Subsections

• 6.4.1 Properties of the R-D Model
  • 6.4.1.0.1 Regime 1:
  • 6.4.1.0.2 Regime 2:
  • 6.4.1.0.3 Regime 3:
  • 6.4.1.0.4 Regime 4:
  • 6.4.1.0.5 Regime 5:
  
  
• 6.4.2 R-D Model vs. Fast Recovery
• 6.4.3 Dispersive Transport
• 6.4.4 Coupling to the Semiconductor Device Equations

6.4 Reaction-Diffusion Model

Figure 6.10: Schematic representation of the reaction-diffusion (R-D) model. Electrically inactive Si-H bonds at the interface are broken and the hydrogen diffuses into the dielectric leaving behind an electrically active interface trap . Here, H$_2$ diffusion is assumed.

The reaction-diffusion (R-D) model is pioneering for the description of NBTI. It was first proposed by Jeppson and Svensson in 1977 [103] and is capable of reproducing the time evolution of device degradation due to negative bias temperature stress for a wide range of measurements.

The model describes the device degradation as a combination of two effects. In the first place, a field-dependent electrochemical reaction at the interface. Electrically inactive, passivated silicon dangling bonds (Section 3.1), Si-H, are broken. Here an electrically active interface state and a mobile, hydrogen related species are formed,

$$\mathrm{Si}-\ensuremath{\textrm{H}} \rightleftharpoons \m... ...+ \textstyle \ensuremath{\textrm{X}}_\mathrm{interface}} .$$ (6.5)

In the second place, the model describes the transport of the hydrogen species away from the interface into the dielectric,

$$\ensuremath{\textrm{X}}_\mathrm{interface} \rightleftharpoons \ensuremath{\textrm{X}}_\mathrm{bulk} .$$ (6.6)

Also the reverse process is possible: transport of a diffusing hydrogen species back to the interface and re-passivation of a dangling bond.

Figure 6.10 gives a schematic illustration of the model. In the figure hydrogen molecules, , are assumed as the diffusing species. The process at the interface is modeled by a rate equation as

$$\frac{\partial \ensuremath{N_\textrm{it}}(t)}{\partial t} = ... ...\ensuremath{N_\textrm{X}}(0,t)^{1/a}}_{\mathrm{annealing}} ,$$ (6.7)

where is the interface-trap generation and the annealing rate. The symbol denotes the initial number of electrically inactive Si-H bonds and $\ensuremath{N_\textrm{X}}(0,t)$ is the surface concentration of the diffusing species. The value of $a$ gives the order of the reaction. In the original publication, neutral hydrogen, H$^0$, was proposed [103] which is obtained with $a=1$. For molecular hydrogen, , $a=2$. In this case the molecule is assumed to be formed in the vicinity of the interface

$$\mathrm{Si}-\ensuremath{\textrm{H}}+ h \rightleftharpoons \mathrm{Si^+ + \textstyle \frac{1}{2} H_2} .$$ (6.8)

The Si-H bond is broken and captures a hole, leading to a positively charged interface state and molecular hydrogen is formed.

The equilibrium of the forward and backward reaction is controlled by the hydrogen density at the interface $\ensuremath{N_\textrm{X}}(0,t)$. Thus, the transport mechanism of the hydrogen species away from the interface characterizes the degradation mechanism, controlling the device parameter shift. The original reaction-diffusion model describes the transport as a purely diffusive mechanism which is described by the diffusion equation

$$\frac{\partial \ensuremath{N_\textrm{X}}(x,t)}{\partial t} ... ...suremath{{\vec{\nabla}}}^2 \ensuremath{N_\textrm{X}}(x,t) .$$ (6.9)

Here, $D$ is the diffusivity of the hydrogen species in the dielectric. The influx of the newly created species has to be considered as

$$a \frac{\partial \ensuremath{N_\textrm{it}}}{\partial t} .$$ (6.10)

For each generated interface trap a hydrogen is released, thus

$$\ensuremath{N_\textrm{it}}= \int_x \ensuremath{N_\textrm{X}} \mathrm{d}x .$$ (6.11)

The R-D model assumes the interface states to be the only contribution to device parameter shift. But especially thick high-voltage oxides also show the generation of oxide charges due to hole trapping [86], leading to an additional parameter shift. Therefore it has been suggested [85] to separate the oxide charge contribution from the measurement results, before the R-D model can put into agreement to them. Suggested methods are the estimation of bulk-trap concentration for every stress voltage, temperature, and oxide thickness or trying to avoid the generation of bulk-traps by optimizing the stress conditions to that aspect.

6.4.1 Properties of the R-D Model

Figure 6.11: The five different regimes for the time exponent $n$, as obtained from the reaction-diffusion model.

For the stress phase the solution of the R-D model can be split up into five different regimes. They are distinguished by different time exponents $n$ (Section 6.2) for the degradation and are depicted in Figure 6.11.

6.4.1.0.1 Regime 1:

In the very early stage of the stress phase the amount of free hydrogen, both, at the interface and in the dielectric is very low. The amount of already broken Si-H bonds at the interface is close to zero. Thus, Equation 6.7 is solely limited by the forward reaction rate and transforms to

$$\frac{\partial \ensuremath{N_\textrm{it}}(t)}{\partial t} \approx \ensuremath {k_\textrm{f}}\ensuremath{N_0} ,$$ (6.12)

with the solution for the interface trap generation

$$\ensuremath{N_\textrm{it}}(t) \approx \ensuremath {k_\textrm{f}}\ensuremath{N_0}t ,$$ (6.13)

when assuming

$$\ensuremath{N_\textrm{it}}(0) = 0 ,$$ (6.14)

having a time dependence of $n=1$.

6.4.1.0.2 Regime 2:

After some time, when the amount of hydrogen at the interface is considerable, the forward reaction reaches a quasi-equilibrium with the backward reaction

$$\ensuremath {k_\textrm{f}}(\ensuremath{N_0}-\ensuremath{N_\t... ...trm{r}}\ensuremath{N_\textrm{it}}\ensuremath{N_\textrm{X}} ,$$ (6.15)

and, as is very large with respect to

$$\ensuremath {k_\textrm{f}}\ensuremath{N_0}\approx \ensuremat... ...trm{r}}\ensuremath{N_\textrm{it}}\ensuremath{N_\textrm{X}} .$$ (6.16)

The diffusion process has not removed a considerable amount of hydrogen from the interface yet, therefore the amount of interface traps equals the amount of hydrogen at the interface

$$\ensuremath{N_\textrm{it}}= \int_x \ensuremath{N_\textrm{X}}... ...x \approx \ensuremath{N_\textrm{X}}{}_\mathrm{,interface} .$$ (6.17)

We therefore get

$$\ensuremath{N_\textrm{it}}\approx \sqrt{\frac{\ensuremath {k_\textrm{f}}\ensuremath{N_0}}{\ensuremath{k_\textrm{r}}}} .$$ (6.18)

As this equation is not time dependent we can write

$$\ensuremath{N_\textrm{it}}\approx \sqrt{\frac{\ensuremath {k... ...xtrm{f}}\ensuremath{N_0}}{\ensuremath{k_\textrm{r}}}} t^0 .$$ (6.19)

with a resulting time dependence of $n=0$.

As long as the diffusion of hydrogen away from the interface has not reached a considerable magnitude, there is no further degradation of the interface.

6.4.1.0.3 Regime 3:

Regime three starts when the diffusion of hydrogen away from the interface sets in and acts as limiting factor for the degradation. In this phase the diffusion front has not reached the poly gate, therefore

$$\ensuremath{t_\textrm{ox}}> \sqrt{4 D t} .$$ (6.20)

An analytical solution has been found to be [103]

$$\ensuremath{N_\textrm{it}}\approx 1.16 \sqrt{\frac{\ensurema... ...suremath{N_0}}{\ensuremath{k_\textrm{r}}}}D^{1/4} t^{1/4} .$$ (6.21)

Numerically the solution is, depending on the group, somewhat larger [85] or smaller [104] then the analytical solution of $n=0.25$ as it is based on some simplifications.

In the reaction-diffusion model, this is the dominating regime in the typical lifetime of a MOSFET. It sets in after some seconds stress and, depending on the exact conditions, is dominating for several orders of magnitude in time, lasting up to several years.

6.4.1.0.4 Regime 4:

When the diffusion front reaches the poly gate contact the time exponent changes again. In the model it is assumed that the gate electrode acts as absorber for the diffusing species or, in other words, the diffusivity is significantly higher in the poly than in the dielectric. For this case a time exponent of $n=0.5$ is derived. Although a slight increase of the time exponent can be found in some measurement data for thin dielectrics, it is way below $0.5$.

6.4.1.0.5 Regime 5:

When, theoretically, all interface bonds are broken and

$$\ensuremath{N_\textrm{it}}\approx \ensuremath{N_0}= \mathrm{const} ,$$ (6.22)

no further degradation can occur in this model. Therefore the change in is zero and so is the time exponent $n$. As this saturation condition would only occur ever at extremely long stress times, or, at very high stress conditions which would lead to other degradation mechanisms as TDDB (Section 5.2), it has not yet been observed experimentally.

6.4.2 R-D Model vs. Fast Recovery

It has been shown that the measured degradation and also the extracted time exponent strongly depend on the time delay during measurement. Using fast measurement methods to spot the full degradation during NBT stress reveals a very low time exponent around $n=0.07-0.1$ (see Section 6.3.6). The reaction-diffusion model, in contrast, predicts a time exponent of $n=0.25$ for neutral hydrogen diffusion, . By assuming molecular hydrogen being the diffusing species a time exponent of $n=0.16$ can be achieved. Here, an additional reaction step at the interface generates molecular hydrogen before diffusion, $2\ensuremath{\textrm{H$^0$}} \rightarrow \ensuremath{\textrm{H$_2$}}$. Only by introducing higher order chemical reactions at the interface the time exponent could be further reduced to fit the model to state-of-the-art measurement results.

The fast recovery effect poses another inconsistency of the pure R-D model with measurement data [99]. Even for very long stress times of more than 1000 seconds the immediate recovery of in the first second after stress is more than 60% using the fast measurement set-up.

Figure 6.12: Hydrogen profiles in the dielectric for 1000 second stress (left) and after the recovery of 60 % of the degradation. It is difficult to argue that the diffusion of more than half of the hydrogen which took 1000 seconds to diffuse into the dielectric can diffuse back to the interface in one second [99].

The R-D model assumes diffusion limited degradation and also diffusion limited relaxation. When considering a stress time of 1000 seconds a certain amount of hydrogen related species diffuses into the dielectric as shown in Figure 6.12. The total amount of hydrogen in the dielectric must equal the number of interface traps (6.11). is in turn directly proportional to the shift of the threshold voltage . To argue a 60% recovery during the first second in the framework of the R-D model, 60% of the hydrogen must diffuse back to the interface and anneal the dangling silicon bonds within this second. This would result in a hydrogen profile in the dielectric, as seen on the right hand side of Figure 6.12, implying that the backward diffusion must be orders of magnitude faster than the forward diffusion [99].

6.4.3 Dispersive Transport

Figure 6.13: Schematic illustration of dispersive transport. Hydrogen is dissociated from the interface and transported into the via a diffusive mechanism. In the dielectric the mobile species sees traps of different energies leading to dispersive transport. The energy diagram on the right hand side shows a possible density-of-states (DOS) energy distribution of the traps.

Instead of using the standard diffusion equation [103,105] to describe hydrogen transport in the dielectric it has been shown [88,106,107,108,109,110,111,112] that correct modeling of transport in an amorphous materials must consider its dispersive [113] nature.

Figure 6.14: At small times the energy distribution of trapped carriers is similar to the density-of-states (DOS) as the trapping probability for each trap is the same. After time the peak of the energy distribution moves towards deeper states. The reason is the higher de-trapping probability for shallow traps.

Dispersion arises when the mobile species experiences different barrier heights at different positions in space. Figure 6.13 gives a schematic illustration where the dielectric contains hydrogen traps of different energy. The trapping probability at each trap is the same. But for de-trapping the deeper traps pose a higher barrier than the shallow ones. This implies that at the beginning of the trapping events the energy distribution of trapped carriers is proportional to the density-of-states (DOS). But after some time, as de-trapping preferably occurs for shallow traps, the peak of the energy distribution moves to deeper traps as illustrated in Figure 6.14. Thus, the equilibrium distribution is totally different from the DOS.

In the modeling approach, the species is separated into two distinct contributions. The conducting, , and trapped, , particles. The trapped particles are distributed in energy where the density at a trap energy-level is given as $\rho({\vec{x}},\ensuremath{E_\textrm{t}},t)$. The trapped particles do not contribute to the transport. To introduce dispersive transport into the reaction-diffusion model, (6.9) transforms to

$$\frac{\partial \ensuremath{N_\textrm{c}}({\vec{x}},t)}{\par... ...ec{\nabla}}}^2{\ensuremath{N_\textrm{c}}({\vec{x}},t)} .$$ (6.23)

At each trap energy-level a rate equation describes the dynamics between trapping and de-trapping as

$$\frac{\partial \rho(\ensuremath{E_\textrm{t}})}{\partial t} ... ...uremath{E_\textrm{t}}) \rho(\ensuremath{E_\textrm{t}}) .$$ (6.24)

Here, $c$ is the capture rate, $r(\ensuremath{E_\textrm{t}})$ the energy-dependent release rate, and $g(\ensuremath{E_\textrm{t}})$ is the trap DOS.

When considering the trap distribution in exponential form [113,92]

$$g(\ensuremath{E_\textrm{t}}) = \frac{\ensuremath{N_\textrm{... ...h{E_\textrm{t}}}{\ensuremath{\ensuremath{E}_0}} \Bigr) ,$$ (6.25)

and with introduction of the dispersion parameter

$$\alpha = \frac{\ensuremath{\textrm{k$_\textrm{B}$}}T}{\ensuremath{\ensuremath{E}_0}} ,$$ (6.26)

can, for neutral species, be expressed with a power-law as [114]:

$$\Delta N_\textrm{it}(t) = \Bigl(a \Bigl(\frac{\ensuremath {... ...)} (\ensuremath{\nu_\textrm{0}}t)^{(1-\alpha/2)/(1+a)} ,$$ (6.27)

with being the release rate coefficient. Here, only hydrogen in the conductive state can contribute to the reverse rate.

When assuming atomic hydrogen ($a=1$) the slope calculates from (6.27) as $n=1/2-\alpha/4$ while when assuming molecular hydrogen ($a=2$) results in $n = 1/3-\alpha/6$. In both cases the time exponent $n$ is increasing for an increasing dispersion parameter $\alpha$. So for an increasing number of deep traps the degradation increases [104,111].

For complex trap distribution functions no analytic formulation can be found. An example is a combination of shallow band-tail traps (the exponential distribution) and deep Gaussian traps describing deep traps (right hand side of Figure 6.13) [111]. For such a DOS only the numerical solution is possible as obtained with the numerical device simulator Minimos-NT.

6.4.4 Coupling to the Semiconductor Device Equations

As NBTI is strongly dependent on the stress conditions, the available models include at least the dependence on the electric field (the bias) and the temperature. But also the hole concentration at the interface might be of interest [85].

In the one-dimensional NBTI models the values of these quantities are accounted for in the model parameters, as and in the reaction-diffusion model for example. For each device, stress voltage, and temperature these parameters are extracted (Section 7.1.2) and are assumed constant for these stress conditions. But as degradation is continuing during stress, the underlying parameters, especially the electric field across the interface, change. So the electric field at the interface is decreasing when the NBTI induced degradation leads to positive charge build up at this interface. Therefore, it might not be appropriate to consider the forward reaction rate as a constant.

Another point is the two-, or three-dimensional investigation of a device structure. Here the electric field and the hole concentration, and therefore also the magnitude of degradation, can change drastically across the channel of the device when the stress conditions are not uniform (Section 6.3.3). Thus, the model parameters are not uniformly distributed at the interface and a one-dimensional consideration of the model might not be accurate enough [115].

To solve these issues it is important to include the NBTI models into a numerical device simulation. The distributed quantities, obtained from the semiconductor device equations (Section 2.1) as the electric field, hole and electron concentration can be directly used in the degradation model. The resulting charge densities from the NBTI model can be included in the device equation leading to a fully self-consistent coupling of those two sets of equations.

The reaction-diffusion model might be linked to the semiconductor device equations with the forward reaction rate as follows [85]

$$\ensuremath {k_\textrm{f}}= \ensuremath{k_\textrm{f,0}} \f... ...emath{E_\textrm{ox}}}{\ensuremath{E_\textrm{ref}}}\right) .$$ (6.28)

Where and are the hole concentration and the electric field at the interface, and are reference values. By using solution variables of the semiconductor equations the NBTI model can be applied to arbitrary device geometries.