Self consistent theory of unipolar charge-carrier injection in metal/insulator/metal systems
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Self consistent theory of unipolar charge-carrier injection inmetal/insulator/metal systems
F. Neumann, Y. A. Genenko, C. Melzer and H. von Seggern
Institut f¨ur Materialwissenschaft, Technische Universit¨at Darmstadt,Petersenstrasse 23, D-64287 Darmstadt, Germany
Abstract
A consistent device model to describe current-voltage characteristics of metal/insulator/metalsystems is developed. In this model the insulator and the metal electrodes are described withinthe same theoretical framework by using density of states distributions. This approach leads todifferential equations for the electric field which have to be solved in a self consistent mannerby considering the continuity of the electric displacement and the electrochemical potential inthe complete system. The model is capable of describing the current-voltage characteristics ofthe metal/insulator/metal system in forward and reverse bias for arbitrary values of the metal/insulator injection barriers. In the case of high injection barriers, approximations are providedoffering a tool for comparison with experiments. Numerical calculations are performed exemplaryusing a simplified model of an organic semiconductor. . INTRODUCTION Injection is an important factor in many electronic devices, especially in applicationswith insulators, where virtually all charge carriers have to be injected from the electrodes.Examples that have been the subject of interest are electronic devices built with organicsemiconductors such as organic-light-emitting-diodes (OLEDs). Organic semiconductorsshow many properties of dielectric materials like relatively large band-gaps or the absenceof intrinsic charge carriers. For the processes describing the conduction in organic semicon-ductors itself, sophisticated models were recently developed [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Theproposed injection mechanisms are taking into account the stochastic, hopping character ofthe transport, the field-dependence of the mobility of charge carriers, the roughness-inducedenergetic disorder at the interface and the specific density of states (DOS) characteristic ofthese materials. However, the space charge effects were accounted so far only concerningthe transport inside the organic materials while the description of injection itself was basedon the classic works of Richardson and Fowler-Nordheim [11, 12] considering injection asa single electron process. A comprehensive numerical model was developed by Tutiˇs et al.[13] comprising the hopping transport in single- and bilayer devices and tunnel injectionfrom electrodes accounting for space-charge effect. However, besides the injection barriersgiven by the bare difference of the chemical potential in the metal and the lowest occupiedorbital (LUMO) in organic material this model involves a range of artifacts like tunnelingfactors, effective attempt frequencies etc. Being a very sofisticated numerical tool and pre-senting good agreement with experiments this approach does not allow analytical fitting ofcurrent-voltage characteristics which gives insight in major mechanisms controlling injection.In this work we develop a device model for a metal/insulator/metal system includingthe description of charge carrier injection at the metal/insulator interfaces. This injectionmodel takes into account the electrostatic potential generated by the charges injected. Thetreatment of this problem faces the problem of self consistency, since the amount of injectedcharges depends on the height of the injection barrier, while the electrostatic potential gener-ated by this charge modifies the height of the injection barrier itself. This problem is solvedby defining the boundary conditions far away from the interface, where the influence of theinterface can be ignored. As a result, charge carrier densities and electric field distributionsin the respective medias are coupled and depend on the conditions of the system. By de-2cribing the electrochemical potential as well as the dielectric displacement continuously allover the entire system the electric field and charge-carrier distributions can be calculated.One can model current-voltage-(IV) characteristics in which either charge injection or thetransport through the insulating layer determines the current. The model includes barrierlowering, but this lowering differs significantly from the lowering due to the image chargepotential which is a direct consequence of the one electron picture.The problem considered in this paper includes the presence of both, injecting and ejectingelectrode with an organic semiconductor in between and, hence, a built-in potential in thesystem. Both interfaces are considered self-consistently involving no additional characteris-tics but the bare injection barriers given by the difference of the chemical potentials in themetals and the LUMO. Electric field distributions in the device are calculated analyticallyfor different barriers heights in equilibrium. Then the field distributions are calculated nu-merically in the presence of a steady-state current. It is shown that these field distributionsare qualitatively different for injection barriers smaller and larger than some characteristicvalue. Using the known solution for the electric field at a fixed current the current-voltagecharacteristics are computed. They exhibit, depending on the barrier heights and bias, dis-tinct areas of linear, exponential and quadratic dependence. In the case of devices withlarge barriers an analytic approximation for the field distribution is derived which describesthe numerical results with high accuracy. The obtained approximation is similar to thewell-known Mott-barrier formula but includes an effective barrier height which results fromthe consistent treatment of the metal/dielectric interface.
II. THE MODEL
Let us consider an insulator of thickness L sandwiched in between two metal electrodes.The insulator is supposed to be extended over the space with − L/ < x < L/
2, whereasthe metal electrodes are extended over the semi spaces with x < − L/ x > L/ . The Electrodes Assuming the free electron model and the Thomas-Fermi approximation [14], the elec-trochemical potentials κ ± m of the metals as a function of the spatial coordinate x read, κ ± m ( x ) = ¯ h m ± (3 π n ± m ( x )) / − eφ ( x ) + E ± b , (1)Here n ± m ( x ) are the electron densities, e is the elementary charge and m ± the effective electronmasses in the metals, φ ( x ) the electrostatic potential and E ± b the bottom of the conductionbands. All quantities with indices ± are assigned to the electrode on the right ( x > L/ x < − L/ j is givenby the conductivity σ and the derivative of κ ( x ) [15]. Since we consider a one-dimensional,monopolar case, the current remains constant across the whole space, j = σe dκ ( x ) dx = const . (2)The conductivities of the metals are proportional to the electron mobilities µ ± m and thecharge-carrier densities far away from the contacts, n ±∞ : σ ± m = eµ ± m n ±∞ . Due to the chargetransfer between the metals and the insulator, space charge regions emerge, which modifythe electric fields F ± m ( x ) also in the metal. According to Gauss’s law, the derivatives of theelectric field are proportional to the excess electron densities δn ± ( x ) = n ± m ( x ) − n ±∞ : F ± ′ m ( x ) = − eǫ δn ± ( x ) . (3)where ǫ is the permittivity of free space.Since electron densities in metals are rather high, the value for the excess electron densitiesis small in comparison with the background electron densities, | δn ± ( x ) | ≪ n ±∞ . Assumingthe linear Thomas-Fermi approximation a straightforward calculation leads to differentialequations for F ± m , which read: jeµ ± m n ±∞ = − ǫ κ ±∞ e n ∞ F ± ′′ m ( x ) + F ± m ( x ) . (4)Here κ ±∞ are the chemical potentials in the metals at an infinite distance from themetal/insulator interfaces. There space charge effects vanish and hence gradients of F ± m ( x ).With this boundary condition the solutions for F ± m ( x ) read, F ± m ( x ) = " F ± m ( ± L/ − jσ ± m exp " ∓ x ∓ L/ l ± T F + jσ ± m , (5)4ith l ± T F = s ǫ κ ±∞ e n ±∞ , (6)being the Thomas-Fermi lengths, defining the typical length scales of the metals ( l T F ≃ − m ). It is clear from expressions (5) that our assumption of the metal semi-spaces is, infact, not necessary. With the very short length l T F the results are valid for any experimentalmetal thickness. In Eq. (5) the electric fields in the metals at the metal/insulator interfaces F ± m ( ± L/
2) remain as the only unknown quantities.
B. The Insulator
The energetic difference between the electrochemical potentials in the metal electrodes,at an infinite distance from the contact, and the bottom of the conduction band in theinsulator is defined as the injection barrier ∆ ± . Accordingly the energetic difference of the∆ ± s is related to the difference between the electrode work functions E ± A , E − A − ∆ − = E + A − ∆ + . (7)The introduction of a bottom of the conduction band in the insulator means that one cancalculate charge carrier densities by using Boltzmann statistics. n s ( x ) = ∞ Z −∞ g s ( E − ∆ − − κ −∞ ) exp κ s ( x ) + eφ ( x ) − EkT ! dE, (8)Here, g s ( E ) is the density of states of the insulator with g s ( E <
0) = 0, T is the absolutetemperature and k is the Boltzmann constant. The energy scale, E , has been adjusted tothe bottom of the conduction band in the left metal electrode. Thus, the electrochemicalpotential κ s can be expressed in terms of the charge-carrier density n s , κ s ( x ) = kT ln n s ( x ) N ! + ∆ − + κ −∞ − eφ ( x ) , (9)where N is defined by N = ∞ Z −∞ g s ( E ) exp( − E/kT ) dE (10)and can be understood as the effective total density of states available in the insulator at agiven temperature T . One should realize that the T dependence of N becomes weak in thecase of a narrow-band material. 5or wide band gap insulators all charge-carriers in the extended states, n s , are excesscharge-carriers appearing in Eq.(9). They have to be considered in Gauss’s law, F ′ s ( x ) = − eǫǫ n s ( x ) , (11)where ǫ is the relative permittivity of the insulator. With Eqs.(2), (9), (11) and σ s ( x ) = eµ s n s ( x ) a differential equation for the electric field in the insulator, F s , is obtained, jµ s ǫǫ = − kTe F ′′ s ( x ) − F s ( x ) F ′ s ( x ) , (12)where µ s and σ s are the electron mobility and conductivity in the insulator, respectively.Alternatively, this differential equation could be derived applying the drift-diffusion modeland the Einstein relation, which relates the mobility and the diffusivity in non-degeneratesystems [14, 16]. C. Boundary conditions and self-consistency
The self consistent treatment of charge transport through an insulator sandwiched be-tween two metals requires continuity of the electrical displacement and of the electrochemicalpotential across the whole system. In particular one may write: κ ( x ) = continuous, (13) ǫF ( x ) = continuous. (14)These conditions have to be fulfilled particularly at the metal/insulator interfaces, eliminat-ing the unknown integration constants. It follows from Eq.(14) that the electric fields in themetals and the insulator at the interface are related by: F s ( ± L/
2) = F ± m ( ± L/ /ǫ . FromEq.(13) nontrivial boundary conditions for Eq.(12) are obtained:ln( − ǫǫ e N F ′ s ( ± L/ ± kT ∓ ǫ el ± T F kT F s ( ± L/
2) = ∓ l ± T F µ ± m n ±∞ kT j, (15)which depend on parameters of both the insulator and the metal electrodes.Indeed, these boundary conditions are virtually independent of the net current density.In Eqs.(15) the current density j is multiplied by small factors l ± T F /µ ± m n ±∞ kT and hence, theboundary conditions are independent of j in most practical cases. Under this approximationthe density of injected electrons at the interfaces reads, n s ( ± L/
2) = N exp " − ∆ ± /kT ± ǫ el ± T F kT F s ( ± L/ . (16)6ence, the dependence of F s ( ± L/
2) or n s ( ± L/
2) on the current is mainly due to Eq.(12).From Eq.(16) it is evident that the height of the injection barrier is modified by the electricfield at the interface. This leads to the definition of an effective injection barrier,∆ ± eff = ∆ ± ∓ ǫel ± T F F s ( ± L/ . (17)The barrier modification is a direct consequence of the charge transfer from the metal tothe insulator and corresponds to the potential energy, electrons gain or lose in the electricfield of the metal. III. PHYSICAL AND NUMERICAL ANALYSIS
We now show the results of analytical and numerical calculations. The section is organizedas follows: In the first part we show results for calculations in equilibrium where chargecarriers diffuse from the metal electrodes into the insulator. In the second part we calculateelectric field distributions in steady state and finally we analyze the resulting current-voltage(IV) characteristics.First, the material parameter of the metal electrodes and the insulator will be specified.As mentioned in the introduction we chose a simplified model of an organic semiconductoras example for an insulator. These materials are characterized by band-gap energies rangingform 1 to 3 eV making thermal excitation to conduction states virtually impossible. As ininsulators, charge transport is therefore dominated by excess charge carriers. The DOS oforganic semiconductors is believed to have a gaussian shape [17], impeding in general theapplicability of Eq.(8) and requiring the use of Fermi-statistics. However, in the case ofweak disorder charge carrier trapping in tail states of the DOS is negligible and Boltzmann-statistics can be applied. We assume that the material specific parameters for the organicsemiconductor adopt the typical values of: N = 10 cm − at T = 300 K , ǫ = 3 and µ s = 10 − cm /V s. (18)An organic semiconductor with a thickness of L = 100 nm will be considered, typical forapplications in OLEDs. As examples for the metal electrodes, calcium, barium and magne-sium are chosen. These metals are characterized by their electron densities in the conductionband n ∞ , their chemical potentials at infinite distance from the contact κ ∞ , which at T = 07qual their Fermi-energies, their electron mobilities µ m and their electron work functions E A .The values for the material parameters are summarized in Tab.I. Here, the barriers ∆ ± aregiven by the energetic difference between the transport level in the organic semiconductorand κ ±∞ of the respective metal electrode. TABLE I:
Parameter values [14] for the different metal electrodes considered in this work. n ∞ [10 cm − ] κ ∞ [ eV] µ m [cm /V s ] E A [ eV]Ca 2.6 4.68 66.7 2.87Ba 3.2 3.65 5.07 2.7Mg 8.6 7.13 16.9 3.68 In the following the solutions for the equilibrium situation will be discussed, where spacecharge zones are formed due to diffusive charge-carrier transport from the electrodes into theinsulator. Then a constant external voltage will be considered, driving a constant currentthrough the insulator, and the calculated IV-characteristics which are crucial for comparisonwith experiment will be presented.
A. Equilibrium
In equilibrium, the current density j vanishes and Eq.(12) can be integrated: kTe F ′ s ( x ) + 12 F s ( x ) = − kTe ! Λ , (19)where Λ is a constant. The derivative of the electric field has to be negative due to Eq.(11).At some position between the contacts the electric field vanishes, thus, Λ > F s ( x ) = − kTe Λ tan (Λ x + λ ) , (20)with λ being the integration constant. This solution describes the following processes: chargecarriers are injected from both metal electrodes and diffuse far into the bulk of the organicsemiconductor. The resulting space charge generates an electric field compensating - inequilibrium - the diffusion current. While the widths of the space charge zones in the metals8re of the order of l T F , they can be extended over the whole insulating layer. This means,that at some point the diffusion currents caused by electrons injected from both electrodescompensate each other: here the electric field is zero.From the analytical solution of Eq.(19), the boundary conditions (15) can be simplifiedto a nonlinear system of equations determining the integration constants Λ and λ :ln ǫǫ N kT Λ e ( ± Λ L/ λ ) ! + ∆ ± kT ± ǫ Λ l ± T F tan( ± Λ L/ λ ) = 0 (21)In general, Eqs.(21) have to be solved numerically. However, for the situation of symmetricelectrodes the solution for the electric field is antisymmetric with respect to x = 0, whichmeans λ = 0. In Fig.1 we show the solution for the electric field of an organic semiconductorsandwiched between two calcium electrodes. The injection is supposed to be barrier free(∆ − = ∆ + = 0). The introduction of nonvanishing injection barriers leads to a reductionof the electric field at the interface, although the dependence of the electric field on thecoordinate is still given by Eq.(20). -600 -550 550 600-2-1012 F ( x ) [ F ] X [l -TF ] FIG. 1:
Distribution of the electric field F in units of F = kT /el − T F as a function of the coordinate x in units of l − T F in case of equilibrium ( j = 0 ) and barrier free injection ( ∆ − = ∆ + = 0 ) fromtwo Ca electrodes. Notice that the x -axis is extended between − L/ l − T F and + L/ l − T F ≃ andinterupted between x = − and x = +490 . Now the electrode at x = + L/ x = + L/ + = 0), the injection9arrier for the calcium/organic interface at x = − L/ − = 0 .
17 eV. The effect of an unbalanced charge-carrierinjection can be seen in Fig.2. A significant asymmetry in the computed distribution of theelectric field can be observed. -600 -550 550 600-2-10 F ( x ) [ F ] X [l TF ] FIG. 2:
Distribution of the electric field F in units of F = kT /el − T F as a function of the coordinate x in units of l − T F in equilibrium ( j = 0 ) with an injection barrier of ∆ − = 0 . eV for a Ca electrodeat x = − L/ and a vanishing injection barrier for the barium electrode at x = + L/ . Notice thatthe x -axis is interupted between x = − and x = +490 . Integration of the electric field over the whole system results in the built-in voltage V BI , − V BI = ∞ Z −∞ F ( x ) dx = F s ( − L/ ǫ l − T F + L/ Z − L/ F s ( x ) dx + F s ( L/ ǫ l + T F (22)using Eqs.(5) and (14). It looks as if the integration runs virtually over the effective lengthof the device L eff = L + ǫ ( l − T F + l + T F ) . (23)This length will be used later by the evaluation of the voltage on the device in a steady-state.The integral in Eq.(22) can be calculated analytically using the solution (20) and boundaryconditions (21): eV BI = E − A − E + A (24)and is independent of the values for the integration constants Λ and λ . One should be awareof that the built-in potential drops over the insulator and the metal electrodes.10 . Steady state Now, the steady state situation is considered, where a constant current j flows through thesystem. In this case, the differential equation (12), describing the electric field distributionin the organic media has to be solved numerically with respect to the boundary conditionsdefined by Eqs.(15).The solution for the electric field of an organic semiconductor sandwiched between twocalcium electrodes is shown in Fig.3. The calculation was carried out for barrier free injection -600 -400 -200 0 200 400 600-0.2-0.10.00.1 F ( x ) [ F ] X [l -TF ] FIG. 3:
Distribution of the electric field F in units of F = kT /el − T F in steady state for currentdensities of | j | = 10 mA / cm (solid line) and mA / cm (dashed line) as a function of the coor-dinate x in units of l − T F . The charge injection is supposed to be barrier free with calcium electrodeson both interfaces. We comment that the maxima of the electric field near the interfaces are cutoff in this plot and do not change in value with respect to the equilibrium. at both metal/organic interfaces (∆ − = ∆ + = 0) and steady state current densities of | j | = 10 mAcm and 100 mAcm . In case of vanishing injection barriers many charge-carriers areinjected into the organic semiconductor. This requires strong electric fields to compensatethe diffusion currents near the interfaces to such an extend that the net current is j . Thisfield is positive on the injecting electrode and negative on the ejecting electrode. Thus,there has to be a position x , often referred to as the virtual electrode [18], where theelectric field changes sign. When the current increases, this position shifts towards theinjecting electrode. It can be seen from Fig.3 that x shifts from approximately 25 nm to110 nm once | j | is increased from 10 to 100 mAcm . In between the two electrodes, the electricfield of the barrier free case follows approximately the electric field of a space charge limitedsystem assuming ohmic boundary conditions [11]. In this situation the dipole layer at theleft electrode acts as a source of charge-carriers, the right electrode can be understood as asink.The introduction of non vanishing injection barriers reduces the amount of charge-carrierspresent in the organic semiconductor and therefore the importance of diffusion. Whenthe injection barrier at − L/ -600 -400 -200 0 200 400 600-1.0-0.50.0 F ( x ) [ F ] X [l -TF ] FIG. 4:
Distribution of the electric field F in units of F = kT /el − T F as a function of the coordi-nate x in units of l − T F in steady state for current densities of | j | = 10 mA / cm (solid line) and mA / cm (dashed line). The organic layer of thickness L = 100 nm is contacted with calciumand barium with injection barriers of ∆ − = 0 . eV and ∆ + = 0 . eV respectively. of a system is shown, where an organic layer with larger band gap forms a contact withcalcium and barium electrodes. The injection barrier heights assigned to the contacts are∆ − = 0 .
35 eV and ∆ + = 0 .
18 eV, respectively. The electric field strength is virtuallyindependent of x while its value depends on the current density. The absence of extendedspace charge zones in the insulator indicates that the system is in an injection limited mode.12 . Current-voltage-characteristics Knowledge about the distribution of the electric field allows one to determine the voltagedrop V for a given current density j and hence to calculate the IV-characteristics. Aftersolving the differential equations for the electric field for a given current density, the voltageapplied to the system can easily be calculated by integration. For the steady-state case,however, one cannot define the voltage drop over the infinite range of integration as inEq.(22) since the electric fields in the metals, Eq.(5), are asymptotically constant and donot vanish. To compare with experimentally observed current-voltage characteristics, onehas to account for the built-in voltage, Eq.(22), which drops virtually over the effectivedevice length, Eq.(23), and can define the voltage drop over the same length: V = − L/ ǫl + TF Z − L/ − ǫl − TF F ( x ) dx − V BI (25)In Fig.5 we present IV-characteristics of a Ca/organic(100 nm)/Ca system. The injectionbarriers are varied from 0 eV to 0 . -3 -2 -1 j [ m A / c m †] V [V]
FIG. 5:
IV-characteristics for an organic layer with thickness L = 100 nm contacted with calciumelectrodes on both interfaces and different injection barriers of ∆ − = ∆ + = 0; 0 .
2; 0 .
3; 0 .
35; 0 . eV. barriers represents a limitation of the current flow through the system. At voltages upto ∼ . V the IV-characteristic shows an ohmic-like j ∼ V dependence. We emphasizethat this ohmic behavior is not due to intrinsic charge-carriers but a result of charge carriers13njected from the electrodes in the organic semiconductor even in equilibrium (Fig.1). Above ∼ . V the system shows the Mott-Gurney j ∼ V dependence of the current on the voltage.Here the system is obviously in a space charge limited mode. The IV-characteristic doesnot change considerably up to a barrier height of ∆crit ≃ .
27 eV. The current flow is forall voltages limited by the transport through the bulk of the organic semiconductor and theinjection process has only a secondary effect. Introducing injection barriers > .
27 eV, thecurrent decreases by orders of magnitude. Nevertheless, all IV-characteristics show the same j ∼ V ohmic-like dependence in the low voltage region like the curve calculated for barrierfree injection. For high ∆s this ohmic regime is followed by an exponential increase of thecurrent on the voltage. For even higher voltages all curves approach the space charge limitand coincide with the barrier free IV-curve.In Fig.6 we introduce a built-in voltage by changing the metal at x = + L/ -14 -11 -8 -5 -2 j [ m A / c m †] V [V]
FIG. 6:
IV-characteristics for an organic layer with thickness L = 100 nm contacted with a calciumelectrode at x = − L/ and a magnesium electrode at x = + L/ . The injection barrier of thecalcium electrode is varied from eV to . eV with an increment of . eV. The work functionof the metals differs by an amount of . eV, accordingly the injection barrier of the magnesiumelectrode is higher by that value. V BI = − .
81 eV. The injection barrier on the calcium electrode at x = − L/ . x = + L/ .
81 to 1 .
21 eV. 14he computed IV-characteristics for high injection barriers can be understood by con-sidering some simple approximations. In the case of high injection barriers the amount ofinjected charges is small and the electric field in the insulator can be conveniently decom-posed into a constant mean value and a space charge part which is assumed to be comparablysmall, F s ( x ) = F s + f ( x ) where | f ( x ) | ≪ | F s | . (26)The mean value of the electric field is determined by the voltage drop on the effective lengthof the metal/insulator/metal system, (25): F s = − V + V BI L eff , (27)which implies that the integral of f ( x ) over the system length equals zero.The decomposition of the electric field represented by Eq.(26) leads to a linearized formof the differential equation for the electric field in the insulator: f ′′ ( x ) + eF s kT f ′ ( x ) + ejµ s ǫǫ kT = 0 . (28)which should be supplemented by the linearized boundary conditions, Eqs.(16). Taking intoaccount that in the considered systems l ± T F /L are as small as 10 − , the solution of Eq. (28)reads f ( x ) = B " kTeF s L sinh (cid:18) eF s L kT (cid:19) − exp (cid:18) − eF s kT x (cid:19) − jµ s ǫǫ F s x, (29)here the quantity B is an abbreviation of B = N kT ǫǫ F s sinh( eF s L/ kT ) " exp − ∆ + eff kT ! − exp − ∆ − eff kT ! . (30)where we assume, approximately, ∆ ± eff ≃ ∆ ± ∓ ǫel ± T F F s . (31)The analytic solution results directly in the IV-characteristic of a system with high in-jection barriers: j = e N µ s F s eF s L/ kT ) " exp eF s L kT − ∆ + eff kT ! − exp − eF s L kT − ∆ − eff kT ! . (32)This relation is symmetric in the injection barriers and is capable of describing the currentflow in both possible directions. In the following we focus on electron transport from the15eft to the right electrode. For such situations it is convenient to examine a nonsymmetricform of Eq.(32) which reads: j = − eµ s V + V BI L eff N exp − ∆ − eff kT ! exp (cid:16) − eVkT (cid:17) − (cid:16) − e ( V + V BI ) kT LL eff (cid:17) − . (33)Equation (33) resembles the current-voltage characteristic for Mott-barriers [12]. An essen-tial difference involves the earlier introduced barrier lowering in the electrodes. In Fig.7we compare the approximate solution of the injection problem, Eq.(33), with the numerical -15 -12 -9 -6 -3 j [ m A / c m †] V [V]
FIG. 7:
IV-characteristics for an organic layer with thickness L = 100 nm contacted with a calciumelectrode at x = − L/ and a magnesium electrode at x = + L/ . The stars and the trianglescorrespond to the the IV-characteristic calculated for forward and reverse bias, while the solid anddashed lines are the corresponding approximate solutions given by Eq.(33). solution. The agreement is perfect for reverse and forward bias until the space charge effectsbecome dominant and Eq.(28) loses its validity.In Fig.8 we compare different regimes of Eq.(33) with the exact numerically calculated IV-characteristic for forward bias. The low voltage part of all the IV-characteristics computedin Figs.(6) and (8) for different values of the injection barrier can be understood in thefollowing way. For voltages V < kT /e ≪ − V BI the expression for the IV-characteristic,Eq.(33), can be approximated by j = eµ s eV BI kT VL eff N exp − ∆ + eff kT ! , (34)16nd shows a linear dependence of the current on the voltage. At zero voltage and current,electrons drift from the magnesium to the calcium electrode through the organic layer asa consequence of the difference in the work functions of the two electrodes. The blockingcharacter of the right electrode restricts the effective electron density in the device as is seenfrom the exponential factor in Eq.(34). -15 -12 -9 -6 -3 j [ m A / c m †] V [V]
FIG. 8:
IV-characteristics for an organic layer with thickness L = 100 nm contacted with a calciumelectrode at x = − L/ and a magnesium electrode at x = + L/ . The stars represent the exactnumerically calculated IV-characteristic for forward bias. The solid line corresponds to the linearpart approximated by Eq.(34), the dashed line to the exponential current increase given by Eq.(35),the dotted line to the injection current described by Eq.(36) and the dashed dotted line to a SCLCgiven by | j | = (9 / ǫǫ µV /L . For voltages from the wide range kT /e ≪ V < − V BI but not very close to − V BI , sothat − e ( V + V BI ) ≫ kT , the IV-characteristic reveals a sharp exponential increase. Such abehavior can also be extracted by an approximation of equation (33) for voltages from thisrange (see Fig.(8)), j = eµ s V + V BI L eff N exp − ∆ + eff kT + eVkT ! . (35)The exponential increase stops when the voltage V approaches the value of the built-involtage, − V BI . At this point, the mean electric field in the device changes sign.Nevertheless, well above the built-in voltage, so that V + V BI >> kT /e , both the com-puted characteristic and Eq.(33) show another exponential dependence of the current on the17oltage, due to the barrier lowering (see Fig.(8)): j = − eµ s V + V BI L eff N exp − ∆ − kT + ǫ l − T F L eff e ( V + V BI ) kT ! . (36)In both equations, (35) and (36), an exponential dependence on the voltage is observed. Inthe second case, however, this dependence is much weaker due to the factor l − T F /L eff in theexponent. The latter exponential increase results from the nonvanishing width of the spacecharge zone in the injecting electrode. The appearance of this zone is a consequence of ourself-consistent description of the device as a whole.After all, when the applied voltage overcomes the barrier at the injecting electrode,∆ − , the calcium contact can supply more charge carriers than the bulk of the organicsemiconductor can transport. That is why all the current-voltage characteristics end up inthe space charge limited regime with j ∼ V as is seen in Figs.(6) and (8). Particularly,when the calcium injection barrier is below the critical value of ∆ crit ∼ .
27 eV, the systemshows a space charge limited current j ∼ V immediately after the built-in voltage is crossed,which can also be seen in Fig.6. IV. CONCLUSIONS
In this work, we have proposed a device model capable of describing ametal/insulator/metal device under injection limited as well as space charge limited condi-tions. The problem of defining the boundary conditions at the metal/insulator interfaces wassolved by a self consistent treatment which fully includes the metal electrodes in a consistentone dimensional description of the device. In this treatment boundary conditions are definedfar into the bulk of the metal electrodes where the respective media is well defined. Thevalues for the electric field and the charge carrier densities at the interface can be calculatedand depend on the condition of the considered system.We applied our model to organic semiconductors sandwiched between metal contacts,being interesting for optoelectronic applications. Though existing models include specificfeatures of organic semiconductors, like energetic disorder or hopping transport, they de-scribe charge-carrier injection mostly in the two limiting cases of high or low injection barri-ers. In the former case charge-carrier injection models are based on the one-electron-picturewithin the framework of the Fowler-Nordheim tunneling model or the Richardson-Schottky18odel of thermionic injection [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], where space charge effects areignored completely. In the latter case the metal/insulator contact is assumed to be ohmic.Here, the contact can supply much more charge-carriers than can be transported throughthe bulk of the insulator and hence bulk properties alone define the characteristics of thesystem with no influence of the injection barrier [19, 20, 21]. As far as we know, the onlymodel accounting consistently for both injecting barriers and space charge effects is that ofTutiˇs et al. [13]. The latter numerical treatment, however, does not allow analytical fittingof the current-voltage characteristics, making the physical reasons for the cross-over frombarrier to space-charge dominated type of device behavior inaccessible.By contrast, the approach described here, offers the possibility to calculate IV-characteristics of metal/insulator/metal systems using experimentally accessible input pa-rameters. Our model is capable to predict IV-characteristics in forward and reverse biasand for all values of the injection barriers at each interface. The description includes thebuilt-in voltage and field dependent effective injection barriers, being direct consequencesof the self consistent approach. Calculated IV-characteristics can be divided in differentregimes for which approximate solutions were derived. For low voltages a linear dependenceof the current on the voltage is observed, followed by an exponential increase. This behavioris not due to intrinsic charge carriers, but due to diffusive charge carrier transport. Oncethe voltage exceeds the built-in voltage a weak exponential current increase is observed as aconsequence of injection barrier lowering. Here, charge carrier injection dominates the IV-characteristic. At very high voltages, barrier lowering becomes so strong that more chargecarriers are injected than can be transported through the bulk of the semiconductor. Thisleads to the SCL dependence, j ∝ V , for all calculated IV-curves. The relative occurrenceof these different regimes thereby depends strongly on the injection barriers at the contactbetween the insulator and the anode and the cathode metals, respectively. As a consequence,rectifying behavior is observed for strongly disparate contacts.A more realistic model incorporating charge-carrier trap states and Gaussian DOS distribu-tions of the organic semiconductor is currently developed.19 . ACKNOWLEDGEMENTS The authors acknowledge the Deutsche Forschungsgemeinschaft for financial support ofthe Sonderforschungsbereich 595. 20
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