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Modeling of Defect Related Reliability Phenomena
in SiC Power-MOSFETs

4.5 Efficient Framework for MOS Gate Leakage Currents

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Figure 4.7: Different gate leakage current mechanisms can be observed in a MOS gate stack. Fowler-Nordheim and direct tunneling currents (green) are the dominating leakage mechanism in thin and mostly defect free di- electrics. In more defective oxides, trap assisted tunneling over multiple (black) or single defects (blue) can be measured in addition to transient charge trapping currents (taken from [CSJ2]).

As stated in Chapter 1, several leakage current mechanisms through the dielectric in a gate stack are possible. In Figure 4.7, the most common tunneling mechanisms are schematically shown, with direct and Fowler-Nordheim Tunneling currents through an energetic barrier and trap assisted currents via one or multiple defects. In this section, a standard model for calculating currents through energetic barriers together with a derivation for the charge hopping mechanism following [225] based on a generalized Ramo-Shockley theorem [226] will be re-framed to be applicable for a MOS gate stack.

4.5.1 Tsu-Esaki Model

The tunneling current through an energetic barrier that separates two electrodes can be computed with the so called Tsu-Esaki formalism  [227], in which the current density at the semiconductor channel for electrons and holes is expressed as

(4.47–4.48) \{begin}{align} J_{\mathrm {TE},e} &= \frac {4\pi m_e q_0}{h^3} \int _{E_\text {CB}}^{\infty } \vartheta _\mathrm {WKB}\left (E\right ) N_e\left (E\right ) \mathrm {d}E \\ J_{\mathrm {TE},h} &= \frac {4\pi m_h q_0}{h^3} \int _{-\infty }^{E_\text {VB}} \vartheta _\mathrm {WKB}\left (E\right ) N_h\left (E\right ) \mathrm {d}E. \label {equ:tsu_esaki_currdens} \{end}{align}

with the effective electron density of states (hole) mass \( m_{e\left (h\right )} \) in the semiconductor in the plane parallel to the interface [228], the elementary charge \( q_0 \), the Planck constant \( h \), a tunneling probability \( \vartheta \) and the so-called supply function \( N \). Assuming Fermi-Dirac statistics at both contacts, the supply function for electron tunneling is given by

(4.49) \{begin}{align} N_e \left (E\right ) = k_\text {B}T \, \text {ln} \left ( \frac {1+\text {exp}\left (-\frac {E-E_\text {F1}}{k_\text {B}T}\right )}{1+\text {exp}\left (-\frac {E-E_\text {F2}}{k_\text {B}T}\right )} \right ). \label {equ:supply_func} \{end}{align}

In order to efficiently calculate the tunneling probability, the WKB approximation can be applied for thin and energetically high barriers as [218]

(4.50) \{begin}{align} \vartheta _\mathrm {WKB}\left (E\right ) = \text {exp} \left ( -\frac {4\pi }{h} \int _{x_1}^{x_2} \sqrt {2m_{e,\mathrm {diel}}\left ( W\left ( x \right ) - E \right )} \dd x \right ) \label {equ:WKB_full} \{end}{align}

with \( m_{e,\mathrm {diel}} \) denoting the effective electron mass in the dielectric.

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Figure 4.8: A triangular energetic barrier with the height \( q_0 \phi \), as typically observed at high field strengths \( F_\mathrm {ox} \) and thicker oxides, eventually leads to FN-like tunneling (left), with the electron tunneling to the oxide conduction band. A trapezoidal barrier, typically seen for thin oxides at medium \( F_\mathrm {ox} \), leads to direct tunneling (DT), where the electron has to tunnel through the oxide layer (right).

Thereby, the integral is carried out from position \( x_1 \) to \( x_2 \) over the energetic barrier with shape \( W(x) \). In the case of a triangular barrier, as shown in Figure 4.8 (left), (4.50) evaluates to the commonly known FN formula for electron tunneling [229]

(4.51) \{begin}{align} \vartheta _\mathrm {WKB}\left (E\right ) = \text {exp} \left ( -\frac {4\sqrt {2m_\text {e}}}{3\hbar q_0 F_\mathrm {ox}} \left ( q_0 \phi - E \right )^{\frac {3}{2}} \right ) \label {equ:WKB_FN} \{end}{align}

with the potential barrier \( \phi \), the electron energy \( E \) and the electrical oxide field strength \( F_\mathrm {ox} = \left ( V_\mathrm {G} - \phi _\mathrm {s} \right ) / t_\mathrm {ox} \). Also, for a trapezoidal barrier (Direct Tunneling (DT)) as shown in Figure 4.8 (right) with the barrier heights \( \phi _2 > \phi _1 \) an analytic expression evaluates to

(4.52) \{begin}{align} \vartheta _\mathrm {WKB}\left (E\right ) = \text {exp} \left ( -\frac {4\sqrt {2m_\text {e}}}{3\hbar q_0 F_\mathrm {ox}} \left ( \left ( q_0 \phi _2 - E \right )^{\frac {3}{2}} - \left ( q_0 \phi _1 - E \right )^{\frac {3}{2}} \right ) \right ). \label {equ:WKB_DT} \{end}{align}

From (4.48) and (4.50) it can be seen that the material parameters, i.e. the energy barrier and effective tunnel masses, within the exponent in the WKB factor strongly influence the current density in the inversion and accumulation regime, in which the supply function (4.49) is non-zero. Together with the exact knowledge of the oxide thickness \( t_\mathrm {ox} \) and the dielectric constant \( \varepsilon \), these parameters determine the accuracy of the Tsu-Esaki computation.

Upon careful parameter calibration, the capability of the model to calculate accurate tunneling currents through defect free dielectrics has been demonstrated. By considering this best case scenario, the suitability of dielectrics for new material combinations has been demonstrated with our implementation in Comphy for a large number of compounds considered as insulators for two-dimensional channel materials in [CSJ7]. It was thereby found that hexagonal boron-nitride (hBN), which was often proposed as dielectric for 2D materials, does not meet the requirements of low gate leakage current for pMOS fabrication even in the defect-free case [CSJ7]. As for real devices the leakage current is eventually further enhanced due to trap-assisted tunneling, the next section outlines the calculation of such a hopping current.

4.5.2 Charge Hopping Model

As for a charge hopping current the local current density does not fulfill the continuity condition, i.e. charges “vanish" at one defect site and “pop up" at another instantaneously. In this case, the current at the gate contact can be written as [225]

(4.53) \{begin}{align} I_{\mathrm {G}} = - \int _{\partial {D_\mathrm {G}}} \bigg ( \vec {J} + \pdv {\vec {D}}{t}\bigg ) \cdot \dd \vec {A} \label {equ:gate_current_int} \{end}{align}

with the integral taken over the device surface at the gate contact area \( \partial {D_\mathrm {G}} \). By selecting a test-function \( h_{i\mathrm {=G}} \) as the solution of the Laplace equation at the gate contact and \( h_i = 1 \) for all other contacts and by using the divergence theorem, the integral over the whole device volume reads [225]

(4.54) \{begin}{align} I_{\mathrm {G}} = - \int _{D} \div { \bigg [ h_i \bigg ( \vec {J} + \pdv {\vec {D}}{t}\bigg ) \bigg ]}\dd V = - \int _{D} \underbrace {\grad {h_i} \cdot \bigg ( \vec {J} + \pdv {\vec {D}}{t}\bigg )}_{= 0} + h_i \div { \bigg ( \vec {J} + \pdv {\vec {D}}{t}\bigg )} \dd V. \label {equ:gate_current_int_vol} \{end}{align}

The left term in the right hand side integral thereby vanishes due to the choice of \( h \) and a conduction current of \( \vec {J} = \vec {0} \) at the contact. Due to the local violation of the continuity condition in the Pauli-Master equation [230], the current is not divergence free and is given on a one-dimensional grid between points \( i \) and \( j \) as

(4.55) \{begin}{align} \div {\bigg (\vec {J} + \pdv {\vec {D}}{t}\bigg )} = q_\mathrm {0} k_{ji} \left ( 1 -f_j \right ) f_i \label {equ:div_current} \{end}{align}

by using (4.10) with \( f_{i/j} \) being the occupations of the two trapping sites. For a bias difference of \( V_\mathrm {G} - \phi _\mathrm {s} \) between gate and channel interface, the test function evaluates to \( h_i = \frac {x_i-x_j}{t_\mathrm {ox}} \) with the oxide thickness \( t_\mathrm {ox} \). Inserting (4.55) into (4.54) the current at the gate contact evaluates to the generalized Shockley-Ramo theorem [231, 226], which yields the displacement currents observed at the gate due to each charge movement within \( \mathcal {D} \), to

(4.56) \{begin}{multline} I_{\mathrm {G,TAT}} = \underbrace {C_\mathrm {ox} \derivative {V_\mathrm {G}}{t}}_{\text {displacement \linebreak current}} + \underbrace {q_0 \sum _i^N k_{\mathrm {e},i,\mathrm {gate}} f_i - k_{\mathrm {c},i,\mathrm {gate}} \left (1-f_i\right )}_{\text {single-TAT current}} \\ + \underbrace {q_0 \sum _{i}^{N} \sum _{j\neq i}^{N} k_{\mathrm {e},ij} f_i \left ( 1 - f_j \right ) \frac {x_{i}-x_{j}}{t_\mathrm {ox}}}_{\text {multi-TAT current}} \\ + \underbrace {q_0 \sum _i^N \left [ k_{\mathrm {c},i,\mathrm {channel}} \left (1 - f_i\right ) - k_{\mathrm {e},i,\mathrm {channel}} f_i \right ] \left ( 1 - \frac {x_i}{t_\mathrm {ox}} \right )}_{\text {charge trapping current}} \label {equ:TAT_cur} \{end}{multline}

with the oxide capacitance \( C_\mathrm {ox} = \varepsilon _\mathrm {ox} W L / t_\mathrm {ox} \). The rates in (4.56) consist of both reservoir to defect interactions for the TAT current and charge trapping current contributions and defect to defect rates for the multi-TAT current contribution. This charge hopping current model has been implemented in Comphy, as will be discussed in the next section.