Random-telegraph-noise by resonant tunnelling at low temperatures

We have found a systematic way to identify the bias conditions to observe the Random-Telegraph-Noise (RTN) in advanced Metal-Oxide-Semiconductor Field-Effect-Transistors (MOSFETs). We measured a p-type MOSFET at 2K, and found narrow bias conditions to observe the RTN presumably caused by charge trapping and de-trapping, which were only observed at low temperatures. It will pave the way to address the nature of a trap, which will be useful to understand the mechanism of RTN to secure the reliability.

reported in our studies on MOSFETs at low temperatures 29 . The regime where current peaks can be observed formed a border at the edge of Coulomb diamonds. Particularly, the density of current peaks was much higher at some narrow bias conditions. We mark one of them as HT. The extended view of stability diagram in HT regime is shown in Fig. 1

Quantum Probabilities of the Trap States
We measured the time domain characteristics of I d at the bias conditions near the current peaks, and observed RTN. One example of the measurement results, which was measured at V g of −640 mV and V d of −13.5 mV, is shown in Fig. 2(a). Two types of RTN could be identified, which were named as RTN1 and RTN2, as shown in Fig. 2(a). RTN1 has larger amplitude and longer average switching time, while RTN2 has smaller amplitude and shorter average switching time. The long switching time and high amplitude imply that the trap corresponding to RTN1 is a deep oxide trap, while RTN2 might come from the shallow interface trap in the SiON/substrate interface. We could roughly estimate the amplitude of RTN1 by estimating the number of carriers inside the channel [32][33][34] , N 0 = k B TC gate /e 2 = k B T ox ε 0 ε WL/t eq e 2 = 11.6, where k B is Boltzmann constant, e is the value of elementary charge, ox ε is the dielectric constant of SiO 2 , 0 ε is the permittivity in vacuum, W is the width of the channel, L is the length of the channel and t eq is the capacitive equivalent thickness of SiON layer [35][36][37] . Then the relative amplitude of RTN1 can be estimated as ΔI/I = 1/N 0 = 8.6%, which was roughly in agreement with the experimental data.
In order to study the statistics of I d , we analysed the frequency with which we observed I d at certain ranges, with the minimum step of 4pA, which was determined by the systematic noise of the system in a bandwidth of 5 Hz, as shown in Fig. 2(b). This is the quantum mechanical probability finding the system under a certain current state, P(I d ), which reveals the information about the wave function of the traps, ψ(I d ), as At higher temperatures, RTN1 could not be observed, which is shown in the supplementary data. This is because the number of carriers significantly increased at higher temperatures 38,39 . As a result, the relative amplitude of RTN dropped, and will be hidden in the 1/f noise.
In order to study the bias dependence of RTN1 and RTN2, the time domain characteristics of I d were measured at the bias conditions marked as red points in Fig. 1(c). The dependence of the time domain characteristics on V g is shown in Fig. 2(c). The corresponding P(I d ) at different bias conditions is shown in Fig. 2(d). RTN1 was only observed in a narrow window of V g , between −635 mV and −650 mV. We observed that the current was more likely to be in the high current state if we increased |V g |. This showed the charge trap shifted from the unoccupied state to the occupied state as we changed V g . It is also in agreement with the I d − V g characteristics near the HT regime as shown in supplementary data.
The V d dependence on time domain characteristics is shown in Fig. 2(e), and the corresponding P(I d ) at different bias conditions is shown in Fig. 2(f). The high current state regarding RTN1 was only observed at a narrow window of V d , between −13 mV and −13.5 mV. This narrow bias window, along with the fact that the current peaks were observed at the edges of the Coulomb diamonds, implies that the RTN1 has originated from resonant tunnelling 40 .

Nature of the trap
We will invesigate on the probability of finding I d at the high current State (P h ) or the low current state (P 1 ) shown in Fig. 2(b) to identify the nature of the charge trap. The way to extract P h and P 1 is shown in the method. The dependence of P h and P 1 on V g and V d are shown in Fig. 3(a) and (b), respectively. From Fig. 3(a), I d was more likely to be in the high current state if |V g | was large, and a sharp transition of the preferred I d state can be observed near V g of −640 mV. This sharp transition showed that the occupancy of the trap, which is revealed from P h and P 1 , was modulated by V g . This implies the charge trap corresponding to RTN1 is located in the SiON layer. The high current state was only observable at V d of −13 mV and −13.5 mV  Fig. 3(b), which implies the presence of resonant tunnelling in the channel due to the narrow bias window of V d . The V g and V d dependence of RTN1 amplitude were shown in Fig. 3(c) and (d), respectively. We can see that the amplitude of RTN1 almost has no dependence on both V g and V d . Since the dependence of I d on V g is non-linear, as shown in supplementary information, RTN1 is unlikely to have come from the impact of the shift of threshold voltage caused by carrier trapping/de-trapping. Otherwise, if the shift of threshold voltage was constant, the amplitude of RTN1 will have dependence on V g and V d , which was not what we observed in Fig. 3(c). We could use a similar method to investigate the characteristics of RTN2. The dependence of RTN2 on biases was complex, which implies RTN2 came from a shallow trap near SiON/substrate interface. The dependence of RTN2 on biases is shown in Supplementary information.
The  single hole as shown in Fig. 2(d) and (f), we must study its bias dependence. We will study the noise coming from wavefunction in order to understand its dependence on biases. The differential current, ΔI d , is defined as where t is time with the unit of second. The dependence of its probability distribution on V g and V d are shown in Fig. 4(a) and (b), respectively. The standard deviation of ΔI d was extracted by fitting the distribution of ΔI d with the Gaussian distribution function. The probability distribution of ΔI d had very weak dependence on V g , as shown in Fig. 4(a), while it showed more significant dependence on V d , as shown in Fig. 4(b). Since the change of I d is similar between Fig. 4(a) and (b), the fact that ΔI d shows more significant dependence on V d implies that the coupling between the quantum dot and energy level created by charge trap was mainly modulated by V d . We observe that the standard deviation of probability distribution showed a peak value if V d was at −13.5 mV, as shown in Fig. 4(b). This reveals that the wavefunction of the single hole in the channel became the broadest at this bias condition, which implies that the strongest correlation between two energy levels in the channel at this bias condition. This is also in agreement with the assumption of resonant tunnelling. The resonant level is likely to have originated from the charge traps. The Johnson noise 41,42 of this system was calculated to be ~20 fA/ Hz and the background noise of the system was ~4pA in a bandwidth of 5 Hz, which were both much smaller than the noise coming from wavefunction.

Two Traps RTN
We could use lag plots 43 to study the correlation behaviour of I d and the fractal nature of the two charge traps 44 . The lag plot of I d , with the time lag, Δt, of 1 s, 10 s and 100 s, were shown in Fig. 5(a),(b) and (c) respectively.
When Δt was 1 s, we could observe strong correlation behaviour, as shown in Fig. 5(a). The fractal shape of the lag plot is diagonal, which reveals the strong positive autocorrelation behaviour of I d in this time scale. When Δt was increased to 10 s, the shape of lag plot remained diagonal, but become rectangular within each small area, as shown in Fig. 5(b). This reveals the lost of I d correlation behaviour regarding RTN2. When Δt was further increased to 100 s, the fractal shape of lag plot became rectangular in both large and small area, as shown in Fig. 5(c). This reveals that the correlation behaviour of I d almost disappeared in this time scale.

Discussion
In the previous sections, we discussed about the characteristics of charge traps. Based on the experimental data, we found that the RTN1 is coming from the opening/closing of a resonant level in the device channel, and the carrier trapping/de-trapping process have originated from the tunnelling process between the Poly-Si gate and charge trap in the SiON layer. The lag plot clearly shows the correlation behaviour and fractal nature of the two charge traps. We also observed the crossover between RTN and 1/f noise, which was previously reported to be observed in magnetic nanodot system 45 , as shown in the Supplementary information.
Based on the information shown above, we could establish a physical model to describe the RTN1, as shown in Fig. 6. A schematic 3-D diagram of the device and its physical model is shown in Fig. 6(a). Two quantum dots in series, marked as QD1 and QD2, were responsible for the drain current. The series quantum dots have presumably originated from the remote surface roughness caused by Poly-Si grains. We simulated the stability diagram of the series quantum dots system based on the master equations 46,47 , which is explained in detail in method part. The parameters are summarised in the supplementary information. The simulated stability diagram is shown in Fig. 6(b), which is roughly in agreement with the experimental data obtained in Fig. 1(a). The simulation result of stability diagram for individual QD1 and QD2 have been summarised in Supplementary information.
The trap corresponding to RTN1 is located in the SiON layer. Considering the V g dependence, the charge trapping/de-trapping process has likely originated from the tunnelling of carriers between the poly-Si gate and the trap inside the SiON, as shown in Fig. 6(c). If |V g | is decreased, the tunnelling barrier between the hole in the poly-Si gate and the energy level in the gate oxide becomes lower, and the trap was easier to be occupied. This corresponds to the trend observed in Fig. 3(a).
The different current states can be explained by the opening/closing of the resonant level [48][49][50] in the channel created by the charge trap. Due to the strong V d dependence, the resonant level should be close to the drain reservoir. We address the resonant level to be located in the tunnelling barrier between QD2 and the drain reservoir. As a result, the resonant level is strongly coupled with QD2. Due to the relatively much larger distance between the resonant level and QD1, the coupling between the resonant level and QD1 is much weaker than the coupling between the resonant level and QD2, and therefore can be neglected.  A schematic diagram of the potential profile across the channel is shown in Fig. 6(d). If no single hole occupies the charge trap, it was not neutral. Since the Poly-Si gate behaves like a metal, the trap generates mirror charge in the Poly-Si gate. As a result, the trap and the corresponding mirror charge forms an electric dipole. The electric dipole therefore forms a potenetial well for the single hole in the channel. Under certain bias conditions, the energy level inside this potential well is aligned with the energy level in the QD2. Under that circumstance, resonant tunnelling is observed and the tunnelling barrier between QD2 and drain reservoir is made more transparent. As a result, I d will increase, therefore we can observe the high current state. If the charge trap is occupied by a single hole, the charge trap will become neutral, and no resonant level will be found in the channel. Under this condition, the tunnelling barrier therefore becomes less transparent, and we can only observe the low current state. This model, which is based on resonant tunnelling, could explain the reason for the narrow V g and V d bias condition to observe RTN. Since the resonant level is in the channel, it is therefore more significantly modulated by V d . Besides, the transmission coefficient and wavefunction were mainly influenced by the V d , which explains the fact that the wavefunction was mainly affected by V d , as shown in Fig. 4. We can estimate the position of charge trap based on the depth of the resonant energy level. From the simplicity point of view, since RTN1 is only observed in the HT regime at the edge of Coulomb diamond, the depth of the potential well, ΔV, is assumed to be 13.5 mV. We could estimate the distance between trap and Poly-Si/SiON interface, d, from e V e d t 2 4 (3) 2 ox 0 eff 2 πε ε Δ = .
From Equation (3), we can roughly estimate d to be ~0.2 nm. This is on the same magnitude with the lattice constant of SiON. This estimation should be correct in magnitude, and is in agreement assuming that the charge trap is located on the top side of SiON. As a result, we think a charge trap located near the Poly-Si/SiON interface, which was presumbaly a boron ion coming from the ion implantation process, is responsible for the resonant level in the channel.
In conclusion, we successfully demonstrate that we could identify the nature of the trap by measuring the RTN and investigating its bias dependence at low temperatures. We could estimate the position of the trap from the experimental data. Our research demonstrate a way to study the characteristics of charge trap systematically, and will pave the way for scientists to understand the detailed nature of the RTN.

Methods
Sample and experiments. The device we measured was a standard bulk-Si p-type MOSFET (p MOSFET), fabricated by a standard 65nm-node technology. The channel of the p MOSFET was 10 μm wide and 75 nm long. Wide-channel devices were chosen to increase the chance to find charge traps. The gate was made of highly-doped poly-crystalline silicon (Poly-Si). The gate oxide was made of SiON, with the equivalent oxide thickness of 2.4 nm. The p MOSFET was wire-bonded with Aluminium wire onto a chip carrier. Then the p MOSFET was put into a cryostat, with the maximum capability to control temperatures down to 2 K. The I d was measured by a B1500A with high resolution current module. The value of current at each bias was obtained after averaging over 10 5 sampling taken with the duration of 2 μs for each point. The background noise was less than 4pA, in a bandwidth of 5 Hz.
Data analysation. The probability to observe high current state and low current state in Figs 2 and 3 are determined by fitting the experimental data with Gaussian distribution functions. The amplitude of RTN is determined by the difference of the peaks. The probability of each state is determined by integrating the corresponding probability distribution function over the current. For example, if the probability distribution function corresponding to the high current state in Fig. 2 is p h (I d ), and the probability distribution function corresponding to the low current state in Fig. 2  which correspond to the shaded areas (magenta and blue, respectively) in Fig. 2(b).

Simulation of Quantum Dots.
In semiconductor quantum dots, the single particle energy spacing is comparable to the charging energy. Therefore, the Hamiltonian of the system does not simply depend on the charging energy, and the effect of single particle spacing must be considered. In order to simulate the quantum dots in a simple way, we used mesoscopic capacitor model, with different effective coupling capacitances in different hole states, as an approximation. Assuming the gate capacitance, drain capacitance, and source capacitance when n holes occupy the quantum dots are C g (n), C d (n) and C s (n) respectively. Considering the circumstance when one less hole occupies the quantum dot, the change in free energy when a hole tunnels out through the electrode drain, Δ − F n ( ) d , or tunnels out through the electrode source, Δ + F n ( ) s , can be expressed as   (6) and (7). Therefore, using Fermi's golden rule, the tunnelling rate through the drain and source can be expressed as In the steady state, the probability is not assosiated with time, so p n t t ( , ) 0 (11) The drain current is therefore expressed as