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3.11 Quasi Fermi Levels 3.12 Continuity Equation 3.12 Poisson Equation 3.14 Summary of Semiconductor Equations References

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3.11 Quasi Fermi Levels 3.12 Continuity Equation 3.12 Poisson Equation 3.14 Summary of Semiconductor Equations References Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp Introduction to Electronic Devices (Course Number 300331) Fall 2006 Electronic Transport Information: http://www.faculty.iu- Dr. Dietmar Knipp bremen.de/dknipp/ Assistant Professor of Electrical Engineering Source: Apple Ref.: Apple Ref.: IBM Critical 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 1 101 dimension (m) Ref.: Palo Alto Research Center Electronic Transport 1 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp Introduction to Electronic Devices 3 Electronic Transport 3.1 Thermal movement of carriers 3.2 Carrier Drift 3.3 Carrier Diffusion 3.4 Einstein Relation 3.5 Current Density Equation 3.6 Thermal equilibrium processes 3.7 Non-thermal equilibrium processes 3.8 Generation Processes 3.9 Recombination Processes 3.9.1 Direct (Band-to-band) recombination 3.9.2 Donor-Acceptor recombination 3.9.3 Trapping 3.9.4 Auger recombination 3.10 Quasi Neutral Semiconductor 3.11 Quasi Fermi Levels 3.12 Continuity Equation 3.12 Poisson Equation 3.14 Summary of Semiconductor Equations References Electronic Transport 2 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3 Electronic Transport Electrons in the conduction band and holes in the valence band are able to move upon thermal activation, a gradient or an applied electric field. In the following the concepts of electronic transport in crystalline materials will be described. 3.1 Thermal movement of carriers Electrons in the conduction or holes in the valence band can essentially be treated as free carriers or free particles. Even in the absence of an electric field the carriers follow a thermally activated random motion. In thermal equilibrium the average thermal energy of a particle (electron or hole) can be obtained from the theorem for equipartition of 3 E thermal = kT Average thermal energy of an electron / hole average 2 The thermal energy of the particle is equal to the kinetic energy of the electron, so that the velocity of the particle can be calculated. The mass of the electron is equal to the effective mass of the electron. Electronic Transport 3 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.1 Thermal movement of carriers Furthermore, the velocity of the electron corresponds to the thermal velocity of the electron, so that the thermal velocity can be determined by: 1 2 E = m v Kinetic energy of an electron / hole kin 2 eff th At room temperature the average thermal velocity of an electron is about 105m/s in silicon and GaAs. 3kT vth = Thermal velocity of an electron meff Thermal motion of free carriers can be seen as random collision (scattering) of the free carriers with the crystal lattice. A random motion of an electron or hole leads to zero net displacement of the free carrier over a sufficient long distance / period of time. The average distance between two collisions within the crystal lattice is called mean free path. Associated to the mean free path we can introduce a mean free time τ. A typical mean free path is in the range of 100nm and the mean free time is in the range of 1ps. Electronic Transport 4 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift When a small electric field is applied to the semiconductor material each free carrier will experience an electro static force Force = −qF So that the carrier is accelerated along the field (in opposite direction of the field). F=0 F Schematic path of an electron in a semiconductor (a) random thermal motion, (b) combined motion due to random thermal motion and an applied electric field. Ref.: M.S. Sze, Semiconductor Devices Electronic Transport 5 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift An additional velocity component will be superimposed upon the thermal motion of the electron. The additional velocity is caused by an applied electric field F. The additional component is called drift velocity. The drift of the electrons can be described by a steady state motion since the gained momentum is lost due to collisions of the electrons and the lattice. P = mnvn P = −q ⋅ F ⋅τ C Based on momentum conservation the drift velocity can be calculated. The drift velocity is proportional to the applied electric field F. qτ C vn = − F Electron drift velocity mn Electronic Transport 6 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift The electric field is multiplied with a prefactor. The prefactor depends on the mean free time, the effective mass of the free carrier, and the charge of the carrier. The prefactor is called the hole or electron mobility µn,p in units of cm2/Vs. Drift velocity as a function of the electric field in silicon and GaAs. Ref.: M.S. Sze, Semiconductor Devices Electronic Transport 7 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift q ⋅τ q ⋅τ C V µ ≡ µ p ≡ Electron and hole mobility n m mn p The mobility is an important electronic transport parameter. The mobility is directly related to the material properties. Rewriting of the expression for the drift velocity leads to v vp µ = n µ = Electron and hole mobility n F p F The mobility is directly related to the mean free time between two collisions, which is determined by various scattering mechanisms. The most important scattering mechanisms are lattice scattering and impurity scattering. Lattice scattering is caused by thermal vibrations of the lattice atoms at any temperature above 0K. Due to the vibrations energy can be transferred from the carriers and the lattice. Electronic Transport 8 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift Since lattice vibrations increase with increasing temperature the influence of lattice scattering is getting the determinant effect at high temperatures. As a consequence the mobility is reduced. Theoretical analysis show that the mobility is decreased in proportion to T-3/2. Impurity scattering is observed when a charged carrier (in our case a free carrier) interacts with dopants / impurities (acceptors or donors). The charged carriers will be deflected due to Coulomb interaction between the two charges. The probability of impurity scattering depends on the total concentration of impurities (sum of the positively and negatively charged ions). Impurity scattering is becoming less dominant for higher temperatures. Theoretical 3/2 calculations reveal that the impurity scattering scales with T /NT, where NT is the total impurity concentration. The probability of a collision can be expressed in terms of the mean free time 1 1 1 = + Probability of a collision τ C τC, lattice τ C, impurity Electronic Transport 9 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift The probability of a collision is proportional to 1/τC. The mobility can be described by 1 1 1 = + µ µL µI where µL is the mobility as a function of the lattice scattering and µI is the mobility as a function of the impurity concentration. Measured temperature dependent electron mobility in silicon for various donor concentrations. The insert shows the limiting electron transport mechanisms for high and low temperatures. Ref.: M.S. Sze, Semiconductor Devices Electronic Transport 10 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift Measured electron and hole mobilities in silicon and gallium arsenide at 300K for various impurity concentrations. Ref.: M.S. Sze, Semiconductor Devices Electronic Transport 11 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift In the next step the influence of electric fields on the energy bands will be discussed. Applying an electric field to a semiconductor leads to a “tilt” of the energy bands. The tilt of the band is called band bending. The contacts are considered to be ohmic (ideal contacts) The behavior of contacts will be discussed as part of the sessions on diodes. F Conduction process in an n-type semiconductor under biasing conditions. Ref.: M.S. Sze, Semiconductor Devices Electronic Transport 12 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift The potential energy of the charges in the bands (electrons in the conduction band and holes in the valence band) can be described in terms of the energy of the conduction band and the valence band. The potential energies are related to electric potential ϕ as follows: EC = −qϕ + const. EV = −qϕ − Eg + const. If an electric potential is applied to the semiconductor the bands get “tilted”. The charges experience a force due the electric field. The force applied to the charges is given by the negative gradient of the potential energy leading to − qF = −grad()EC = −q ⋅ grad(ϕ ) Therefore, the electric field applied to an electrons is equal to the negative gradient of the electric potential. Electronic Transport 13 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift If we reduce the problem to one dimension the equation simplifies to E dϕ − qF = − C = −q dx dx The bottom of the conduction band corresponds to the electric potential of the electron. Since we are interested in the gradient of the energy and not the energy itself, we can use any energy level of the band diagram. We will therefore simply use the intrinsic energy level. 1 dE 1 dE F = C = i q dx q dx The potential energy of homogenous semiconductor increases linearly with the distance. Therefore, the electric field distribution throughout the semiconductor is constant. Electronic Transport 14 Introduction to Electronic Devices, Fall 2006, Dr. Dietmar Knipp 3.2 Carrier Drift The transport of carriers under the influence of an applied electric field produces a current called drift current.
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