Electronic Conduction Mechanisms in Insulators

The current density–electric field <inline-formula> <tex-math notation="LaTeX">${(}{J}-\xi {)}$ </tex-math></inline-formula> characteristics of four insulators of dramatically different electrical qualities are assessed in terms of their operative electronic conduction mechanisms. Conduction in the two high-quality insulators is dominated by Ohmic conduction and Fowler–Nordheim tunneling, whereas conduction in the two low-quality insulators involves Ohmic conduction and space-charge limited current (SCLC). Ohmic conduction and SCLC are somewhat puzzling mechanisms for contributing to insulator leakage current since they require the existence of an Ohmic contact at the cathode. Our conventional understanding of an Ohmic contact makes it difficult to ascertain how an Ohmic contact could be formed to a wide bandgap insulator. This Ohmic contact dilemma is resolved by formulating an equivalent circuit appropriate for assessing the <inline-formula> <tex-math notation="LaTeX">${J}-\xi $ </tex-math></inline-formula> characteristics of an insulator and then recognizing that an insulator Ohmic contact is obtained when the injection-limited current density from the cathode electrode is greater than that of the operative bulk-limited current density, i.e., Ohmic or SCLC for the four insulators under consideration.


I. INTRODUCTION
S INCE very few electrons are present in the conduction band of an insulator in equilibrium, electrons usually must be supplied from a cathode contact in order for an appreciable leakage current to flow through the insulator. (In this contribution, electron transport is assumed to dominate so that hole transport can be ignored.) Leakage current due to this type of electronic transport is referred to as injectionlimited conduction. As listed in Table I and illustrated in Fig. 1, electron injection can be accomplished over a barrier (thermionic emission), through a barrier (Fowler-Nordheim tunneling), or a combination of thermal excitation and subsequent tunneling through a thinner barrier (thermionic-field emission).
Given this, it is very puzzling that steady-state electrical assessment of real insulators often leads to the conclusion that bulk-limited conduction (see Table I since Ohmic and space-charge limited current (SCLC) require that the cathode contact be Ohmic while Frenkel-Poole (FP) emission, i.e., conduction arising from the thermal emission of traps lying below the conduction band minimum, can only be sustained in a steady-state manner if the cathode functions as an Ohmic-like contact so that FP traps may be continuously refilled after emission [1]. Thus, the following question arises: How is an Ohmic or Ohmic-like contact formed to an insulator? The objective of this contribution is to offer a framework for assessing leakage current behavior in insulators. We accomplish this by first formulating analytical expressions for electronic conduction mechanisms commonly observed in insulators. Then, we examine the electrical properties of four insulators of dramatically differing electrical quality. Conduction mechanisms are identified for each insulator and their measured electrical characteristics are accurately simulated using an appropriate analytical expression for each conduction mechanism. Subsequently, an insulator equivalent circuit is formulated in order to serve as a basis for development of a simulator, which can, in principle, account for all six of the bulk-or injection-limited conduction mechanisms under consideration (Table I). Finally, the nature of an Ohmic contact to a wide bandgap insulator is elucidated.

II. EXPERIMENTAL DETAILS
Metal-insulator-metal (MIM) test devices were fabricated using degenerately doped p-type (p + ) Si as the bottom 'metal' contact. Insulators were formed on top of the p + -Si substrate This work is licensed under a Creative Commons Attribution 3.0 License. For more information, see http://creativecommons.org/licenses/by/3.0/ using the following deposition methods. The 100 nm of SiO 2 was grown via thermal oxidation. The 10 and 64 nm of Al 2 O 3 was deposited by plasma-enhanced atomic layer deposition (PEALD) and solution deposition (SD), respectively. The 98 nm of lanthanum zirconium oxide (LZO) was deposited by SD. PEALD (SD) Al 2 O 3 and SD LZO films were subjected to a post-deposition anneal at 400 (500)°C in air for 1 h. Aluminum contacts with 500-μm diameter were deposited on top of each insulator via thermal evaporation and patterned via a shadow mask. MIM devices were completed by etching through each insulator and making contact to the underlying p + -Si substrate using indium solder. MIM devices were electrically characterized using an Agilent 4155C semiconductor parameter analyzer in the dark at room temperature. All electrical testing was accomplished using the Al top metal as a cathode. Insulator thicknesses were confirmed using ellipsometry. Table II summarizes the analytical equations describing steady-state current density-electric field (J − ξ) behavior for all six bulk-and injection-limited electronic conduction mechanisms found in Table I. The parentheses notation used in Table II specifies model parameters and (separated by a semicolon) temperature (when appropriate), the only physical operating parameter of relevance in this simulation. For example, the Ohmic conduction current density, J (m * e , n 0 ; T ), reveals that the model parameters are the effective mass, m * e , and the equilibrium free electron concentration, n 0 , while the physical operating parameter temperature T is implicitly relevant since it is used to model the mobility μ as discussed below.

III. J − ξ MODELING
In our formulation, calculation of all of the six conduction mechanisms requires specifying seven model parameters, i.e., electron effective mass, m * e ; equilibrium free electron concentration, n 0 ; low-frequency (static) dielectric constant, ε s ; high-frequency (optical) dielectric constant, ε ∞ ; acceptor-like conduction band tail state characteristic (Urbach) energy, W TA ; trap energy, qφ T ; and cathode metal-insulator barrier energy, qφ B ; as well as the physical operating parameter temperature, T . Several terms used in the current density expressions shown in the top of Table II are defined as auxiliary parameters, as shown in the middle of Table II, i.e., electron mobility, μ; conduction band effective density of states, N C ; peak density of acceptor-like conduction band tail states, N TA ; effective Richardson constant, A * * ; and barrier lowering due to the Schottky, E, or the FP effect, E FP . Displacement current, J DPL , can be misinterpreted as a dc current, but is actually a transient effect. J DPL is dependent on the insulator capacitance density, , and the voltage ramp rate, (dV /dt) (V s −1 ), where h is the insulator thickness (cm). When measuring a leakage current density of less than 10 −8 A cm −2 at low electric fields of less than 2 MV/cm, a voltage ramp rate of less than 10 mV s −1 is recommended in order to minimize displacement current artifacts [2]. Note that we model μ as a diffusive mobility, as exemplified by Brownian motion, in accordance with the amorphous nature of a practical insulator [3]- [5]. The auxiliary parameter equation for N TA is new. It is derived by setting the slope of the band tail state density (dg TA /d E) equal to the slope of the conduction band density of states (dg C /d E) at the mobility edge in order to ensure that the density of states is continuous at the mobility edge [6]. The equation used to describe SCLC in Table II pertains to SCLC involving an exponential trap distribution; this type of trap distribution is relevant to the case of band tail states associated with an amorphous insulator [1], [3]- [6]. Series resistance, R S ( − cm 2 ), is included in the SCLC equation of Table II. All nine model parameters (including R S ) are listed at the bottom of Table II. The first five are used to simulate the measured J − ξ characteristics of thermal SiO 2 , PEALD Al 2 O 3 , SD Al 2 O 3 , and SD LZO. In our simulation, ε s and m * e (color-coded blue) are fixed values obtained from the literature, while n 0 , qφ B , W TA ; and R S (color-coded green) are varied to optimize simulated J −ξ curves in order to obtain a best fit to the measured data. Contact area variation due to shadow mask processing and thickness nonuniformity across the substrate accounts for minor variations in estimated n 0 and qφ B values. ε ∞ and qφ T are not declared in the simulation since thermionic-field emission and FP emission do not appear to appreciably contribute to the leakage currents of the four insulators under consideration.  match quite well over most of the range of applied electric field. Fig. 3 shows log(J ) − log(ξ ) curves for all four insulators under consideration. When plotted on a log-log scale, it becomes more apparent that there are two dominant regimes of J − ξ behavior for each insulator. Fig. 3(a) and (b) indicates that the J − ξ behavior in thermal SiO 2 and PEALD Al 2 O 3 is dominated by Ohmic and Fowler-Nordheim tunneling conduction mechanisms. In contrast, Fig. 3(c) and 3(d) demonstrates that SD Al 2 O 3 and SD LZO exhibit much larger Ohmic leakage current densities at low fields than thermal SiO 2 and PEALD Al 2 O 3 and also suffer from a much earlier onset of an increase in current density due to SCLC.  [22]. Also, the m auxiliary equation included in Table II shows that m is related to the conduction band Urbach energy, W TA . The W TA [SD LZO] = 51 meV value is relatively large, but is well within the range normally expected for an amorphous insulator. However, the W TA [SD Al 2 O 3 ] = 220 meV value is extraordinarily large. Since the SD Al 2 O 3 insulator was deposited in such a manner to intentionally increase its porosity [23], it is likely that its anomalously large Urbach energy is attributable, at least in part, to its porosity. Porous silicon Urbach energies in the range of 150-300 meV have been reported [24]. Peak conduction band tail state density N TA is calculated from W TA , as shown in Table II. In addition, using the equation for the total conduction band tail state density n TOTAL listed in Table II, or recognizing that n TOTAL = N TA W TA [6], n TOTAL = 2.0 × 10 20 cm −3 , and n TOTAL = 8.2 × 10 18 cm −3 for SD Al 2 O 3 and SD LZO, respectively. Note that, the n TOTAL relationship indicates that an increase in the Urbach energy results in a corresponding increase in the total amount of conduction band disorder which is primarily assocated with the cation sublattice [3] [4]. Finally, in Fig. 3(c), the rollover behavior at an electric field greater than ∼ 2 MV/cm is attributed to series resistance, i.e., R S = 400 − cm 2 , which is modeled as a parasitic voltage drop, as shown in the equation for exponential SCLC in Table II. Fig. 4 is an equivalent circuit model used to simulate insulator J −ξ curves. Each conduction mechanism is modeled as a voltage-controlled current source since the electric field is assumed to be uniform across the insulator. The total insulator current density J is given by

V. EQUIVALENT CIRCUIT CONSIDERATIONS
Note that injection-limited current densities, i.e., J TE , J FN , and J TFE , contribute directly to J . In contrast, bulk-limited current densities, i.e., J , J SCLC , and J FP , only contribute directly to J when J OI J + J SCLC + J FP so that Eq. 1 simplifies to J = J DPL + J TE + J FN + J TFE + J + J SCLC + J FP . In Eq. 1, J OI denotes an Ohmic injection current density, as discussed in Section VI.
Satisfying the less restrictive inequality J OI > J + J SCLC + J FP corresponds to the cathode contact being able to supply a sufficient amount of current so that the cathode functions as an Ohmic contact. Thus, satisfying the inequality J OI > J + J SCLC + J FP constitutes our definition of what constitutes an Ohmic contact. This definition of an Ohmic contact is quite different from the normal view in which the cathode metal contact is required to maintain an unlimited concentration of charge carriers [1] [25].
Finally, a capacitor with a capacitance density, C I , is connected in parallel in Fig. 4 in order to account for the displacement current density, J DPL , associated with the ramp rate of the applied voltage waveform.
All conduction mechanisms included in Fig. 4 have now been discussed within the context of Table II, except for Ohmic injection, J OI , as considered in Section VI. A potential source of confusion is our use of the term "Ohmic" in two different ways: 1) denoting the flow of Ohmic current in the insulator (J ) or 2) referring to the injection of electrons from the cathode contact in order that the contact function as an Ohmic contact (J OI ).

VI. NATURE OF AN OHMIC CONTACT TO
A WIDE BANDGAP INSULATOR Steady-state bulk-limited conduction is observed for all of the insulators included in Fig. 2. This is surprising since it implies that an Ohmic contact is formed to the insulator at the cathode. Rose asserts that such an Ohmic contact is "an electrode that supplies an excess or a reservoir of carriers ready to enter the insulator as needed" [25]. An energy band diagram illustrating the type of Ohmic contact envisaged by Rose is shown in Fig. 5. In equilibrium, as indicated in   , between E F and the conduction band minimum of the insulator, E C . As a negative bias is applied to the cathode metal, as shown in Fig. 5(b), a virtual cathode forms, in which the original energy barrier, qφ B , is reduced due to image force barrier lowering, E. For this virtual cathode to function as an Ohmic contact, it must be capable of supplying a larger current density than that required for bulk-limited conduction due to Ohmic conduction or SCLC, for the insulators considered herein. In turn, the current density that this virtual cathode can supply depends critically on the magnitude of qφ B . Rose's picture of an insulator Ohmic contact implies that thermionic emission or thermionic emission with barrier lowering is the operative mechanism responsible for Ohmic injection. In the following, this hypothesis is examined for the case of thermal SiO 2 with an Al metal contact, an important test case since its injection barrier is so large, i.e., qφ B = 3.1 eV. Fig. 6 is a simulation [at an electric field of 0.3 MV/cm, corresponding to the approximate onset of Ohmic current, as shown in Fig. 3(a)] of the leakage current density expected for thermal SiO 2 due to thermionic emission (J TE , black curve) or thermionic emission with barrier lowering (J TE,BL , black dashed curve) versus the barrier (qφ B ). These simulated trends are compared to the measured Ohmic current density of J (SiO 2 ) ≈ 4 × 10 −10 A cm −2 (blue solid line), and the  expected barrier, qφ B (Al − SiO 2 ) = 3.1 eV (green dashed line). Fig. 6 is revealing. It shows that thermionic emission (thermionic emission with barrier lowering) is a viable Ohmic injection mechanism only when qφ B ≤ 0.9 eV (qφ B ≤ 1.1 eV). Since qφ B (Al − SiO 2 ) ≈ 3.1 eV, J TE and J TE,BL are clearly not the operative Ohmic injection mechanisms giving rise to the Ohmic current density measured for SiO 2 . Thus, Rose's model for an insulator Ohmic contact cannot account for Ohmic injection in the Al-SiO 2 system.
We believe that Ohmic injection in thermal SiO 2 is trapmediated. Fig. 7 shows an energy band diagram for the Al-SiO 2 system with two trap distributions included within the bandgap of SiO 2 [26] [27]. It is reported that there exists a shallow, Gaussian-like trap distribution (green) with a peak density at ∼1 eV and an extended tail down to ∼1.7 eV below E C [26]. Additionally, a Gaussian-like deep trap distribution (blue) is found to have a peak density centered at ∼2.5 eV with symmetrical tails extending to ∼2 and ∼3 eV [27]. Fig. 8(a) illustrates a simplified energy band diagram of the Ohmic injection mechanism associated with the trap distributions pictured in Fig. 7. Note that, the shallow and deep Gaussian-like trap distributions shown in Fig. 7 are replaced by two discrete trap states in Fig. 8(a) in order to focus on the primary current density components that are likely to give rise to the Ohmic injection current density, J OI . Initially, electrons are injected via trap-mediated thermionicfield emission, J TFE,T , (blue) into the deep trap distribution, E DT , followed by trap-to-trap emission, J TT , (red) from deep trap states into shallow trap states, E ST . Finally, electron emission occurs from shallow states into the SiO 2 conduction band, J ST , (green) in order to supply enough electrons to the SiO 2 conduction band to sustain steady-state bulk-limited conduction. Fig. 8(b) shows an equivalent circuit revealing how the Ohmic injection current density, J OI , is determined from the three trap-mediated contributions. From the equivalent circuit shown in Fig. 8(b), it is evident that J OI is equal to the reciprocal sum of J TFE,T , J TT , and J ST and is, therefore, dominated by the smallest current density contribution. Table III lists the equations used to very crudely model, J TFE,T , J TT , and J ST using a discrete trap approximation. Trapmediated thermionic-field emission J TFE,T is modified from the equation for thermionic-field emission given in Table II by incorporating the energy of the deep trap state, qφ T , thus accounting for electron tunneling into a trap state, rather than E C [9] [10]. The trap-mediated thermionic emission current density, J TE,T , is similarly formulated. Current density due to electron emission from a deep trap state to a shallow trap state, J TT , is formulated using (11) from [31], but ignoring hole emission, e p , and evaluating electron emission, e n , using the expression given on [31], pp. 1620. The filled deep trap density, n DT , and empty shallow trap density, p ST , are crudely estimated to be ∼ 10 16 cm −3 . Also the term exp(−2R/R 0 ) is included in J TT in order to account for tunneling from a deep trap state to a shallow trap state [29] [30]; R is the localized state separation distance and R 0 is the localization length. Note that, the adiabatic limit, i.e., exp(−2R/R 0 ) ≈ 1, is assumed in the simulation shown in Fig. 9 in order to circumvent complications associated with estimating R and R 0 as well as including high electric field effects, barrier lowering, and polarization corrections [29]. The current density due to electron emission from the shallow trap state to E C , J ST , Both J TT and J ST utilize an electron capture cross section for a neutral trap, σ n = 10 −15 cm 2 , similar to the reported value of 10 −16 cm 2 for low-temperature electron emission from a shallow trap [26]. Fig. 9 shows a simulation of the current density as a function of energy for the three Ohmic injection contributions listed in Table III, including the energy for trap-mediated thermionic-field emission, (q(φ B − φ T )), trap-to-trap emission (E DT − E ST ), and emission from a shallow trap to E C (E ST ). The measured Ohmic current density for thermal SiO 2 at an electric field of 0.3 MV/cm, J (SiO 2 ) ≈ 4 × 10 −10 A cm −2 (black), is also shown in Fig. 9. Recall that J OI > J (SiO 2 ) is our condition that the cathode metal is properly functioning as an Ohmic contact. Considering each contribution to J OI separately, Fig. 9 reveals that q(φ B − φ T ) 0.4 eV, E DT − E ST 0.6 eV, and E ST 0.7 eV for the J TFE,T , J TT , and J ST contribution, respectively, to meet the Ohmic contact requirement. All three of these Ohmic contact requirements are met using the trap distribution for SiO 2 given in Fig. 7. Thus, Ohmic injection in Al-SiO 2 appears to involve a complicated process of trap-mediated thermionic-field emission, trap-totrap emission, and trap emission to the SiO 2 conduction band to provide enough conduction band electrons to sustain Ohmic current flow.
It is important to re-emphasize that the Ohmic injection model outlined in Table III is very crude and as currently formulated is not capable of quantitatively accounting for the Ohmic injection trends. For example, the adiabatic limit assumption in which exp(−2R/R 0 ) ≈ 1 implies that R R 0 , which is not true unless n DT and p ST are orders of magnitude larger than their assumed value of ∼ 10 16 cm −3 . Thus, the Ohmic injection model of Table III needs to be refined and enhanced by accounting for high electric field effects, barrier lowering, and polarization corrections [29].

VII. CONCLUSION
The J − ξ characteristics of four insulators are assessed. High-quality insulators (thermal SiO 2 and PEALD Al 2 O 3 ) exhibit low leakage current densities involving Ohmic conduction and Fowler-Nordheim tunneling, whereas low-quality insulators (SD Al 2 O 3 and LZO) exhibit high leakage current densities involving Ohmic conduction and SCLC. Formulation of an equivalent circuit for insulator conduction mechanisms leads to the condition J OI > J + J SCLC + J FP , which constitutes a new definition of what constitutes an Ohmic contact. When the insulator injection barrier is large, i.e., qφ B 1 eV, satisfying the Ohmic contact condition can only be achieved when Ohmic injection is trap-mediated.