Evaluation of stability and robustness of poly-Si resistors with different dopant concentrations

Typical I–V curves for poly-Si resistors (250 °C) are shown in Fig. 2(a). The HR sample has a sheet resistance of approximately 2k Ω sq⁻¹, which is four times that of the LR sample. The temperature (T) dependence of resistance shift is shown in Fig. 2(b). The TCR of the HR and the LR are found to be negative and positive, respectively. Different resistivity mechanisms will lead to different types of temperature dependence for poly-Si resistors. The resistivity (ρ) of the poly-Si resistor can consist of grain resistivity (ρ_grain) and GB resistivity (ρ_gb), ²⁷⁾ which are expressed as follows:

$$\begin{eqnarray}&&{\rho }_{\mathrm{grain}}\sim {T}^{m},\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\rho }_{\mathrm{gb}}\sim {e}^{Eg/kT},\end{eqnarray} \tag{ 2 }$$

where Eg is the GB barrier energy and m is a constant.

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The resistance of LR has a linear T dependence, whereas that of HR has a 1/kT dependence, as shown in Fig. 3. The temperature dependences of poly-Si resistors indicate that LR and HR exhibit grain conduction and GB conduction, respectively. The exponential dependency of 1/T in Eq. (2) can be explained on the basis of the inter-grain resistance with thermionic-emission conduction for a barrier potential between GBs. For HR samples with a low dopant concentration, the carriers have a large barrier potential between the GBs owing to their low Fermi energy level. Such a large barrier potential dominates the conduction characteristics and is the cause of the inter-grain conduction characteristic of HR. ^15,28) The inter-grain resistance has a negative TCR, which can be shown by taking the derivative of Eq. (2). The GB barrier height of the HR was evaluated to be approximately 0.01 eV from the data in Fig. 3, which is similar to the reported barrier height of doped poly-Si film in the literature. ¹⁶⁾ For LR samples with a higher carrier density and Fermi energy level, the GB barrier is insignificant, whereas the intra-grain carrier mobility plays an important role. Intra-grain conduction is impacted by the phonon scattering of carriers, which is more prominent as the temperature increases, resulting in a positive TCR and the power-law dependence of resistivity on temperature. ²⁸⁾

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The bias temperature stress test can be used to monitor the stability of interconnects or resistor structures. ²⁹⁾ Resistance stability tests of LR and HR were performed using low current (Low_I ∼ 2 mA) and high current (High_I ∼ 8 mA) stress levels at 150 °C, as shown in Fig. 4. Both HR and LR samples showed high stability with minimal change in resistance under Low_I stress. LR had a very small Rshift of <0.02%. The Rshift of HR increased to 0.2% and then saturated. Under High_I stress, LR still exhibited high stability with a minimal change of 0.1%, whereas HR displayed a power-law increment of 4% at the end of the test.

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The resistance shift (%) observed at different temperature steps during the stability test is shown in Fig. 5. The stress currents are turned on in step (2). When the stress current is turned off in step (3) and when the temperature returns to the initial value of 25 °C, the resistances of HR and LR under Low_I stress recover to their initial values at this temperature. Only HR under High_I stress has a permanent resistance shift after the stress current turned off in step (4), as shown in Fig. 5(b). Based on these stability stress results, a new resistance stability model that can explain the time dependences under low and high stress levels needed to be developed.

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A critical stress current to make a permanent resistance increment is observed in the interconnect metal structure due to electromigration. ^30,31) Electromigration is defined as the migration of atoms of conduction line due to the momentum exchange owing to collisions with conduction electrons. ³²⁾ A void or hillock of the conduction line may occur due to the nonuniformities of atom migration induced by defects in the lattice or the grain structure. ^33–36) The PMOS FET results in permanent degradation and induces recovery effects under NBTI stress. ³⁷⁾ NBTI degradation is primarily due to carrier trap generation and its recovery effect can be explained as the de-trapping of trapped carriers in the absence of stress. ^38,39) The carrier-trapping model was proposed to explain the electrical transport properties of poly-Si at different dopant concentrations. ⁸⁾

Reviewing the aforementioned degradation models, we developed a novel carrier-trapping model for the poly-Si resistance shift evolution. Under stress, the carriers gain energy to overcome the barrier and are confined to the trap sites. However, new traps are also generated as a result of stress. The available trap density, Dt, which is time (t) dependent, can be defined as follows:

$$\begin{eqnarray}&&Dt\left(t\right)=Dt\left(0\right)+Dts\left(t\right)-{N}_{t}\left(t\right),\end{eqnarray} \tag{ 3 }$$

where Dt(0) is the initial available trap density and Dts is the stress induced new trap density, and N_t is the trap site density occupied by the carrier.

The trap occupation rate is assumed to be proportional to the available trap density Dt, with rate constant ra

$$\begin{eqnarray}&&\displaystyle \frac{d{N}_{t}}{dt}=ra\cdot Dt.\end{eqnarray} \tag{ 4 }$$

If the stress condition is low and Dts can be ignored, N_t can be derived as an exponential function of t.

$$\begin{eqnarray}&&Dt\left(t\right)=Dt\left(0\right)-{N}_{t}\left(t\right),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\displaystyle \frac{dDt}{dt}=-\displaystyle \frac{d{N}_{t}}{dt}=-ra\cdot Dt,\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&Dt\left(t\right)=Dt\left(0\right)\cdot \exp \left(-ra\cdot t\right),\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{N}_{t}\left(t\right)=Dt\left(0\right)\cdot \left(1-\exp \left(-ra\cdot t\right)\right).\end{eqnarray} \tag{ 8 }$$

Conductivity σ is assumed to be proportional to the carrier density Na(t), and σ can be derived as

$$\begin{eqnarray}&&\begin{array}{l}\sigma \sim Na\left(t\right)=Na\left(0\right)-{N}_{t}\left(t\right)=N{a}^{{\prime} \left(0\right)}\\ \,+\,Dt\left(0\right)\,\cdot \,\exp \left(-ra\,\cdot \,t\right),\end{array}\end{eqnarray} \tag{ 9 }$$

where Na'(0) ≡ Na(0)−Dt(0).

Given that resistance R is proportional to the inverse of the conductivity and Rshift under evaluation is $\ll$ 1, we suppose that Na(t) $\gg$ Dt and Na'(0) $\gg$ Dt(0), and derive Rshift(t) as follows:

$$\begin{eqnarray}&&R\sim 1/\sigma \sim {\left(\left(Na^{\prime} \left(0\right)+Dt\left(0\right)\cdot \exp \left(-ra\cdot t\right)\right)\right.}^{-1},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\approx \,N{a}^{{\prime} }\left(0\right)-Dt\left(0\right)\,\cdot \,\exp \left(-ra\,\cdot \,t\right),\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{R}_{\mathrm{shift}}\left(t\right)\equiv \frac{R\left(t\right)-R\left(0\right)}{R\left(0\right)}\approx \frac{Dt\left(0\right)}{Na^{\prime} \left(0\right)}\cdot \left(1-\exp \left(-ra\cdot t\right)\right).\end{eqnarray} \tag{ 12 }$$

The physical meaning of Eq. (12) is that the resistance shift increases exponentially and has a saturation point given by the ratio of initially available trap density to the total carrier density, because the number of available trap sites is limited. This ratio was 0.2% for HR, as shown in Fig. 4(a).

Under high-stress conditions, the new trap density is assumed to be larger than the occupied trap density. Assuming that Dts increases as a power law with time under such conditions, N_t of Eq. (4) will also exhibit a power-law time dependence

$$\begin{eqnarray}&&Dt\approx Dts\sim {t}^{n-1},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{N}_{t}\left(t\right)=ra\cdot \displaystyle \int Dt\sim {t}^{n}\equiv rs\cdot {t}^{n},\end{eqnarray} \tag{ 14 }$$

where rate constant rs and exponent n are related to the test structure and stress conditions. Rshift(t) under a high-stress condition is then derived as follows:

$$\begin{eqnarray}&&\sigma \sim Na\left(t\right)=Na\left(0\right)-{N}_{t}=Na\left(0\right)-rs\,\cdot \,{t}^{n},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&R\left(t\right)\sim 1/\sigma \approx {\left(Na\left(0\right)-rs\,\cdot \,{t}^{n}\right)}^{-1}\approx Na\left(0\right)+rs\,\cdot \,{t}^{n},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{R}_{\mathrm{shift}}=rs\cdot {t}^{n}/Na\left(0\right),\end{eqnarray} \tag{ 17 }$$

Rshift in Eq. (17) increases as a power law with time and is proportional to the ratio of the occupied trap density to the initial carrier density.

LR exhibits higher stability under the same stress current levels, as shown in Fig. 4. This implies that the generation of new LR carrier traps requires a higher stress current. The resistance stability models of Eqs. (12) and (17) predict that resistance will increase with different time dependences according to the different stress conditions. These models successfully explain the data of the poly-Si Rshift of HR and fit them well, as shown in Fig. 6. For the medium-stress condition, the time dependence of Rshift is a transition function between Eqs. (12) and (17).

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Stability tests with higher constant stress currents at higher temperatures were performed to clarify the current and temperature effects on ploy-Si resistance shifts, as shown in Figs. 7 and 8. The test oven temperatures were 250 °C–300 °C, and the sample temperature was estimated using the resistance and temperature relation shown in Fig. 2(b). The stress currents of HR and LR were approximately about 1.5 and 1.8 times the values of High_I respectively. Under these high-stress conditions, more trap sites may be generated, and both HR and LR exhibited power-law Rshift increments, as expected.

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In general, stress time when catastrophic failure occurs is defined as the time to failure (TTF) of a metal line under electromigration stress. ^40,41) A stability lifetime or TTF may be defined as the time at which Rshift reaches a specific criterion for more reliable resistance control. Based on this definition and the results of Figs. 7 and 8, a stability TTF model can be developed to predict other TTFs under different stress conditions. To demonstrate the TTF model, we adopted a 4% resistance change as the TTF criterion for the poly-Si resistors, which is the maximum resistance shift in Fig. 4(b). The TTF results for different stress currents and temperatures show that Ln(TTF) has a linear dependence on 1/T and stress current, as shown in Fig. 9. Based on these TTF results, we utilized the free energy of activation approach for the stress-induced Rshift model. ⁴²⁾ According to this approach, a reaction rate Kr is related to the free energy of activation G which is a sensitive function of stress parameters (electrical current I in this study) as follows:

$$\begin{eqnarray}&&Kr\sim \exp \left(\displaystyle \frac{-G}{kT}\right),\end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray}&&G={E}_{a}-n\,\cdot \,kT\,\cdot \,I,\end{eqnarray} \tag{ 19 }$$

where E_a is the activation energy and n is the acceleration factor of the stress current. ⁴³⁾ Given that the TTF is inversely related to the reaction rate, we proposed the time to failure of the poly-Si resistor as follows:

$$\begin{eqnarray}&&\mathrm{TTF}\sim 1/Kr=A\,\cdot \,\exp \left(-n\,\cdot \,I\right)\,\cdot \,\exp \left(\frac{{E}_{a}}{kT}\right).\end{eqnarray} \tag{ 20 }$$

Based on the free energy model of Eq. (20), Ln(TTF) has linear a dependence on I and 1/T, as demonstrated in Fig. 9. The E_a of LR and HR are found to be 1.9 and 0.78 eV, respectively. A high-resistivity poly-Si resistor was reported to have E_a of 0.68 eV, consistent with that of HR found here. ²¹⁾ The grain or lattice migration energy is larger than the GB migration energy. ^44–47) The disorder at GBs in polycrystalline silicon allows for high-diffusivity paths along which dopant atoms can easily move. If the carrier trap site generation is related to the migration of atoms or dopants, an HR with GB conduction is expected to have a lower E_a than that of LR with grain conduction. This expectation is consistent with the E_a results obtained using the proposed free energy model. The n values of LR and HR were 3.0 and 11.4 cm² MA⁻¹, respectively. A higher n of HR implies a more severe TTF decrement when the operation current increases for HR poly-Si resistors. We estimated the TTF of the HR under the lower stress condition of Fig. 4(a) using Eq. (20), as shown in Fig. 10. This result shows that the free energy TTF model can serve as a reliable stability model for evaluating the long-term performance of poly-Si resistors.

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The TLP test can be used to assess the robustness of the resistor under a pulse current at high power. TLP current pulses with different pulse widths were applied to the HR and LR poly-Si resistors in this study. The critical failure current, I_fail, was defined as the TLP current that caused the breakdown of the resistor. I_fail exhibited different TLP time width, tw, dependence as shown in Fig. 11(a) with I_fail ² as the vertical axis and 1/tw as the horizontal axis. I_fail decreased as the pulse time width increases, and started to saturate when tw was larger than 1 μs.

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The correlation between I_fail and tw involves the breakdown energy (Eb) and conduction energy during the TLP test. When breakdown happens during a TLP pulse step, the total TLP input energy, E_tlp, includes Eb of resistor and the thermal conduction energy as follows:

$$\begin{eqnarray}&&{E}_{\mathrm{tlp}}={{I}_{\mathrm{fail}}}^{2}R\cdot tw=Eb+kc\cdot {\rm{\Delta }}T\cdot tw,\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&{{I}_{\mathrm{fail}}}^{2}=Eb/\left(R\cdot tw\right)+kc\cdot {\rm{\Delta }}T/R,\end{eqnarray} \tag{ 22 }$$

where ΔT is the temperature difference between the resistor and environment and kc is the thermal conductance.

When tw is smaller than the thermal time constant of the resistor, the heat energy dissipated from the resistor to the environment by conduction is negligible, and the TLP breakdown process can be regarded as adiabatic. Under such conditions, Eq. (22) can be simplified as follows:

$$\begin{eqnarray}&&{{I}_{\mathrm{fail}}}^{2}=Eb/\left(R\cdot tw\right).\end{eqnarray} \tag{ 23 }$$

Under the adiabatic conditions of Eq. (23), I_fail ² is proportional to 1/tw. The linear fitting lines of I_fail ² versus 1/tw (solid lines) are shown in Fig. 11(a). The fitting results are very good in the range of small tw values.

When tw increases and exceeds the thermal time constant of the resistor, I_fail first decreases and eventually saturates to the square root of kc·∆T/R [Fig. 11(a)], according to Eq. (22).

Given that the resistance of LR is lower than that of HR, I_fail of LR will be larger than that of HR if the Eb and kc of these two resistors are similar, according to Eq. (22). This conclusion is validated by the I_fail results for HR and LR, as shown in Fig. 11(a). The TLP input energy for breakdown increases as tw increases according to Eq. (21), which is confirmed by the TLP test results in Fig. 11(b). Given that the TLP energy approaches the value of Eb as tw decreases, the TLP energies in the small tw range in Fig. 11(b) also confirm that the Eb values of LR and HR are similar.

Based on the proposed TLP breakdown model, the specification of the pulse current for the poly-Si resistor can be accurately defined with different pulse time widths. Because I_fail of LR is higher than that of HR, the LR poly-Si resistor can sustain a larger pulse current than the HR poly-Si resistor during electrostatic discharge or electrical overstress events. SEM images of the silicide-block LR poly-Si resistors after low current stability stress tests did not depict significant failure in our previous study, as shown in Fig. 12(a). ²⁰⁾ When the stress current was larger than the breakdown current, the observed failure sites of the silicide poly-Si resistor were usually near the contact area, where the current crowding and heat generation were severe, as shown in Fig. 12(b).

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