1. Introduction

The current conduction through SiO₂ gate dielectric is crucial for its successful application in metal–oxide–semiconductor (MOS) devices. Because of the large electron energy barrierΦ_B at the Si/SiO₂ interface and low trap density in the oxide forbidden gap, the dominant current conduction mechanism in silicon based MOS devices was believed to be the electrode-limited conduction^[1] via Fowler–Nordheim (FN)^[2–5] or direct tunneling (DT)^[6] depending on the oxide thickness t_ox and applied oxide voltage V_ox. The electronic current conduction through SiO₂ films thicker than 5 nm and at oxide electric fields E_ox above 6 MV/cm satisfying V_ox > Φ_B is widely accepted to be solely due to FN tunneling of electrons from the inversion or accumulation layer of silicon substrate or polysilicon gate through the triangular energy barrier at the cathode/oxide interface^{[4, 6]}. FN tunneling current is a major concern for the reliability of MOS structures as well as for the functionality of electrically erasable programmable read-only memory (EEPROM) devices at room temperature^[4] and also at elevated temperatures^[7]. Therefore, since early 1970s FN tunneling mechanism has been received much attention by several researchers^[2–10].

In earlier reports^[2–5], current conduction mechanism through SiO₂ dielectric in MOS structures at room temperature was established as the FN tunneling of electrons from the excellent linear fit of the measured ln (J_G $E_{\rm ox}^{2}$ ) versus 1/E_ox data as demanded by the Lenzlinger–Snow (LS) equation^[2].

$${J_{\rm{G}}} = {A_{{\rm{FN}}}}\, E_{{\rm{ox}}}^2 \, {\rm{exp}}\left( {-{B_{{\rm{FN}}}}/{E_{{\rm{ox}}}}} \right),$$

(1)

where J_G is the current density, E_ox is the oxide electric field, B_FN and A_FN are FN constants given by^[4]

$${B_{{\rm{FN}}}} = \frac{4}{3}\sqrt {\frac{{2q \, {m_{{\rm{ox}}}}\varPhi _{\rm{B}}^3}}{{{\hbar ^2}}}},$$

(2)

$${A_{{\rm{FN}}}} = \frac{{{q^2}\, {m_{{\rm{Si}}}}}}{{16{\pi ^2}\hbar {\varPhi _{\rm{B}}}{m_{{\rm{ox}}}}}},$$

(3)

where m_ox = 0.42m₀ and m_Si = 0.916m₀ are the effective masses of the electron in oxide^[2] and in silicon normal to the interface^[3], respectively, m₀ is the free electron mass, $\hbar$ is the reduced Planck’s constant, q is the magnitude of electron charge, and Φ_B is the effective electron barrier height^[7] at the cathode/oxide interface measured from the Fermi level E_F of the cathode. Φ_B is related to the conduction band offset^[11] $\varPhi _{\rm{e}}^{\rm{c}}$ by ${\varPhi _{\rm{B}}} = \varPhi _{\rm{e}}^{\rm{c}}-(\mathcal{E}_{\rm{F}}-{\mathcal{E}_{\rm{c}}})$ , where $\mathcal{E}$ _c is the conduction band edge in eV as shown in the inset in Fig. 1(a). Incorporating the temperature variations empirically in B_FN and A_FN, the above LS Eq. (1) has also been used by other researchers to explain the current conduction in MOS devices as FN tunneling at elevated temperatures^[7–10]. However, the validity of the LS Eq. (1) originally developed^[2] for electron emission from the metal electrode at 0 K becomes questionable^[11] while used in explaining tunnel injection of electrons from the semiconductor surface. To resolve the above debate, we therefore have developed a closed form expression for FN tunnel injection from the accumulation layer of the semiconductor surface at a wide range of temperatures. Moreover, the excellent numerical fit of the measured data with Eq. (1) cannot ensure the conduction mechanism solely due to FN tunneling of electrons as will be elaborated below.

Furthermore, understanding the current conduction mechanism is crucial in studying hot-electron induced anode hole injection (AHI) and related oxide degradation particularly in thin SiO₂ films down to 15 nm as originally proposed by Samanta and co-worker^{[12, 13]} for 4H-SiC devices. The present work is therefore a first time attempt to critically investigate the mechanism(s) of oxide thickness t_ox dependent current conduction through thin SiO₂ films in n⁺-polySi/SiO₂/p-Si diodes at high fields and temperatures with an aim to theoretically study the time-dependent dielectric breakdown (TDDB) in silicon basedMOS devices.

2. Device details

The devices studied here were n⁺-polySi gate tunnel oxide MOS capacitors having 5.4-, 7-, 9- and 12-nm thick SiO₂ films thermally grown on (100) oriented p-Si. Acceptor dopant concentrations of the substrates were 1.2×10¹⁵ cm⁻³ for 5.4-, 7- and 12-nm thick and 6×10¹⁷ cm⁻³ for 9-nm thick SiO₂ films, respectively. Phosphorous donor concentration (N_D) in n⁺-polySi gate was 10²⁰ cm⁻³. Details of device process flow are found elsewhere^[8]. Current–voltage measurements were carried out by Hadjadj and co-workers^[8] keeping p-Si in accumulation condition (negative bias on n⁺-polySi gate) at a wide range of temperatures between 25 and 300 °C.

3. Analysis and discussions

3.1 Current conduction mechanism

The excellent fit of the measured^[8] J_G versus E_ox data with the conventional LS Eq. (1) irrespective of oxide thickness and temperature as shown in Fig. 1(a) apparently indicates FN tunneling of electrons to occur from the n⁺-polySi gate at a wide range of temperatures. However, FN tunneling current density J_G modeled by the LS Eq. (1) is essentially oxide thickness independent when considered at a given applied electric field E_ox. Therefore, FN tunneling is unlikely the primary conduction mechanism to explain the observed t_ox dependence of J_G at a given applied electric field E_ox and temperature in Fig. 1(a). The effective electron barrier height Φ_B at the injecting electrode/oxide interface is usually estimated from the slope B_FN of the experimental ln(J_G/E_ox²) versus 1/E_ox linear FN plot and using Eq. (2). Both theoretical and experimental results of the temperature variations of Φ_B are shown in Fig. 1(b). Contrary to the results of Φ_B estimated using experimental J_G–E_ox data, our theoretical calculation clearly showsthat Φ_B increases with increasing temperature due to thecombined effects of increasing $\varPhi _{\rm e}^{\rm c}$ and the Fermi level with increasing temperature for a given dopant concentration inthe n⁺-polySi cathode. Moreover, there is no physicalreason of the t_ox dependence of Φ_B as estimated at a giventemperature [Fig. 1(b)]. tox independent theoretical valuesof Φ_B were calculated using temperature variation of theconduction band offset $\varPhi _{\rm e}^{\rm c}$ schematically depicted in Fig.2(a) and incomplete ionization^[14] of the dopant usingFermi-Dirac (FD) carrier statistics in n⁺-polySi.

Temperature dependence of $\mathop {\varPhi }\nolimits_{\rm{e}}^{\rm{c}}$ occurs due to unequal downshifts of the CBs of the n⁺-polySi and the SiO₂ caused due to spreading of the Urbach tail towards the midgap without any experimentally observed^[16] shift of the valence band (VB) top as schematically depicted in the inset (right) in Fig. 2(a). Taking both thermal narrowing^{[14, 17]} of intrinsic gaps and temperature dependent ionized dopant-induced bandgap narrowing^[18], the estimated bandgaps of n⁺-polySi and SiO₂ are listed in Table 1 along with the electron barrier height $\mathop {\varPhi }\nolimits_{\rm{e}}^{\rm{c}}$ (also called the CB offset) at the cathode (n⁺-polySi)/oxide interface. In order to estimate $\mathop {\varPhi }\nolimits_{\rm{e}}^{\rm{c}}$ , we have taken its measured^[15] value of 3.29 eV for nominal doping at room temperature. This value of CB offset for nominal doping is close to 3.25 eV reported in Ref. [14]. The small increase in $\mathop {\varPhi }\nolimits_{\rm{e}}^{\rm{c}}$ with increasing temperature (Table 1) is due to the smaller temperature-induced CB downshift in SiO₂ than that in n⁺-polySi, consistent with their measured^[14] thermal conductivities, which are correlated to the relative bond length change due to thermal expansion.

The above discussions critically examine the limitations of the widely used LS Eq. (1) in identifying the thickness and temperature dependent conduction mechanism through thin SiO₂ films used in MOS devices, even though a large class of J_G versus E_ox data can be empirically fitted with Eq. (1) yielding unphysical parameters. One should note that the above LS Eq. (1) originally derived^[2] for electron emission from the metal electrode at 0 K is no longer valid for field emission from a semiconductor at room temperature and above^[11] where a significant contribution in tunneling comes from the electrons residing below the CB edge due to band bending at the interface as schematically shown in Fig. 2(b). This physical idea necessitates redefining the electron barrier height by the CB offset $\mathop {\varPhi }\nolimits_{\rm{e}}^{\rm{c}}$ . Because of small band bending of the accumulation layer at the semiconductor (cathode)/oxide interface with an assumption of the classical 3D Fermi gas model, FN tunneling current density J_FN can be calculated by

$$\begin{array}{l}{J_{{\rm{FN}}}} = \displaystyle {\frac{{q{m_{{\rm{Si}}}}{k_{\rm{B}}}T}}{{2{\pi ^2}{\hbar ^3}}}} \displaystyle \int_{{\mathcal{E} _x} = {\mathcal{E}_{{\rm{c, 0}}}}}^\infty {{\rm d}{\mathcal{E} _x}} {T_{{\rm{WKB}}}}\left( {{\mathcal{E} _x}} \right) \\[23pt]\;\;\;\;\;\;\;\;\;\;\;\;\;\times\ln \left[ {1 + \exp \left( {{\mathcal{E} _{\rm{F}}} - {\mathcal{E} _x}} \right)/{k_{\rm{B}}}T} \right],\end{array}$$

(4)

where $\mathcal{E}$ _F and $\mathcal{E}$ _{c, 0} are the semiconductor Fermi level and the CB edge at the semiconductor (cathode)/oxide interface, respectively as shown in Fig. 2(b), k_B is the Boltzmann constant and T is the absolute temperature. Taking image force barrier lowering as schematically shown in Fig. 2(b) into account in calculating the tunneling transmission probability T_WKB( $\mathcal{E}$ _x) using Wenzel-Kramers-Brillouin (WKB) approximation, after involved algebraic manipulation Eq. (4) reduces to the following analytical form

$${J_{{\rm{FN}}}} = A'f\left( c \right)E_{{\rm{ox}}}^2\exp \left( {-B'/{E_{\rm ox}}} \right),$$

(5)

$$c = t\left( y \right)\frac{{2{k_{\rm{B}}}T\sqrt {2{m_{{\rm{ox}}}}\Phi _{\rm{e}}^{\rm{c}}/q} }}{{\hbar {E_{{\rm{ox}}}}}},$$

$$A' = A_{{\rm{FN}}}'/{t^2}\left( y \right){\rm{, }}\quad B' = B_{{\rm{FN}}}'v\left( y \right),$$

$$f\left( c \right) = {e ^{ - \eta c}}\left[ {_2{F_1}\left( {1, - c;1 - c; -e^\eta } \right) - c\ln \left( {1 +e^\eta } \right) - 1} \right],$$

where $B'_{_{\rm FN}}$ and $A'_{_{\rm FN}}$ are obtained upon replacing Φ_B by $\mathop {\varPhi }\nolimits_{\rm{e}}^{\rm{c}}$ in Eqs. (2) and (3), respectively, $\hbar$ is the reduced Planck constant. t(y) and v(y) are the correction factors due to image force barrier lowering ^[19], where y is the image force induced barrier lowering Δ $\mathop {\varPhi }\nolimits_{\rm{e}}^{\rm{c}}$ normalized to $\mathop {\varPhi }\nolimits_{\rm{e}}^{\rm{c}}$ and is given by^{[14, 20]}

$$y = {\rm{\Delta \varPhi _{\rm{e}}^{\rm{c}}}} /\varPhi _{\rm{e}}^{\rm{c}}\;, \quad\Delta \varPhi _{\rm{e}}^{\rm{c}} = \sqrt {\frac{{{q^3}{E_{{\rm{ox}}}}}}{{4\pi {\varepsilon_{\rm ox}^{\infty }}}}} .$$

(6)

$\varepsilon_{ox}^{\infty }$ is the high-frequency permittivity of SiO₂, ₂F₁ is the Hypergeometric function^[21] and η = ( $\mathcal{E}$ _F– $\mathcal{E}$ _{c, 0})/k_BT. For a semiconductor, ( $\mathcal{E}$ _F– $\mathcal{E}$ _{c, 0}) changes as a function of the electric field across it, which in turn is related to the oxide electric field by the continuity condition of the electric displacement vector across the semiconductor/oxide boundary. Solving 1D Poisson’s Equation in a semiconductor^[14], η can be evaluated as a function of oxide field by solving

$$\displaystyle \frac{{\varepsilon _{{\rm{ox}}}^2E_{{\rm{ox}}}^2}}{{2q{\varepsilon _{{\rm{sc}}}}{k_{\rm{B}}}T}} = {N_{\rm c}}{F_{3/2}}\left( \eta \right),$$

(7)

where ε_sc and N_c are the permittivity and density of states in the CB of the semiconductor, respectively and F_j is the complete Fermi-Dirac integral of fractional order j and is given by^[14]

$${F_j}\left( \eta \right) = \frac{1}{{\varGamma \left( {j + 1} \right)}}\int_0^\infty {\frac{{{x^j}{\rm d}x}}{{1 + \exp \left( {x-\eta } \right)}}, }$$

(8)

where Γ(j) is the Gamma function^[21].

Our modified formula for FN tunneling from the accumulation layer governed by Eq. (5) is valid for c < 1 depending on T and E_ox as long as the cathode electric barrier remains triangular. In order to satisfy the condition c < 1, the lower limit of E_ox increases from 4.14 to 6.4 MV/cm when the temperature changes from 100 to 300 °C. As expected, the new analytical expression for J_FN is universal (independent of t_ox at a given temperature) for electron injection from the accumulation layer of a semiconductor having a given resistivity.

Using the CB offset values from independent band structure understanding as listed in Table 1 and employing Eq. (5), theoretically calculated J_FN from n⁺-polySi is observed consistently much smaller than the measured^[8] J_G irrespective of temperatures and oxide thicknesses as illustrated in Fig. 3(a). Similar results are also presented in Fig. 3(b) during electron injection^[3] at room temperature from non-degenerately doped n-type Si into SiO₂ films having varying thicknesses (20 to 105 nm). This was found to be true also with the CB offset values calculated using its value^[14] of 3.25 eV for nominal doping at room temperature (not shown here). Oxide thickness independent J_FN shown in Fig. 3 exhibits a smaller thermal activation compared to that of the measured^{[3, 8]} J_G. The observed effect was distinct and reproducible in several different occasions as reported during the write operation^[20] of EEPROM devices under positive gate bias as well as during electron injection^[13] from n-4H-SiC into thin SiO₂ and hence is not incidental.

The enhanced thermal activation of electron injection from the n⁺-polySi/SiO₂ interface and oxide thickness dependency of the leakage current J_G indicate the presence of a thermally activated trap-assisted mechanism via Poole-Frenkel (PF)^{[22, 23]} or Schottky emission (SE)^[14] or two-step generalized trap-assisted tunneling (GTAT)^[1] in these thin SiO₂ films. The GTAT model^[1] is unlikely to be a candidate in explaining the strong temperature dependent J_G data because the trap concentration in as-grown devices does not change with ambient temperature as reported experimentally by Hadjadj et al.^[8]. Moreover, the difference between the magnitudes of J_G and J_FN does not fit (not shown) with the equation^[14] of SE. On the other hand, it is evident from Fig. 4 that J_G–J_FN versus E_ox data fit well with the bulk-limited PF emission equation given by^{[13, 22]}

$${J_{{\rm{PF}}}} = {C_{{\rm{PF}}}}{E_{{\rm{ox}}}}{\rm{exp}}\left( {-\frac{{q\varPhi_{\rm T}-\xi {\beta _{{\rm{PF}}}}\sqrt {{E_{{\rm{ox}}}}} }}{{{k_{\rm{B}}}T}}} \right),$$

(9)

$${\beta _{{\rm{PF}}}} = \sqrt {{q^3}/\pi {\varepsilon _0}{\varepsilon _\infty }},$$

where J_PF(= J_G–J_FN) is the PF current density, J_G is the measured leakage current density and J_FN is theoretically calculated FN current density using Eq. (5) at an identical applied field. C_PF is a constant depending on the density of electron traps and hence with t_ox as shown in Fig. 5(a), β_PF is the PF constant, ε₀ is the free space permittivity and ε_∞ = 2.15 (square of refractive index) is the high-frequency dielectric constant of SiO₂ and Φ_T is the trap depth relative to the bottom of SiO₂ CB. The acceptor compensation factor ξ varies between 1 and 2 depending upon the amount of acceptor compensation^[22] as depicted in Fig. 5(b). ξ was found to increase linearly with increasing oxide thickness and temperature. Indeed, during oxidation, near-interfacial neutral electron traps are created in the oxide. These traps emit electrons into the oxide CB under the application of an electric field at room temperature and above via the PF mechanism^[23]. Φ_T can be determined from the Arrhenius plot of the intercepts of individual ln (J_PF/E_ox) versus $\sqrt {{E_{{\rm{ox}}}}}$ plots (also called PF plots) at a given T as illustrated in Fig. 5(a). The estimated trap depth of 0.93 eV for 9 nm thick SiO₂ film agrees well with that reported by Harrell and co-worker^[23] for an identically thick MOS diode during negative gate injection. The estimated range of Φ_T between 0.6 to 1.12 eV is similar to the range of the metastable puckered configuration of oxygen vacancies in thermal SiO₂ as reported by Lu et al.^[24] from density functional theory. The results illustrated in Fig. 5(a) clearly shows that PF emission occurs from the deeper electron traps as well as the concentration of electron traps decreases resulting in C_PF being smaller with increasing oxide thickness. These cause a decrease in J_PF at a given E_ox and at a given T with increasing t_ox. We therefore speculate that very thick SiO₂ samples exhibit a universal (thickness independent) J_G–E_ox characteristic due to FN tunneling at a given temperature, where the contribution from PF electron emission is insignificant from energetically deep traps.

The classical PF Eq. (9) is physically meaningful till all the traps are ionized (emptied) and this happens at the saturation field E_sat obtained from the solution^[23] of ${\partial ^2}{J_{{\rm{PF}}}}/\partial {T^2} = 0$ . Beyond E_sat, PF current is due to the drift motion of the carriers under the applied field and can be written as^[23] J_PF = Kn_r^maxE_ox, where K is a constant, n_r is the free electron density in the conduction band relative to the effective density of states N_c and it attains maximum n_r^max at E_sat. With increasing temperature, saturation in PF emission occurs at a lower field as can be seen in Fig. 6. Moreover, at a given temperature the saturation field E_sat exhibits a weak t_ox dependence(not shown). As evident from Fig. 6, E_sat is well above the intrinsic breakdown field of good quality SiO₂. A similar inference is drawn throughout the SiO₂ thickness range studied here between 25 and 300 °C. Therefore, measurement data cannot be extended up to the saturation field.

4. Conclusion

In conclusion, a comprehensive analysis is presented to understand the mechanisms of oxide thickness dependent electronic current conduction in thin SiO₂ films at a wide range of temperatures under negative bias on the n⁺-polySi gate. We propose that the measured leakage current from the n⁺-polySi gate is due to PF emission of electrons from the process-induced near-interfacial electron traps located in the oxide gap at energy levels between 0.6 and 1.12 eV below the SiO₂ CB for 5.4 to 12 nm-thick SiO₂ films in conjunction with FN tunneling of electrons from the accumulation layer of n⁺-polySi. We address that fitting the measured J_G versus E_ox data with the widely used LS Eq. (1) incorrectly explains the conduction mechanism solely due to FN tunneling at room temperature and above, where the thermally activated PF emission of trapped electrons from the near-interfacial neutral electron traps is a dominant factor responsible for the increase in leakage current.

A new modified expression has been proposed to calculate the FN tunneling current density as a function of temperature and applied electric field during electron injection from the accumulation layer of the semiconductor. A method is developed to segregate the above two current components I_FN and I_PF from the measured leakage current I_G, where I_FN from the accumulated semiconductor substrate can be estimated using our modified FN equation without any adjustable parameters and taking the small variation of CB offset $\varPhi _{\rm{e}}^{\rm{c}}$ with temperature from the independent understanding of band structure physics instead of apparent thermal lowering of Φ_B obtained by fitting the measured J_G–E_ox data with LS Eq. (1) even exhibiting an excellent fit numerically. Our present work speculates the oxide thickness independent universal J_G–E_ox characteristics for very high quality thick oxides where the FN tunneling component significantly dominates the PF emission from energetically deep traps in the temperature range studied here. The present analysis can be extended to study the conduction mechanism under positive gate bias in MOS capacitors on n-type semiconductor substrate.