validity check of the SCLC reconstruction. If we approximate the temperature statistical shift of the Fermi level EF(T) with respect to the temperature independent transport energy Etr at a certain temperature T by the linear extrapolation to zero temperature, then we may write for the activation energy Ea E tr  E F þ gF kT ¼ E tr  E F0 ¼ E a ;

(17)

where EF0 is the extrapolation of the Fermi energy to zero temperature and gF is the linear coefficient of the Taylor expansion of the temperature dependence of Fermi level position EF(T). Then, using Eq. (1) we can write for the SCLC conductivity [6] s ¼ sM egF eE a =kT ¼ s0 eE a =kT ;

(18)

where we formally put s0 ¼ sM egF : The plot of preexponential factor lnðs0 =sM Þ (or gF ) vs. activation energy Ea is sometimes called the Meyer–Neldel rule. We want to show that this dependence gives a validity test of the SCLC method, i.e. the energy range where the reconstruction of the DOS from the measured data is reliable. We may put the criterion defining this energy range somewhat arbitrarily as jgF jo4; i.e. as the energy range where the distance of the Fermi level position from its zero temperature extrapolation is jE F  E F0 jo4 kT (and it is easy to show that also the energy distance of the states that are filled by injected space-charge from the Fermi level). This criterion seems to be applicable not only when the statistical shift is operative, but also in measurements with the inferior contacts or presence of the mobility field dependence, etc. [25]. To explain the SCLC resolving possibilities we resort to modelling of SCLC (modelling procedure is described in Ref. [7]). We use two typical DOS distributions, occurring in organic semiconductors, i.e. the Gaussian distribution and the double exponential (bell-shaped) distribution. The first distribution is typical for the defects and impurities in otherwise disordered systems characterized by the parameter S; whereas the second one is typical for the rising tailing of the energy

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10

5

~U2

2

~U

-5

2

Et + Σ /kT

-10

0.6

Et

-15

0.8

2

Et - Σ /kT

0.4

Ea, EF (eV)

log (j (a.u.))

0

0.2

-20

0.0 -25 -4

-2

0

2

4

6

log ( U(a.u.)) Fig. 1. The model TM-SCLC dependences, i.e. current–voltage j(U) and activation energy–voltage Ea(U) for the Gaussian DOS distribution function h(E), Eq. (19), E tr  E t ¼ 0:5 eV; S ¼ 50 meV; Nt ¼ 1018 cm3, T ¼ 150 K; the dependence of the Fermi energy-voltage EF(U) is included (dotted line).

bands due to the increased topological disorder characterized by the increased effective temperature Tt. In Fig. 1 there is the model TM-SCLC current–voltage dependence j(U) and activation energy–voltage dependence Ea(U) for the Gaussian distribution of the DOS h(E) (given by dotted line in Fig. 2a) with the total concentration of states Nt in the form 2 2 Nt hðEÞ ¼ pffiffiffiffiffiffi eðE t EÞ =2S ; 2pS

(19)

where S ¼ 50 meV and temperature T ¼ 150 K. In Fig. 1 (dotted line) is also given the position of the Fermi energy EF on the applied voltage. The transport energy was supposed to be the reference energy, putting Etr ¼ 0 eV. The concentration of free carriers in thermodynamical equilibrium was neglected, supposing nf0 ¼ 0, so the Ohmic part of the current j U1 is absent and the first part of the j(U) dependence is j U2, corresponding to the filling of the states near the first inflection point of the Gaussian distribution E t þ S2 =kT ¼ 0:7 eV (see the activation energy voltage dependence, Ea(U), where Ea 0.7 eV), then comes the part of the j(U) characteristic

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(a)

log ( h(E)), a.u.

20

18

16

0

(b)

γF

-5

Et + Σ /kT

Et - Σ /kT

= 0.7eV

= 0.3 eV

-15 -20

2

2

-10

0.0

0.2

0.4

0.6

0.8

1.0

energy E (eV) Fig. 2. (a) The reconstructed DOS, h(E), from the data in Fig. 1, using Eqs. (13) and (14) (bold squares), compared to input h(E) (dashed curves), (b) the dependence of the coefficient gF (defined in Eq. (17)), on energy E, see the marked reliability energy interval where jgF jo4:

corresponding to the filling of states of the Gaussian distribution itself, and finally there is again j U2 dependence, corresponding to the filling of states of the transport path at Etr ¼ 0 by the space-charge; this fact is reflected in the activation energy Ea ¼ 0 eV. In Fig. 2a the reconstruction of the DOS from the data in Fig. 1, using expressions (13) for the DOS h(E) function and Eq. (14) for the energy E, we can see that the reconstruction is very satisfactory in the energy interval from the first inflection point of the Gaussian distribution, E t þ S2 =kT ¼ 0:7 eV; through the maximum of the distribution Et and as far as the second inflection point, E t  S2 =kTð0:3 eVÞ: The point in DOS, where the filling of states of the transport path near E ¼ 0 eV takes over, depends on the energy position of the Gaussian distribution, its concentration and temperature, as we showed in Ref. [7]. In Fig. 2b there is the dependence of gF on energy E (in fact EEEa, the differential correction factor in Eq. (14) may be neglected for the purpose of Fig. 2b). We want to show that the reliability of the SCLC reconstruction correlates with the dependence gF vs energy E. We can see that the reconstruction of the SCLC data is correct in the energy interval where jgF jo4 (see the hatched interval of gF in Fig. 2b), outside this range the reconstruction error is increasing. This rule seems to have general validity for SCLC DOS reconstruction.

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10

3.0 ~U

5

2

2.5 0 2.0 ~ U ( Tt /T+1)

-10 1.5 -15 1.0

-20

Et

-25

0.5 -30 -35 -15

Ea (eV)

log (j (a.u.))

-5

0.0 -10

-5

0

5

log (U(a.u.)) Fig. 3. The influence of the energy position of the DOS distribution EtrEtA(0.2,0.7) eV: the SCLC model for states distributed in energy according to Eq. (20) with Nt ¼ 1 1014 cm3, Tt ¼ 300 K and T ¼ 100 K. The model TM-SCLC dependences, i.e. current–voltage j(U) and activation energy–voltage Ea(U).

To find the influence of parameters characterizing the DOS distribution and temperature on the energy interval of reconstruction of TM-SCLC data we also modelled in Figs. 3–6 the TM-SCLC for double exponential (bell–shaped) DOS distribution defined as hðEÞ ¼

Nt eðEE t Þ=kT t kT t ð1 þ eðEE t Þ=kT t Þ2

(20)

with the total concentration of states Nt and with rising and descending parts, both characterized by the effective temperature Tt. In Fig. 3 there is the model TM-SCLC current–voltage dependence j(U) and activation energy–voltage dependence Ea(U) for the double exponential distribution (defined by Eq. (20) and depicted in Fig. 4a by dotted lines). The concentration of free carriers in thermodynamical equilibrium was again neglected, supposing nf0 ¼ 0, so the Ohmic part of the current j U1 is absent and the first part of the j(U) dependence is j U(Tt/T+1), corresponding to the filling of the states rising (decreasing) exponentially with energy and distributed with the characteristic temperature Tt and finally there is the dependence j U2, corresponding to the filling of states of the transport path at Etr ¼ 0 by the spacecharge; this fact is reflected in the value of the activation energy Ea ¼ 0 eV. In Fig. 3 (and Fig. 4) the influence of the energy position of traps is examined, in Fig. 5 the influence of their concentrations, and in Fig. 6 the influence of the temperature. Figs. 4–6 are organized in a similar way, in parts (a) are given the input h(E)

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(a)

20 log ( h(E)), a.u.

247

Εt

18 16 14 12

6 (b)

4

γF

2 0 -2 -4 -6 0.0

0.2

0.4 0.6 energy E (eV)

0.8

1.0

Fig. 4. The influence of the energy position of the DOS distribution EtrEtA(0.2,0.7) eV: the SCLC model for states distributed in energy according to Eq. (20) with Nt ¼ 1 1014 cm3, Tt ¼ 300 K and T ¼ 100 K: (a) the reconstructed DOS, h(E) from the data in Fig. 3 using Eqs. (13) and (14) (bold curves), compared to input h(E) (dashed curves), (b) the dependence of the coefficient gF (defined in Eq. (17)), on energy E, see the marked reliability energy interval where jgF jo4:

functions (dotted curves) and corresponding reconstructed DOS h(E) functions (using the Eqs. (13) and (14)) and in parts (b) are the calculated plots of the statistical shift of the Fermi level gF ¼ lnðs0 =sM Þ on energy E. The ranges jgF jo4; defining the energy intervals of correct reconstruction of the TM-SCLC data is given by the dashed lines. It is clear from these model calculations that the SCLC measurements results are more reliable with the increased depths of the states (Fig. 4), with their increased concentration (Fig. 5) and with decreased temperature (Fig. 6). 4.4. TM-SCLC in organic semiconductors with correlated defects For completeness, the peculiarities of the TM-SCLC in organic semiconductors with correlated (amphoteric) defects with both negative and positive correlation energy may be found in Ref. [26].

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(a)

log (h(E)), a.u.

20 18 16 14 12

4

(b) 0

γF

-4 -8 -12 -16 -20 0.0

0.2

0.4

0.6

0.8

energy E (eV)

Fig. 5. The influence of the total concentration of DOS NtA(1012, 1018) cm3: the SCLC model for states distributed in energy according to Eq. (20), E tr  E t ¼ 0:2 eV; T t ¼ 200 K and T ¼ 100 K: (a) the reconstructed DOS, h(E) using Eqs. (13) and (14) (bold curves), compared to input h(E) (dashed curves), (b) the dependence of the coefficient gF (defined in Eq. (17)), on energy E, see the marked reliability energy interval where jgF jo4:

5. Conclusions (1) The TM-SCLC technique represents a sensitive spectroscopical method for establishing the DOS of localized electron states, suitable in general for low conductivity semiconductors and insulators, originally devised for inorganic semiconductors. Though the charge transport in organic semiconductors differs from that prevailing in covalently bonded inorganic semiconductors, it is shown that under some, not so strict conditions, the method may be applicable to organic semiconductors, especially

 

introduction of temperature and injection independent effective transport energy Etr, introduction of the concept of the field and temperature independent hopping mobility at the transport path mðE tr Þ;

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input h(E)

log (h(E)), a.u.

18

Τ=150Κ 200Κ

16

250Κ 300Κ

14

12

(a)

4 2

γF

0 -2 -4 -6 -8 -10 0.0

(b)

0.2

0.4

0.6

0.8

energy E (eV)

Fig. 6. The influence of the temperature TA(150,300) K: the SCLC model for states distributed in energy according to Eq. (20) with EtrEt ¼ 0.3 eV, Nt ¼ 1.1015 cm3, Tt ¼ 200 K: (a) the reconstructed DOS, h(E) using Eqs. (13) and (14) (bold curves), compared to input h(E) (dashed curves), (b) the dependence of the coefficient gF (defined in Eq. (17)), on energy E, see the marked reliability energy interval where jgF jo4:



existence of the injection contact, creating the reservoir of mobile charges, with negligible influence on the bulk electrical transport in the semiconductor.

(2) The visibility of the localized states measured by TM-SCLC i.e. the power of the TM-SCLC method to distinguish the localized electron states, is increased by their increased energy position, increased concentration and decreased temperature. (3) The reliability criterion for the spectroscopical method TM-SCLC for the DOS reconstruction is the correlation between the measured SCL current and its activation energy on applied voltage, expressed by the dependence of preexponential factor of conductivity s0 on activation energy, gives the possibility to determine the energy range where the reconstruction of density of localized states function is reliable. The criterion for the validity of the TM-SCLC reconstruction is jgF j ¼ jln s0 =sM jo4; where gF is the linear coefficient of the Taylor expansion of the temperature dependence of Fermi level position EF(T).

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