Since in a semiconductor the local carrier density depends on the position of the Fermi level, and hence on the local electrostatic potential, even a simple parallel plate capacitor structure with one plate replaced by a bulk semiconductor (MOS) exhibits strongly non-linear behavior [10, 52]. The charge distribution of minority and majority carriers in such a MOS structure varies as a function of the applied gate voltage. Depending on this charge distribution, different operating regimes can be identified. When going from accumulation towards inversion, the depletion region inside the semiconductor is formed first. Here, the majority carriers are driven away from the interface between the oxide and the semiconductor. The only remaining charges within this depletion region are fixed ionized acceptors (p-type) or donors (n-type), which build up a depletion charge. In combination with the insulator this results in a decrease of the total capacitance.
Charge
Strong Inversion Weak Inversion Depletion Accumulation
Figure 2.8: Left: The frequency dependence of the MOS-structure capacitance. While for low frequencies the minority carriers in the inversion layer are fast enough to contribute to the signal, the increase in charge can only be compensated by an increase of the depletion width (majority carriers) at higher frequencies.
Right: AC(V)-curve of a pMOSFET taken at 1 MHz and normalized to the capacitance of an ideal capacitor of the same size. To be able to distinguish between the different operating regimes more easily, the graph is also shifted by the workfunction difference. The flatband capacitanceCfb is then obtained atVG= 0 V.
With the onset of inversion the minority carriers exceed the majority carriers at the interface and create the inversion layer. At that point the depletion region with its ionized impurities virtually stops to increase in width and any increase in gate charge is only balanced by an increase of the inversion charge. Whether the minority carriers are able to follow the signal or not influences the capacitance in this inversion regime, i.e. the contribution of the inversion layer charge to the total capacitance depends on the frequency. Only at low frequencies the recombination-generation rates of the minority carriers can keep up with small signal variations leading to a charge exchange with the inversion layer. With this additional inversion charge the capacitance signal increases as the depletion width remains constant [10]. At high frequencies on the other hand only the majority carrier response is measured. Hence the incremental charge in deeper inversion is put at the edge of the depletion region, while the inversion regime is not altered. This causes the capacitance to remain constant when going from depletion into inversion. Both cases (low and high frequency) are depicted in Fig. 2.8 (left).
When adding source and drain regions to form a MOSFET, minority carriers are provided independently of the frequency. Therefore low-frequency C(V)-characteristics of MOSCAPs and C(V)-characteristics of MOSFETs look alike. Exemplarily, a C(V)-characteristics of a pMOSFET measured at 1 MHz is depicted in Fig. 2.8 (right). For a better understanding the curve shown is shifted by the flatband voltage5 and the different operation regimes of the MOSFET are marked.
Based on the above mentioned findings the C(V)-characteristic provides valuable information of the semiconductor structure and its interface. For example, present interface states stretch the C(V)-characteristic along the VG-axis, because additional charge is necessary to fill these traps.
5The flatband voltage results from the workfunction difference of the materials used.
Chapter 2. Measurement Methods 17
Oxide charges on the other hand are independent of the appliedVG and cause a mere parallel shift of theC(V)-characteristic towards higher or lowerVG[10]. Furthermore, with the knowledge of the capacitance as a function of VG, the oxide electric field can be calculated. This is necessary when the degradation caused by NBTI is compared with that caused by PBTI for the same device type, e.g. for a pMOS. Due to the nonzero flatband voltage it is not possible to apply just the opposite VG to achieve the opposite electric field. Moreover, the different behavior of the capacitance during accumulation and inversion yields an asymmetric C(V)-characteristic. This as well influences the value of the proper (exact opposite) field. An application of C(V)-characteristics to obtain the required stress voltage VG,str for a certain NBTI stress and its corresponding PBTI stress is given in Chapter 7.
Chapter 3
Previous Modeling Attempts
In the 1960’s, the investigations of the Si-SiO2 interface revealed a close-coupling of the increase of surface traps sitting at the interface and a phenomenon which will be later on known as the bias temperture instability. Both effects were known to cause a negative shift of the threshold voltage [3, 4, 53]. Though this phenonmenon was already known not to cause real device failure as for example time dependent dielectric breakdown (TDDB) [35, 54], the creeping shift of VTH alerted the industry and the scientific community to develop a model which is capable of describing the mechanisms behind BTI. In order to judge such a model as functional, clear definitions of its applicability but also potential limits have to be listed. The following review summarizes the existing modeling efforts, including their advantages and disadvantages.
3.1 Reaction Diffusion Model
As the fabrication of a semiconductor device today as back then consists of more and more single layer depositions, often being followed by an annealing step, hydrogen as “the” passivation agent was suggested to play a key role in the first modeling attempt dating back to 1977 [13]. In the so-called reaction-diffusion model Jeppson et al. assumed the breaking of hydrogen-bonds at the interface via a thermally and field-activated process under stress. This reaction-limited stress phase is schematically depicted in Fig. 3.1 (taken from [55]) and can be described with the kinetic rate equation at the interface after [31, 56–58] as
∂Nit
∂t =kf(N0−Nit)−krNitXit1/a (3.1) where N0 denotes the total amount of interface states, Nit the fraction of dangling bonds thereof (not yet passivated), and Xit the interfacial hydrogen concentration. The rates kf and kr describe the forward (depassivation) and reverse (passivation) process with a kinetic exponentaconsidering the “size” of the diffusing species.
3.1.1 Stress Phase
As long as the bond-breaking dominates the rate equation, the reverse rate is negligible because there is simply not enough free hydrogen. Thus the degradation within this initial stress phase is only proportional to the stress time tstr.
19
Equilibrium
Stress Reaction−
Limited
Oxide Bulk Hydrogen Diffusion Interface
Nit∼0
Nit∼tstr
Si–H⇋Si•+ H
Relaxation Stress
Balance
H Diffusion
H Back Diffusion Limited
Diffusion−
Diffusion−
Limited Nit∼t1/4str
Nit∼ 1
1+
rtrel
tstr
Figure 3.1: FromToptoBottom: Schematic view of the atomic hydrogen reaction-diffusion model during stress and relaxation. During the short initial phase, the interface region enters equilibrium. When in equilibrium, the degradation is dominated by the diffusion of hydrogen. As the stress is removed, the hydrogen diffuses back to the interface.
Once the interface reaction reaches equilibrium, the previously released hydrogen species diffuse towards the oxide. Under the assumption of a well passivated interface with only a few initial dangling bondsNit,0 and atomic hydrogen H0 as diffusion species the diffusion-limited stress regime can be approximated by a power-law of the formt1/4str. The mathematics behind this diffusion-limited process can be found in Appendix C.
However, not all measurement results are consistent with the predictions of the reaction-diffusion theory using atomic hydrogen. Quite to the contrary, the extracted exponents were found to depend on the measurement delay time; the exponents of 1/4 were obtained for measurement data with large delay time only [7,59]. Shorter delay times on the other hand yielded exponents of around 1/6.
Chakravarthiet al.now interpreted this weaker time dependence by introducing instant dimerization of the released atomic hydrogen at the interface via H0 + H0 ⇋H2 and subsequent diffusion of the created H2[60]. This theoretical assumption yields a smaller exponent of approximately 1/6 because of the larger kinetic exponent (a= 2 for H2 instead of a= 1 for H0, cf. Appendix C). When per-forming even faster measurements with delay times around a micro-second, e.g. OTF-measurements
Chapter 3. Previous Modeling Attempts 21
done by Reisinger et al., exponents of 0.12 were obtained [12] which does not correspond to the stress behavior predicted by the reaction-diffusion theory.
3.1.2 Back Diffusion of Hydrogen during Recovery
The phase after switching off a device after a certain stress time or between switching on and off is called recovery or relaxation. During this recovery phase the degraded parameters start to revert to their initial values, but within times that depend on the prior stressing conditions. Now the question arises how the interplay between BTI stress and recovery during the operation of a MOSFET does affect the reliability of the device. This is important because only when charactising and modeling both stress and recovery a realistic lifetime extrapolation of the device is possible.
Within the RD theory the recovery is explained via back diffusion of hydrogen. Once the stress is removed, the quasi-equilibrium at the interface causes all interfacial hydrogen to immediately passivate the dangling bonds. By the time all interfacial hydrogen is bound and no more hydrogen is available at the interface, hydrogen located deeper in the oxide is assumed to diffuse back and control the entire recovery as the limiting process. During recovery the back diffusion can be approximated by the empirical expression
Nit∼ 1 1 +
rtrel tstr