Drift Velocity

6 Key excerpts on "Drift Velocity"

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  eBook - ePub

  Fundamentals of Ionized Gases

  Basic Topics in Plasma Physics

  • Boris M. Smirnov(Author)
  • 2012(Publication Date)
  • Wiley-VCH

    (Publisher)

  (3.26) we have

  (3.31)

  where now ε =

  me

  v 2 /2 is the current electron energy.

  This consideration is valid when we consider evolution of an individual electron in a gas in an external electric field and ignore a chaotic change of its energy, which corresponds to the criterion

  (3.32)

  Here ε is the current electron energy and Δε is the uncertainty in the electron energy or the width of its distribution function. Hence, here we consider the electron energy to be strictly determined in the course of its evolution. Let us determine the character of the variation of the electron energy ε if at the beginning it is relatively small. We multiply the kinetic equation (3.4) by the electron energy ε =

  me

  v 2 /2 and integrate over electron velocities, which gives

  This equation has a simple physical sense: the first term on the right-hand side of this equation is the energy transferred from the field to the electron, and the second term accounts for collision of this electron with atoms. We take into account elastic electron–atom collisions only and use (3.18) for the collision integral. Because of criterion (3.32) , one can replace the average electron energy by its current value ε and ignore the first term in (3.18) for the collision integral. As a result, we obtain

  (3.33)

  This equation exhibits the character of variation of the electron energy when an electron travels in an atomic gas in an external electric field and has low energy at the beginning. We have that the electron energy increases in time, and in the course of its increase the relative energy portion given to gas atoms from an electron increases until the maximum electron energy ε max

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  eBook - ePub

  Quantum Waveguide in Microcircuits

  • Jian-Bai Xia, Duan-Yang Liu, Wei-Dong Sheng(Authors)
  • 2017(Publication Date)
  • Jenny Stanford Publishing

    (Publisher)

  d can be obtained from the following equations:

  Figure 2.13 . Average velocity and average energy of electron in GaAs. The impurity concentration N

  D

   = 0 and T  = 77K subjected to a time‐electric‐ field step as functions of time. (a) At t  = 0 an electric‐field step ▵ E  =  7kV /cm is applied; (b) an electric field pulse ▵ E  = 70kV /cm with the duration ▵ t  = 0.06 ps is applied.

  2.23

  d =

  ∫

  0

  T

  v

  ( t )

  d t , v

  ( d )

  =

  d T

  Figure 2.13 shows the average velocity and average energy of electrons in GaAs. The impurity concentration N

  D

   = 0 and T  = 77K are subjected to a time‐electric‐field step as functions of time [8 ]. The results are obtained by the Monte Carlo simulation. (a) At t  = 0 an electric‐field step ▵ E  = 7kV /cm is applied; (b) an electric field pulse ▵ E  = 70kV /cm with the duration ▵ t  = 0.06 ps is applied, shown by dashed curves in figures. The former case is called an overshoot, and the latter case is called ballistic by the authors.

  Figure 2.14 . Electron average velocities v as functions of traveling distance d :(a ) overshoot case and (b) ballistic case.

  When the electron energy is smaller than the polar optical phonon energy (case 1) or smaller than the intervalley energy of Γ and L valleys

  △

  ε

  Γ L

  (case 2), the scattering probability is small. Thus, it is possible to obtain very high instantaneous drift velocities 3.5 × 107 and 108 cm /s , respectively in the two cases.

  The distance and average velocity of electrons in the time period T can be calculated by Eq. 2.2 3. For the two cases the calculated v (d ) curves are shown in Fig. 2.14 . From the figure we see that in the case of an overshoot the maximum average velocity is only half of that in the ballistic case, which occurs at d  = 375nm . At this distance the average energy reaches the polar optical phonon energy In the ballistic case, the average velocity decreases monotonously with the distance, and at d  = 0 it has the maximum value 7.3 × 107 cm /s . This is because after the electric field pulse the electron does not have energy supplement and the farthest distance is about 200 nm

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  eBook - PDF

  Plasma Processes and Plasma Kinetics

  580 Worked Out Problems for Science and Technology

  • Boris M. Smirnov(Author)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  From the analogy of these equations it follows the analogy of the values v x ϕ 1 and eE T Φ . Therefore, according to formula (8.65) the diffusion coefficient is D i = T eE v 2 x ϕ 1 d v , where the gradient of the ion number density is directed along x . On the other hand, the ion Drift Velocity according to its definition is w x = v 2 x ϕ 1 d v = EK i , where K i is the ion mobility. Comparing these expressions, we find the Einstein relation K i = eD i T . (8.66) Problem 8.30 Find the Drift Velocity of ions whose mass is much greater than the mass of the gas atoms. The width of the ion velocity distribution is determined either by the thermal ve-locity of ions or by the mean variation of the ion velocity in one collision with a gas atom. The thermal velocity of ions is low compared with the thermal velocity of atoms and the mean ion velocity variation is of the order of mg / M . Hence the width of the ion velocity distribution is always smaller than the relative velocity g of the ion–atom collision. Therefore, if we use the distribution function of ions in formula (6.33) in the form f ( v ) = δ ( v − w ) , we obtain eE µ N = m 2 π T 3/2 exp − m v 2 a 2 T g x g σ ∗ ( g ) d v a , where m is the atom mass, v a is the atom velocity. This gives eE µ N = 1 w 2 exp − m w 2 2 T 2 T π m 1/2 ∞ 0 exp − mg 2 2 T g 2 σ ∗ ( g )( y cosh y − sinh y ) dg , y = m w g T . (8.67) 248 8 Transport of Ions and Atoms in Gases and Plasmas When the ion Drift Velocity is small compared to a thermal velocity of gas atoms, it follows from the above w = 3 π eE 8 N σ ( 2 Tm ) 1/2 , (8.68) where σ = ∞ 0 σ ∗ ( x ) exp ( − x 2 ) x 2 dx , x = mg 2 2 T . This result corresponds with the first Chapman–Enskog approximation, but it holds true in a wider velocity range ( w T / m ) than the Chapman–Enskog approximation is valid ( w T / M ). Note that the method used of replacing the distribution function with the delta function is applicable if w T / M .

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  Nanoelectronics

  Quantum Engineering of Low-Dimensional Nanoensembles

  • Vijay Kumar Arora(Author)
  • 2018(Publication Date)
  • CRC Press

    (Publisher)

  345 10 Drift-Diffusion and Multivalley Transport A theory that makes an explicit connection between scattering-limited ohmic mobility and quantum-emission-limited saturation velocity is presented. The theory is applied to electrons in bulk silicon by taking a quantum equal to the energy of an optical phonon. Because this quantum emission is indi-cated, a modification in the mfp appears only in the high-field regime. This modification is shown to lead to electric-field-induced degradation of the diffusion coefficient. The theory presented agrees with the drift-diffusion experimental data and empirical relations utilized in modeling devices. The theory makes connections with an alternate description in terms of elec-tron temperature under ac and dc conditions. As drift-diffusion processes are central in the performance evaluation of submicron-scale devices where high fields are necessarily present, these results contribute significantly in reshaping thinking processes in the high-field regime. Transfer to higher valleys in multivalley band structure is also discussed. 10.1 Primer Considerable interest in the high-field transport will continue to exist because of scaled-down dimensions of devices that are now of submicron scale in the quasi-free direction of the carrier flow. As we have seen, with a high-electric field present, the familiar linear velocity-field characteristics become sublinear and the velocity eventually saturates at an applied high-electric field. This nonlinear response of the carrier velocity to a high electric field has been extensively explored, both theoretically and experimentally (for a review, see Reference 1). However, not much work has been reported on diffusive transport in the presence of a high electric field. An acceptable procedure for studying such processes is that of the Monte Carlo simula-tion with input parameters varied to get a desired output.

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  Gaseous Electronics

  Theory and Practice

  • Gorur Govinda Raju(Author)
  • 2005(Publication Date)
  • CRC Press

    (Publisher)

  The electrodes are made of nichrome wire so that they can be individually heated by dc to a slightly higher temperature than the vapor pressure temperature, thus preventing the condensation of vapor on the electrodes. The drift distance is 2 cm and over this distance the vapor pressure was found to be constant. Selected references for Drift Velocity data of electrons in Cs are: Chanin and Steen 193 ; Postma 194 ; and Saelee and Lucas. 195 Figure 6.32 shows the transport parameters. The Drift Velocity is a power-law function of E / N . The earlier results of Chanin and Steen 193 are considerably higher than the subsequent data obtained by Saelee and Lucas. 195 The characteristic energy and the mean energy ( ) are obtained by the Boltzmann distribution method. It is noted that D r / is marginally higher than at the same E / N . If the electron energy distribution is Maxwellian, then D r / ¼ 0.67 , Equation 6.39. Since this condition is not satisfied the electron energy distribution is non-Maxwellian, as is indeed found by the Boltzmann analysis. Saelee and Lucas 195 show that at the very low E / N investigated by them (1.41 Td) the electrons are in thermal equilibrium with gas molecules and that departure from a Maxwellian distribution occurs as the mean energy becomes larger. The cross section data shown in Figure 6.33 are reproduced from Zecca et al. 196 The Cs atom shows the Ramsauer–Townsend minimum at 0.08 eV. 197 The agreement between the total collision cross section data of various workers is less than satisfactory. The references for cross sections shown in Figure 6.33 are as follows. Total collision cross section: Brode 198 ; Visconti et al. 199 ; Jaduszliwer and Chan. 200 Momentum transfer: Nighan and Postma 201 ; Stefanov. 202 Elastic: Bartschat. 203 Excitation (2 p 6 2 P): Chen and Gallagher. 204 ; Ionization: Brink 205 ; McFarland and Kinney 206 ; Nygaard.

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  Principles of Electric Methods in Surface and Borehole Geophysics

  • Alex Kaufman, B. Anderson, Alex A. Kaufman(Authors)
  • 2010(Publication Date)
  • Elsevier Science

    (Publisher)

  But, the medium itself “resists” this motion, decreasing the acceleration. The mechanism producing this resistance varies from material to material. In metals, for instance, resistance to charge acceleration occurs because of the “collision” of moving electrons with ions of the crystal lattice. In electrolytes, the resistance is caused by viscous forces in the fluid. We consider such “frictional” force resisting charge motion from the macroscopic point of view. Regardless of its origin, it generally increases as the charge velocity increases and eventually becomes equal and opposite to the force caused by the electric field. At that moment, the resultant force acting on a charge vanishes and its acceleration also vanishes. In other words, the velocity of charge ω reaches a maximum value and then remains constant, giving rise to a steady current density j. If the forcing electric field is increased, the charge accelerates to a higher velocity before reaching again a balance of forces. Thus, it seems reasonable to assume that the velocity of charge in steady current flow is directly proportional to the Coulomb electric field E c, that is, ω + = u + E c and ω − = − u − E c, [2.8] where u + and u − are positive coefficients called the mobilities of the positive and negative charges, respectively. Several further comments are appropriate: 1. The process of establishment of a constant current described above uses a classical model of a charged particle and Newton's Second Law. In reality, this process involves the interaction of electromagnetic fields with the atomic structure of a conductive medium, and a detailed microscopic model requires a quantum mechanical description. 2. The values of ω + and ω − in Eqs

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