Carrier Transport Phenomena Chapter 4[Neamen]

Carrier Transport Phenomena

Adersh Miglani

IIT Delhi

August 07, 2013

Adersh Miglani Carrier Transport Phenomena

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1 The process by which these charged particles - holes andelectrons, move is called transport.

2 There are two basic transport mechanism.1 Drift: Movement of charge due to electric field2 Diffusion: Movement of charge due to density gradients

3 The temperature gradients in a semiconductor can also leadto carrier movement. However, as the semiconductor sizebecomes smaller, this effect can usually be ignored. Why?

4 IMPORTANT:The carrier transport phenomena are thefoundation for determining the current-voltage characteristicsof any semiconductor device.

5 ASSUMPTION: Thermal equilibrium will not be substantiallydisturbed while understanding transport phenomena.

6 Non-equilibrium processes are discussed in the next chapter.


Carrier Drift

1 An electric field, when applied to a semiconductor, willproduce a force on the electrons and holes so that they willexperience a net acceleration and net movement.

1 Acceleration is caused by electric field and net movementmeans charge carriers have finite resultant displacement withinthe device even after collision with ionized atoms.

2 Also, there should be available energy states in conductionband (n-type) and valence band (n-type).

2 Say, volume charge density is ρ in Coulomb/cm3 and averagedrift velocity is vd in cm/sec, then drift current density isgiven by Jdrift = ρvd in Coulomb/cm2-sec or ampere/cm2.

3 Say, charge on a hole carrier is q and volume density is n incm−3, current density due to holes is given by

Jp,drift = qnvd (1)


Carrier Drift Contd..

1 A hole (effective mass m∗) accelerates due to electric field(E ), the relation between force (F ) and acceleration (a) isgiven by

F = m∗a = m∗ dvddt

= qE ⇒ vd =qE

m∗ (2)

2 If electric field is constant, the velocity to increase with time.But, charge carriers face collisions or scattering events andtheir velocity characteristics are altered. So, we consideraverage velocity and effective time of movement between twoconsecutive collisions.

3 After every collision, charge loses most of its energy and gainenergy again in the form of drift velocity due to electric field.Therefore, the drift velocity of charge (say, hole) isproportional to applied electric field. The constant ofproportionality is called mobility (of hole) µp.

vdp = µpE (3)


Mobility

1 Mobility decides how well a charge carrier will move due to anelectric field.

2 Unit of mobility is cm2/V-sec.3 Unit of electric field is V/sec.4 By combining relationship between drift current density and

velocity and relationship between velocity and mobility, we get

Jp,drift = qpvd = qpµpE (4)

5 Drift current due to holes is in the direction of electric field.And the drift current due to electron is in the oppositedirection of electric field.

Jn,drift = qnµnE (5)

6 The total current density is sub of current density of electronsand holes

Jdrift = q(µpp + µnn)E (6)


Mobility Effects

1 There is a mean time between collisions which may bedenoted by τ .

m∗ dvddt

= qE ⇒ m∗vd = qEt ⇒ vd =qEt

m∗ ⇒ vd = µE ⇒ µ =qt

m∗

2 Hole and electron mobility is given as follows

µp =qτpm∗

p

µn =qτnm∗

n

(7)

3 The parameters τp and τn are mean time between collisionsfor an hole and electron, respectively.

4 If m∗n < m∗

p, the electron mobility is higher than the holemobility.

5 There are two types of collision or scattering mechanisms thatdominate in a semiconductor and affect the carrier mobility:phonon or lattice scattering and ionized impurity scattering.


Phonon or Lattice Scattering

1 The atoms in the semiconductor crystal have a certainamount of thermal energy at temperature above absolute zerothat causes the atoms to randomly vibrate about their latticeposition within the crystal.

2 Without these vibrations of atoms, the electrons would moveunimpeded or with no scattering. But, thermal vibrationscause disruption in the movements of electrons due tovibrating lattice atoms. This lattice scattering is termed asphonon scattering.

3 The lattice scattering is related to atom’s thermal motionwhich is a function of temperature.

4 Let us denote µL the mobility that would be observed due tolattice scattering alone. As per the scattering theory therelation between µL and temperature, T is given by,

µL ∝ T− 32 (8)


Ionized Impurity Scattering Contd..

1 Therefore, mobility due to lattice scattering increases astemperature decreases.

2 The impurity atoms are ionized at room temperature so that acoulomb attraction exists between the electrons and holes andionized impurity atoms.

3 Let us denote µI the mobility that would be observed due toionized impurity scattering alone.

4 If temperature increases, the random thermal velocity ofcarriers increases. This reduces the time a carrier spend in thevicinity of ionized impurity center. Lesser time spend in thevicinity of Coulomb force, lesser would be the scattering effectand, thus, larger the expected value of µI .

5 If the number of ionized impurity centers increases, theprobability of a carrier encountering the ionized impuritycenter increases. This implies smaller value of µI .


Joint effect of µL and µI

1 Probability of a lattice scattering event occurring in adifferential time dt is dt

τL, where τL is mean time between

collision due to lattice scattering.2 Similarly, probability of an ionized impurity scattering event

occurring in a differential time dt is dtτI

, where τI is mean timebetween collision due to ionized impurity scattering.

3 If these two events are independent, the total probability is

dt

τ=

dt

τL=

dt

τI(9)

4 We know that τ = µm∗q , so we can write

1

τ=

1

τL=

1

τI(10)

5 With two or more independent scattering mechanisms, the netmobility decreases. Independent mobilities possess relationlike parallel resistors.


Conductivity

1 The total drift current density can be written as

Jdrift = q(µnn + µpp)E = σE (11)

2 The σ is the conductivity (Ω-cm−1) of the semiconductormaterial.

3 Since, mobility is function of impurity concentrations;conductivity is a complicated function of impurityconcentration. Resistivity (Ω-cm) is given by

σ =1

q(µnn + µpp)(12)

4 Suppose, we apply a voltage V on a bar of length L andcross-section area A. The current density and electric field (Ein Volt/cm)is given by the following expression

J =I

Aand E =

V

L(13)


Conductivity

1 Use J and E in equation of conductivity

I

A= σ

V

L⇒ V =

ρL

AI ⇒ V = IR (14)

2 The V-I characteristics is linear upto a voltage V. This linearregion is called ohmic region where current increases withincrease in voltage. After a voltage threshold, currentsaturates because of velocity saturation.


Velocity Saturation

1 We assumed that drift velocity is not a function of electricfield. It means that the drift velocity increases linearly until acollision occurs. At low electric field, drift velocity varieslinearly with applied field.

2 The slope of drift velocity and applied electric field defines themobility. In case of Si, the velocity increases but saturatesafter a certain value of applied field. If drift velocity saturates,drift current also saturates.

3 For GaAs, the velocity-field characteristics shows differentbehavior. After a certain applied field, velocity does notsaturate but it starts decreasing. This produces negativedifferential mobility and hence negative resistance. Thisparticular characteristic is used in the design of oscillators.

4 TO BE READ: Understand the E-k diagram for furtherclarifications on negative resistance.


V-I Characteristics

1 The first two regions are ohmic and saturation. If appliedvoltage is further increase even after reaching saturationregion, electrons gain very large energy. When they collidewith semiconductor atoms, they release more electrons. Thesecarriers are called excess carriers. This process is known asimpact ionization.

2 For same donor and acceptor concentrations, the drift currentis higher in case of n-type semiconductor as compared top-type semiconductor. This is due to higher electron mobilityand lower electron effective mass.


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Carrier Transport Phenomena Chapter 4[Neamen]