Noise and correlation functions of hot carriers in semiconductors

We present a unifying theory of electronic noise appropriate to semiconductor materials in the presence of electric fields of arbitrary strength. In addition to thermal noise, a classification scheme for excess noise indicating different microscopic sources of fluctuations responsible for number and mobility fluctuations is provided. On the basis of simple two-level models, numerical calculations using a Monte Carlo technique are performed for the case of p-type Si at 77 K. The primary quantity which is evaluated by the theory is the auto-correlation function of current fluctuations which, subsequently, is analyzed in terms of correlation functions of the relevant physical variables. Accordingly, the corresponding current spectral-densities are determined and then compared with direct experimental results and/or analytical expressions. Important subjects which have been investigated are: (i) the effect of field assisted ionization on generation-recombination noise from shallow impurity levels; (ii) the contribution to the total noise spectrum of cross-correlation terms coupling fluctuations in velocity with those in energy and number; (iii) the current random telegraph signal and the corresponding spectral density associated with a mobility fluctuator. In all cases the numerical calculations are found to be in satisfactory agreement with experiments and/or analytical expressions thus fully supporting the physical reliability of the theoretical approach here proposed.List of the Symbols Used e Absolute value of the electron charge - f Frequency - f Distribution function - g₁ Scattering strength with the scatter in state 1 - g₂ Scattering strength with the scatter in state 2 - Reduced Planck constant - j Total current density - j_c Conduction current density - j_d Displacement current density - j_x Component along the x direction of the total current density - k Carrier wavevector - m Carrier effective mass - m₀ Free electron mass - r Position vector - s Average sound velocity - t Time - u Fraction of ionized carriers - u_i Random telegraph signal related to carrier state - u_m Random telegraph signal related to scatterer state - v_d Ensemble average of the free carrier drift-velocity - v_i Carrier group velocity - v_t Ensemble average of the carrier velocity in the direction transverse to the applied field - v_ix Component along the x direction of the carrier group velocity - v_d^r Ensemble average of the reduced drift-velocity - v^r_i Reduced velocity component in the field direction of the i-th particle - v_ix^j Reduced velocity component along the x axis of the i-th particle in band j - v^r_ix Reduced velocity component along the x axis of the i-th particle - x_d Ensemble average of the carrier displacement along the x direction from the initial position - x_i Displacement along the x direction of the i-th carrier from the initial position - y_i i-th stochastic parameter - A Cross-sectional area of a homogeneous sample - C_I Auto-correlation function of the total current fluctuations - Auto-correlation function of the total current fluctuations due to mobility fluctuations - D Diffusion coefficient - D_tK Optical deformation potential - E Electrical field strength - E Electric field - E_x Component of the electric field along the x direction - E₁⁰ Acoustic deformation potential - G Conductance - I Total current - I₀ Total current in the voltage noise operation - I_m Total current associated with mobility fluctuations - I^V Total current in the current noise operation - K_B Boltzmann constant - L Length of a homogeneous sample - N Number of free carriers which are instantaneously present in the device - N_A Acceptor concentration - N_I Total number of carriers inside the device participating in the transport (here assumed to be constant in time) - N_T Total number of carriers which are instantaneously present in the device - S_I Spectral density of current fluctuations - S_V Spectral density of voltage fluctuations - Spectral density of current fluctuations associated with the mobility fluctuations - Spectral density of current fluctuations due to correlations between fluctuations in number and velocity - Spectral density of current fluctuations due to generation-recombination processes - Spectral density of current fluctuations due to free carrier drift-velocity fluctuations - S_I^l Longitudinal component with respect to the applied field of the current spectral-density - S_I^t Transverse component with respect to the applied field of the current spectral-density - T Absolute temperature - T_e Electron temperature - V Electrical potential - V^I Electrical potential in the voltage noise operation - W Collision rate - Z Small signal impedance - Poole-Frenkel factor - Equilibrium generation rate - _E Field dependent generation rate - Typical energy for thermally escaping from the impurity level - v_d⁽⁰⁾ Fluctuation of the ensemble average of the driftvelocity associated with Brownian-like motion - v_d^r(0) Fluctuation of the ensemble average of the reduced drift-velocity associated with Brownian-like motion - Carrier energy - ₀ Vacuum permittivity - _a Energy of the acceptor level - _r Relative static dielectric constant - Angle between initial and final k states - _op Optical phonon equivalent temperature - Mobility - ₀ Chemical potential - ₁ Mobility with the fluctuating scatterer in state 1 - ₂ Mobility with the fluctuating scatterer in state 2 - ₀ Crystal density - _E Field dependent volume recombination rate - _eq Equilibrium volume recombination rate - Conductivity - _g Cross-section for impact ionization - _c Average scattering time - _g Generation time - _l Carrier lifetime - _m Scatterer lifetime - _m1 Mean value of the time spent by the fluctuating scatterer in state 1 - _m2 Mean value of the time spent by the fluctuating scatterer in state 2 - _r Average recombination time - _T Transit time - Scattering rate - _AB Correlation function of the two variables A and B