Energy Bands and Charge Carriers in Semiconductors
Drift of Carriers in Electric and Magnetic Fields
  • In addition to the knowledge of carrier concentrations, the collisions of the charge carriers with the lattice and with the impurity atoms (or ions) under electric and/or magnetic fields must be accounted for, in order to compute the current flow through the device.
  • These processes will affect the ease (mobility) with which carriers move within a lattice.
  • These collision and scattering processes depend on temperature, which affects the thermal motion of the lattice atoms and the velocity of the carriers.
Conductivity and Mobility
  • Even at thermal equilibrium, the carriers are in a constant motion within the lattice.
  • At room temperature, the thermal motion of an individual electron may be visualized as random scattering from lattice atoms, impurities, other electrons, and defects.
  • There is no net motion of the group of n electrons/cm3 over any period of time, since the scattering is random, and there is no preferred direction of motion for the group of electrons and no net current flow.
  • However, for an individual electron, this is not true the probability of an electron returning to its starting point after time t is negligibly small.
  • Now, if an electric field is applied in the x-direction, each electron experiences a net force q from the field.
  • This will create a net motion of group in the x-direction, even though the force may be insufficient to appreciably alter the random path of an individual electron.
  • If is the x-component of the total momentum of the group, then the force of the field on the n is
    (2.18)

    Note: this expression indicates a constant acceleration in the x-direction, which realistically cannot happen.

  • In steady state, this acceleration is just balanced by the deceleration due to the collisions.
  • Thus, while the steady field does produce a net momentum , for steady state current flow, the net rate of change of momentum must be zero when collisions are included.
  • Note: the collision processes are totally random, thus, there is a constant probability of collision at any time for each electron.
  • Consider a group of electrons at time t = 0, and define N(t) as the number of electrons that have not undergone a collision by time t


    Fig.2.7 The random thermal motion of an individual electron, undergoing random scattering.
  • The rate of decrease of N(t) at any time t is proportional to the number left unscattered at t, i.e.

    (2.19)

    where is the constant of proportionality.

  • The solution is an exponential function
    (2.20)

    and represents the mean time between scattering events, called the mean free time.

  • The probability that any electron has a collision in time interval dt is dt/ , thus, the differential change in due to collisions in time dt is
    (2.21)

  • Thus, the rate of change of due to the decelerating effect of collisions is
    (2.22)

  • For steady state, the sum of acceleration and deceleration effects must be zero, thus,
    (2.23)

  • The average momentum per electron (averaged over the entire group of electrons) is
    (2.24)

  • Thus, as expected for steady state, the electrons would have on the average a constant net velocity in the -x-direction
    (2.25)

  • This speed is referred to as the drift speed, and, in general, it is usually much smaller than the random speed due to thermal motion .

  • The current density resulting from this drift
    (2.26)

  • This is the familiar Ohm's law with being the conductivity of the sample, which can also be written as , withis defined as the electron mobility
    (in ), and it describes the ease with which electrons drift in the material.


  • The mobility can also be expressed as the average drift velocity per unit electric field, thus with the negative sign denoting a positive value for mobility since electrons drift opposite to the direction of the electric field.

  • The total current density can be given by (2.27) when both electrons and holes contribute to the current conduction; on the other hand, for predominantly n-type or p-type samples, respectively the first or the second term of the above equation dominates.

    Note: both electron and hole drift currents are in the same direction, since holes (with positive charges) move along the direction of the electric field, and electrons (with negative charges) drift opposite to the direction of the electric field.

  • Since GaAs has a strong curvature of the E-k diagram at the bottom of the conduction band, the electron effective mass in GaAs is very small the electron mobility in GaAs is very high since is inversely proportional to .

  • The other parameter in the mobility expression, i.e., (the mean free time between collisions) is a function of temperature and the impurity concentration in the semiconductor.

  • For a uniformly doped semiconductor bar of length L, width w, and thickness t, the resistance R of the bar can be given by where is the resistivity.
Effects of Temperature and Doping on Mobility
  • The two main scattering events that influence electron and hole motion (and, thus, mobility) are the lattice scattering and the impurity scattering.
  • All lattice atoms vibrate due to temperature and can scatter carriers due to collisions.
  • These collective vibrations are called phonons, thus lattice scattering is also known as phonon scattering.
  • With increasing temperature, lattice vibrations increase, and the mean free time between collisions decreases mobility decreases (typical dependence ).
  • Scattering from crystal defects and ionized impurities dominate at low temperatures.
  • Since carriers moving with low velocity (at low temperature) can get scattered more easily by ionized impurities, this kind of scattering causes a decrease in carrier mobility with decreasing temperature (typical dependence ).
  • Note: the scattering probability is inversely proportional to the mean free time (and to mobility), hence, the mobilities due to two or more scattering events add inversely:
    (2.28)
  • Thus, the mechanism causing the lowest mobility value dominates.
  • Mobility also decreases with increasing doping, since the ionized impurities scatter carriers more (e.g., for intrinsic Si is 1350 at 300 K, whereas with a donor doping of , n drops to 700 ).
High Field Effects
  • For small electric fields, the drift current increases linearly with the electric field, since is a constant.
  • However, for large electric fields (typically > ), the current starts to show a sublinear dependence on the electric field and eventually saturates for very high fields.
  • Thus, becomes a function of the electric field, and this is known as the hot carrier effect, when the carrier drift velocity becomes comparable to its thermal velocity.
  • The maximum carrier drift velocity is limited to its mean thermal velocity (typically ), beyond which the added energy imparted by the electric field is absorbed by the lattice (thus generating heat) instead of a corresponding increase in the drift velocity.
2.4.4 The Hall Effect
  • An extremely important measurement procedure for determining the majority carrier concentration and mobility.

    Fig.2.8 The experimental setup for the Hall Effect measurement.

  • If a magnetic field is applied perpendicular to the direction of carrier flow, the path of the carriers get deflected due to the Lorentz force experienced by the carriers, which can be given by
    F = q(E + v x B) (2.29)
  • Thus, the holes will get deflected towards the -y-direction, and establish an electric field along the y-direction, such that in steady state
  • The establishment of this electric field is known as the Hall effect, and the resulting voltage is called the Hall voltage.
  • Using the expression for the drift current, is called the Hall coefficient.
  • A measurement of the Hall voltage along with the information for magnetic field and current density gives the majority carrier concentration
  • Also, the majority carrier mobility can be obtained from a measurement of the resistivity
  • This experiment can be performed to obtain the variation of majority carrier concentration and mobility as a function of temperature.
  • For n-type samples, the Hall voltage and the Hall coefficient are negative a common diagnostic tool for obtaining the sample type.
  • Note: caution should be exercised for near intrinsic samples.

EXAMPLE 2.4: A sample of Si is doped with In . What will be the measured value of its resistivity? What is the expected Hall voltage in a 150 m thick sample if ?

SOLUTION:


Equilibrium Condition
  • In equilibrium, there is no external excitation except a constant temperature, no net transfer of energy, no net carrier motion, and no net current transport.

  • An important condition for equilibrium is that no discontinuity or gradient can arise in the equilibrium Fermi level EF.

  • Assume two materials 1 and 2 (e.g., n- and p-type regions, dissimilar semiconductors, metal and semiconductor, two adjacent regions in a nonuniformly doped semiconductor) in intimate contact such that electron can move between them.

  • Assume materials 1 and 2 have densities of state N1(E) and N2(E), and F-D distribution functions f1(E) and f2(E) respectively at any energy E.



  • The rate of electron motion from 1 to 2 can be given byrate from 1 to 2 N1(E)f1(E) . N2(E)[1 f2(E)] (2.30)and the rate of electron motion from 2 to 1 can be given byrate from 2 to 1 N2(E)f2(E) . N1(E)[1 f1(E)] (2.31)" At equilibrium, these two rates must be equal, which gives f1(E) = f2(E) => EF1 = EF2 => dEF/dx = 0; thus, the Fermi level is constant at equilibrium, or, in other words, there cannot be any discontinuity or gradient in the Fermi level at equilibrium.

Practice Problems
    2.1 Electrons move in a crystal as wave packets with a group velocity where is the angular frequency. Show that in a given electric field, these wave packets obey Newton's second law of motion, i.e., the force F = m*a, where m* is the effective mass and a is the acceleration.

    2.2 Some semiconductors of interest have the dependence of its energy E with respect to the wave vector k, given by is the effective mass for E = 0, k is the wave vector, and is a constant. Calculate the dependence of the effective mass on energy.

    2.3 Determine the equilibrium recombination constant r for Si and GaAs, having equilibrium thermal generation rates of respectively, and intrinsic carrier concentrations of respectively. Comment on the answers. Will change with doping at equilibrium?

    2.4 The relative dielectric constant for GaP is 10.2 and the electron effective mass is Calculate the approximate ionization energy of a donor atom in GaP.

    2.5 Show that the probability that a state above the Fermi level is occupied is the same as the probability that a state below is empty.

    2.6 Derive an expression relating the intrinsic level to the center of the band gap and compute the magnitude of this displacement for Si and GaAs at 300 K. Assume respectively.

    2.7 Show that in order to obtain maximum resistivity in a GaAs sample it has to be doped slightly p-type. Determine this doping concentration. Also, determine the ratio of the maximum resistivity to the intrinsic resistivity.

    2.8 A GaAs sample (use the date given in Problem 2.7) is doped uniformly with out of which 70% occupy Ga sites, and the rest 30% occupy As sites. Assume 100% ionization and T = 300 K.
    a) Calculate the equilibrium electron and hole concentrations
    b) Clearly draw the equilibrium band diagram, showing the position of the Fermi level with respect to the intrinsic level , assuming that lies exactly at midgap.
    c) Calculate the percentage change in conductivity after doping as compared to the intrinsic case.

    2.9 A Si sample is doped with donor atoms. Determine the minimum temperature at which the sample becomes intrinsic. Assume that at this minimum temperature, the free electron concentration does not exceed by more than 1% of the donor concentration (beyond its extrinsic value). For

    2.10 Since the event of collision of an electron in a lattice is a truly random process, thus having a constant probability of collision at any given time, the number of particles left unscattered at time t, Hence, show that if there are a total of i number of scattering events, each with a mean free time of then the net electron mobility can be given by where is the mobility due to the ith scattering event.

    2.11 A Ge sample is oriented in a magnetic field (refer to Fig.2.8). The current is 4 mA, and the sample dimensions are w = 0.25 mm,
    t = 50 m, and L = 2.5 mm. The following data are taken: Find the type and concentration of the majority carrier, and its mobility. Hence, compute the net relaxation time for the various scattering events, assuming

    2.12 In the Hall effect experiment, there is a chance that the Hall Probes A and B (refer to Fig.2.8) are not perfectly aligned, which may give erroneous Hall voltage readings. Show that the true Hall voltage can be obtained from two measurements of with the magnetic field first in the +z-direction, and then in the z-direction.