Research on Carrier Correlation in Ballistic Transport Nano-MOSFETs

In this paper,the experiment proved that shot noise is suppressed by Fermi and Coulomb interaction correlation. Meanwhile, the establishment of shot noise suppression factor (Fano) in ballistic transport nano-MOSFETs the Coulomb interaction correlation and the combination of the two effects are derived from separately considering the Fermi interaction correlation. And on this basis the variation of Fano with voltage, doping concentration and temperature are investigated.


INTRODUCTION
With the shortening of the channel length of MOSFET, the mechanism of the carrier has been changed as it passes through the channel.The carriers are drift-diffusive transport in long channel MOSFET, which is the result of a large number of carriers scattering in the field of electric field, thus the current noise is mainly thermal noise [1].When the channel length is in the range of sub 100 nm, carrier transport mechanism shifts from drift diffusive to quasi-ballistic or even to ballistic transport (Quasi-ballistic transport means less carrier scattering times; ballistic transport is limiting case that the carrier shot noise [2 -9].The research has demonstrated the role of the Fermi and Coulomb correlations in the nano-MOSFET, and the simulation has been carried out [6,7,10].The Fermi correlation effects carrier injection in the contact side, and the higher carriers degenerate, the stronger the suppression of Fermi on the shot noise;the Coulomb correlation can change carriers, and the stronger the space charge effect, the stronger the suppression of Coulomb on the shot noise .
Although it can be concluded from the experimental results and the simulation results that shot noise is suppressed in nano-MOSFET [3,4,6,11], and the Fano is closely related to the device structure parameters and working parameters, there are still no suppression mechanism and shot noise suppression strength in-depth theoretical study.The research showed that noise expression in nano-MOSFET, and considering the existence of Fermi and Coulomb at the same time, only the Fermi or Coulomb interaction, or neither with simple analysis [7,10]; The research established the total suppression factor formula of shot noise in ballistic MOSFET, with a simple explain and discussion on the relationship between the suppression factor and the gate voltage, but the formula for Coulomb and Fermi suppression was not analyzed respectively [6, 7, 10, 12 -14].At present, there is no comprehensive analysis of the carrier correlation.
In this paper, firstly, the experimental results showed that the shot noise is affected by the Fermi and Coulomb interaction correlation.Then, Fano models were established with theoretical analysis of the Fermi and the Coulomb correlation.The suppression of shot noise in different situations is discussed with the variation of the bias voltage, doping concentration and temperature.Finally, the suppression results are compared with the experimental results the macroscopic shot noise suppression and the existing suppression of shot noise in the ballistic transport MOSFET.

EXPERIMENTAL
Select the threshold voltage of 90nm n-MOSFET device was 0.7V, the width of the device(W) is 2×10 -6 m, the thickness of the oxide layer is 1×10 -9 m, the source and drain doping concentration(N SD ) is 1×10 26 m -3 ,and the noise power spectrum(S) of the device was tested under low temperature.The device works in the sub-threshold region, which is the gate voltage (V GS ) is 0.25V.The variation of S with source-drain voltage (V DS ) was shown in Fig. (1).Then, the device works in the linear region and saturation region, which is V GS is 1.2V, as shown in Fig. (2), the S changed with V DS .As shown in figures, when the V DS is small,S is not obvious; and V DS increases, we can hear shot noise clearly, this is because the electric field increases in the channel, and lead to the barrier decreases, which means shot noise was suppressed by Fermi and Coulomb interaction correlation.Jia and He

SHOT NOISE SUPPRESSION FACTOR MODEL
The leakage current fluctuation is the common result of the Fermi effect and potential fluctuations, which leads to the suppression of shot noise.Based on the reference [7], Fano is derived in ballistic transport of nano-MOSFET.
Considering the minimum sideband has been occupied, in order to simplify the formula, the following notation of is defined: Where, h is function of kinetic energy in transport direction E. Here, N D is the electron density in the subband.The electron concentration per unit area can be expressed as: Where, f S (E) and f D (E) are the Fermi Dirac occupation factors.We assume that the fluctuations of the propagating states occupation factor affect have a direct impact on n D through electrostatic effect on E M .From (2), we can derive: The gate capacitance per unit area C G was used to simplify the approximation of total electrostatic effects, thus the relationship between the electron density and the maximum barrier value of the side band can be expressed as: From formula (3) and ( 4), it is concluded that Where, When the side band barrier is maximum, the ballistic current density I B at the top of barrier is expressed as: As can be seen from the above formula, the fluctuations of f s , f D and E M have direct impacts on the Ballistic current.Considering the far from equilibrium case, in which f D <<1, thus f D can be discarded.

(9)
From formula (5) and (9), it is concluded that where is a weighted mean of the velocity: Current is the sum of the current pulses induced by all charged q, therefore , and the occupation factor of different modes are not related.So under Fermi and Coulomb interaction correlation, the Fano is 2 is Fermi interaction.When f s <<1, the small amount of high-end can be ignored, hence the formula mainly expresses Coulomb interaction correlation.Fano of the Coulomb interaction can be expressed as: where, is Coulomb interaction.When , the Fano of the Fermi interaction correlation can be expressed as: It can be seen from the derivation of the formula that in order to get each Fano in ballistic nano-MOSFET, the top barrier energy E M should also be obtained.The gate voltage is the sum of the surface potential and the voltage drop across the oxide in ballistic transport nano-MOSFET.

(15)
Where 1/2 / C Ox -2Φ F is the flat-band voltage, C Ox is the oxide capacitance per unit area, and Φ F = (kT/q)ln(N A / n i ), N A is channel doping concentration, n i is the intrinsic carrier density.V GS-B is the gate voltage of ballistic transport MOSFET; Ψ S is the work function; Q is the charge at the top of barrier [15].Eq. ( 15) can be deformed to: E M and V ' GS-B , Q are linked in formula (16).The carrier energy at the top barrier is generated by the positive emission rate of carriers in the source region and the negative emission rate of carrier in the drain region.So the top the barrier charge is also related to the top barrier potential as where, is the effective density of states, E F is the Femi energy.For a given gate and drain bias, Eq. ( 16) and ( 17) define a nonlinear equation for n S (V ' GS-B , V DS ).Fig. (3) shows the relationship between the gate voltage with top barrier potential.It can be seen from the figure that with the increase of the gate voltage, the top barrier energy decreases.

THE ANALYSIS OF MOEEL RESULTS
Under the Fermi interaction correlation, the Coulomb interaction correlation and the combination of the two effects, the Fano with bias voltage, doping concentration and temperature are studied separately.The channel length of the MOSFET chosen was 90nm, V DS is 0.8 V , V GS is 0.4V, temperature (T) is 300K, the source and drain doping concentration (N DS ) is 1×10 26 m -3 .

Analyze the Effect of Voltage on Fano
The variation of Fano with V DS was researched under the interaction of the Fermi interaction correlation, the Coulomb interaction correlation and the combination of the two effects.As shown in Fig. (4),it can be seen that each Fano decreases with the increase of source-drain voltage, but the change is not very obvious, and the total suppression(F) is stronger than Fermi (F F )and Coulomb(F C ) alone.Experiment figure showed that when V DS , the total shot noise suppression factor changed slightly the research papers [5],which is consistent with the changing trend of our theoretical results, and the theoretical explanation of the experimental results is also given.The variation of Fano with V GS is shown in Fig. (5).As the gate voltage increases, the longitudinal electric field increases, resulting in the increase of carrier surface scattering, the device transport deviates from the ballistic limit; while the electrons increase in inversion layer, and the carrier degeneracy increases, thereby enhancing the Fermi correlation to shot noise suppression [16,17].On the other hand, with the increase of the gate voltage,the channel barrier height reduces, the number of carriers increase, thus increasing the number of carrier inelastic scattering and the Coulomb correlation [18,19], leading to the suppression of shot noise.The model not only follows the shot noise suppression in mesoscopic conductors, but also the experimental trend of total suppression factor with the gate voltage [5].

Analyze the Effect of T and V DS on Fano
The increase in temperature will stop shot noise becoming the main component of noise, because the noise suppression is enhanced by temperature, which is consistent with our model.The variation of each Fano with temperature is shown in Fig. (6).As shown in figure, the Fano decreases with the increase in T, and which is consistent with the conclusions in the research papers [3].F F and F C decrease with the increase in T in the range of 150-300K, where the channel carrier scattering is mainly acoustic wave scattering and optical wave scattering, both scattering probability of which decreases with temperature.So with temperature increase, the number of phonon scattering increases, the average number of phonons increases, leading to the enhancement of shot noise suppression [3, 20 -22].And above 300K, Fano increases with temperature,this is because of the high temperature, the device transportation is changed from ballistic to drift-diffusion,and the device is mainly thermal noise.