CARRIER TRANSPORT PHENOMENA 1. Drift Current 2. Diffusion Current (Diff. Current) Figure 4.1 Typical random behavior of a hole in a semiconductor (a) without an electric field and (b) with an electric field Drift Current: If a positive volume charge density ρ is moving at an average drift velocity vd, the drift current density can be expressed as Jdrf = ρ . vd [4.1] Where J = current density (Amps/cm2 or coul/cm2-sec), ρ = charge density and vd = average drift velocity. If the volume charge density is due to the positively charged holes, then Jpdrf = (ep) . vdp [4.2] The charge density can be defined as ρ = e .p [4.3] Where, Jpdrf = holes current density and vdp = average drift velocity of the holes The equation of motion of a positive charged hole in the presence of an electric field is F = m*p a = e E [4.4] Where e = electronic charge a = acceleration E = electric field For low electric field, the velocity of the carrier is directly function of the electric field. Vdp = µpE [4.5] Where µp = holes mobility By recombining Equations (4.2) and (4.4), the drift current density due to holes may be expressed as The drift current density due to holes is in the same direction as the electric field The drift current density due to the electrons: Where Jn/drf = drift current density of electrons and vdn = average drift velocity vdn is directly function of the electric field and the net transport of the electron is opposite of the electric field. So we can write Where µn = electron mobility. The equation [4.6] may be expressed as The electron and hole mobility (µn and µp) are function of temperature and doping concentrations. Since both electrons and holes contribute to the drift current, the total drift current density is the sum of the individual electron and hole drift current densities expressed as: The equation [4.4] of motion (Lorentz Equation) can be expressed by following equation by substituting the holes acceleration: Where v is the velocity of the hole influenced the electric field. The equation of motion is the Lorentz equation: If we assume that the effective mass and electric fields are constant, the integration of equation [4.10] yields: VELOCITY SATURATION: The total velocity of a particle is the sum of the random thermal velocity and drift velocity. Total Velocity = Random thermal velocity + Drift velocity At T = 300K, the average thermal energy can be expressed as: Thermal velocity or thermal speed is a typical velocity of the thermal motion of particles that make up a gas, liquid, etc. Thus, indirectly, thermal velocity is a measure of temperature. It is a measure of the width of the peak in the Maxwell– Boltzmann particle velocity distribution If is defined as the root mean square of the velocity in any one dimension (i.e. any single direction), then The energy translates into a mean thermal velocity of approximately 1017 cm/sec for an electron in silicon. The two basic types of scattering mechanisms that influence electron and hole mobility are lattice scattering and impurity scattering. In lattice scattering a carrier moving through the crystal is scattered by a vibration of the lattice, resulting from the temperature. The frequency of such scattering events increases as the temperature increases, since the thermal agitation of the lattice becomes greater. Therefore, it is expected the mobility to decrease as the sample is heated (Fig. 3-22). On the other hand, the scattering from crystal defects such as ionized impurities becomes the dominant mechanism at low temperatures. Since the atoms of the cooler lattice are less agitated, lattice scattering is less important. However, the thermal motion of the carriers is also slower. Since a slowly moving carrier is likely to be scattered more strongly by an interaction with a charged ion than is a carrier with greater momentum, impurity scattering events cause a decrease in mobility with decreasing temperature. As Fig. 3-22 indicates, the approximate temperature dependencies are T3/2 for lattice scattering and T3/2 for impurity scattering. Since the scattering probability of Eq. (3-32) is inversely proportional to the mean free time and therefore to mobility, the mobilities due to two or more scattering mechanisms add inversely: The mobility affects by (1) ionized impurity scattering effect (µI) and (2) acoustic phonon scattering effect (µL). The mobility µL (acoustic phonon scattering) can be evaluated by qh 4 C11 8 L ( m * ) 5 / 2 T 3 / 2 *5 / 2 3/ 2 3E ds m (kT ) Where C11 = average longitudinal elastic constant of the semiconductor, and Eds = displacement of the edge of the band per unit dilation of the lattice. The mobility µI (ionized impurity scattering) can be evaluated by 64(2kT ) 3 / 2 S2 12 S kT 2 I ln 1 ( ) N I q 3 m *1 / 2 q 2 N I1 / 3 1 (m * ) 1 / 2 N I1T 3 / 2 Where NI is the total ionized impurity concentration in the semiconductor NI = ND+ + NA Phonon Scattering: Acoustic Phonon and Optical Phonon. o The electron and hole mobilities have a similar doping dependence: For low doping concentrations the mobility is almost constant and is primarily limited by phonon scattering. At higher doping concentrations the mobility decreases due to ionized impurity scattering with the ionized doping atoms. The actual mobility also depends on the type of dopant. The above figure is for phosphorous and boron doped silicon and is calculated using: The drift current density evaluated in Equation [4.9] may be expressed as: Where σ is the conductivity of the semiconductor material. The reciprocal of conductivity is resistivity (unit ohm-cm). If we have a bar of semiconductor material as shown in Figure 4.5 applying a voltage to generate the current. So we can write: J = σ E, J = I/A Equation [4.22b] is ohm’s law for a semiconductor. The resistance is a function of resistivity or conductivity as well as the geometry of the semiconductor. If p-type semiconductor with acceptor doping concentration NA and ND = 0, so NA >> ni And assume electron mobility µn = hole mobility µp The conductivity can be expressed as: The conductivity and resistivity of an extrinsic semiconductor are function of majority carrier parameters. RESISTIVITY DIFFUSION CURRENT: It is the process whereby the particles flow from a region of high concentration toward low concentration region. The diffusion of electrons from a region of high concentration to a region of low concentration creates a flux of electrons flowing in the negatives x directions. Since electrons have a negative charge, the conventional current direction is in the positive x direction. The diffusion current density may be expressed by following equation: Where Dn = electron diffusion coefficient (cm2/sec) The holes diffusion current can be expressed by Where Dp = holes diffusion coefficient (cm2/sec) -----------------------------------------End-------------------------------------------------------------- INDUCED ELECTRIC FIELD Consider the semiconductor is in thermal equilibrium. The Fermi energy level is constant through out the crystal. The doping concentration decreases with as x increases. Concentration gradient is formed Electrons are coming from donors atoms making the ionized donors atoms. Separation of +ve and –ve charge induces an electric field and that will oppose the migration of electrons. The electric potential Φ is correlated to the EPE (electron potential energy) by the charge (-ve), so we can express: If we assume a QN (quasi neutral) condition, in which the electron concentration is equal to the donor impurity concentration, then we can write as: