Lecture 3

Electron and Hole Transport in

In this lecture you will learn:

• How and holes move in semiconductors

• Thermal motion of electrons and holes

• via drift

• Electric current via

• resistors

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

Review: Electrons and Holes in Semiconductors

holes +

electrons + + As Donor A crystal lattice + impurity + As atoms

+

● There are two types of mobile charges in semiconductors: electrons and holes

In an intrinsic (or undoped) semiconductor density equals hole density

● Semiconductors can be doped in two ways:

N-: to increase the electron density P-doping: to increase the hole density

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

1 Thermal Motion of Electrons and Holes In thermal equilibrium carriers (i.e. electrons or holes) are not standing still but are moving around in the crystal lattice because of their thermal energy

The root-mean-square velocity of electrons can be found by equating their kinetic energy to the thermal energy:

1 1 m v 2  KT 2 n th 2 KT  vth  mn

In pure Silicon at room temperature:

7 vth ~ 10 cm s

Brownian Motion

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

Thermal Motion of Electrons and Holes In thermal equilibrium carriers (i.e. electrons or holes) are not standing still but are moving around in the crystal lattice and undergoing collisions with:

• vibrating Silicon atoms • with other electrons and holes • with dopant atoms (donors or acceptors) and other impurity atoms

Mean time between collisions = c

In between two successive collisions electrons (or holes) move with an average velocity which is called the thermal velocity = vth 12 In pure Silicon, c  0.110 s  0.1 ps 7 vth  10 cm s

Brownian Motion Mean distance traveled between collisions is called the    vth  c

In pure Silicon,   107  0.11012  10-6 cm  0.01m

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

2 Drift: Motion of Electrons Under an Applied

Silicon slab

E

L V E  L +-V

• Force on an electron because of the electric field = Fn = -qE

• The electron moves in the direction opposite to the applied field with a constant equal to vdn

• The electron drift velocity vdn is proportional to the electric field strength vdn   E  vdn   n E cm2 • The constant n is called the . It has units: cm2 V - s • In pure Silicon,   1500 n V - s

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

Drift: Motion of Holes Under an Applied Electric Field

Silicon slab

E

L V E  L +-V

• Force on a hole because of the electric field = Fp = qE

• The hole moves in the direction of the applied field with a constant drift velocity equal to vdp

• The hole drift velocity vdp is proportional to the electric field strength vdp  E  vdp  p E cm2 • The constant p is called the hole mobility. It has units: cm2 V - s • In pure Silicon,   500 p V - s

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

3 Derivation of Expressions for Mobility Electrons: Force on an electron because of the electric field  Fn  qE F q E Acceleration of the electron  a  n   mn mn Since the mean time between collisions is  c , the acceleration lasts only for a time period of  c before a collision completely destroys electron’s velocity

q c So in time  c electron’s velocity reaches a value  a c   E mn

q c This is the average drift velocity of the electron, i.e. vdn   E mn q c Comparing with v dn    n E we get, n  mn Holes: q c Similarly for holes one gets, p  mp

Special note: Masses of electrons and holes (mn and mp) in Solids are not the same as the mass of electrons in free space which equals 9.11031 kg

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

Mobility Vs Doping

More doping (n-type of p-type) means more frequent collisions with charged donor and acceptor impurity atoms and this lowers the carrier mobility /V-s) 2 Mobility (cm Mobility

Dopant concentration (1/cm3)

Note: Doping in the above figure can either be n-type or p-type

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

4 Density of Electrons

Consider electrons moving under an applied electric field: E

vdn   n E

Flux Density: Flux density is the number of particles crossing a unit area surface per It has units cm-2-s-1

Unit area surface

Density: n

Velocity: vdn

Area Time Flux density: nvdn Volume = 1 x (vdn x 1)

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

Drift of Electrons

Electrons Drift Current Density: Check directions Electron flux density from drift  n v dn E

vdn Electron drift current density drift is, Jn drift Jn

drift vdn   n E Jn  q  electron flux density drift  q n vdn  q n n E Jn  q n n E

drift Coulombs Amps Jn has units:  cm2 - s cm2

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

5 Drift Current Density of Holes

Holes Drift Current Density: Check directions The hole drift current density is J drift , p E drift v Jqp  hole flux density dp drift qpvdp  qp p E Jp

vdp  p E drift Jp  q p p E

drift Coulombs Amps Jp has units:  cm2 - s cm2

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

Conductivity and Resistivity

Total Drift Current Density: drift drift drift The total drift current density J is the sum of J n and Jp drift drift drift J  Jn  Jp

 q n n  p p E   E The quantity  is the conductivity of the semiconductor:

  q n n  p p 

Conductivity describes how much current flows when an electric field is applied. Another related quantity is the resistivity  which is the inverse of the conductivity, 1    Units of conductivity are: Ohm-1-cm-1 or -1-cm-1 or S-cm-1 Units of resistivity are: Ohm-cm or -cm or S-1-cm

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

6 Example: A Semiconductor Resistor Silicon slab For a resistor we know that, V I  1 R L - V We also know that, + Area  A I  Jdrift A   E A I V  A   A  V 2 L L L  L 1 1  2  R   where    q n n  p p  A A  Lessons: • Knowing electron and hole densities and mobilities, one can calculate the electrical resistance of semiconductors

• n-doping or p-doping can be used to change the conductivity of semiconductors by orders of magnitudes

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

Diffusion

Diffusion of ink in a glass beaker

Why does diffusion happen?

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

7 Diffusion and Diffusivity

There is another mechanism by which current flows in semiconductors …….

• Suppose the electron density inside a semiconductor is not uniform in space, as shown below n(x) electron flux in +x direction electron flux in -x direction d nx slope  dx x • Since the electrons move about randomly in all directions (Brownian motion), as time goes on more electrons will move from regions of higher electron density to regions of lower electron density than the electrons that move from regions of lower electron density to regions of higher electron density d nx • Net electron flux density in +x direction   dx d nx  D n dx 2 -1 • The constant Dn is called the diffusivity of electrons (units: cm -s )

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

Diffusion Current Density

Electrons Density: d nx Check directions Electron flux density from diffusion  Dn dx nx

diff Electron diffusion current density J n is, diff x Jn  q  electron flux density d nx Elec. flux  q Dn dx diff Jn Holes Diffusion Current Density: d px Check directions Hole flux density from diffusion  Dp dx px

diff Hole diffusion current density J p is, diff x Jp  q  hole flux d px Hole flux  q Dp dx diff Jp

diff diff Coulombs Amps Jn and J p has units  cm2 - s cm2 ECE 315 – Spring 2005 – Farhan Rana – Cornell University

8 Einstein Relations

Einstein worked on other things besides the theory of relativity……..

• We introduced two material constants related to carrier transport: 1) Mobility 2) Diffusivity

•Both are connected with the transport of carriers (electrons or holes)

•It turns out that their values are related by the Einstein relationships

Einstein Relation for Electrons: D K T Example: n  2 In pure Silicon, n  1500 cm V - s n q   500 cm2 V - s Einstein Relation for Holes: p 2 Dp K T This implies, Dn  37.5 cm s  D  12.5 cm2 s p q p

• K is the Boltzman constant and its value is: 1.38x10-23 Joules K • KT has a value equal to 0.0258 at room temperature (at 300 oK) q

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

Total Electron and Hole Current Densities

Total electron and hole current densities is the sum of drift and diffusive components Electrons:

drift diff Jn x  Jn x  Jn x d nx  q nx  E x  q D n n dx

Holes:

drift diff Jp x  Jp x  Jp x d px  q px  E x  q D p p dx

Electric currents are driven by electric fields and also by carrier density gradients

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

9 Thermal Equilibrium - I There cannot be any net electron current or net hole current in thermal equilibrium ……… what does this imply ??

Consider electrons first:

drift diff Jn x  Jn x  Jn x  0 d n x  q n x  E x  q D o  0 1 o n n dx d logn x  q 1 can also be written as: o   Ex dx KT dx Since the electric field is minus the gradient of the potential: Ex   dx d logn x  q dx We have: o  dx KT dx qx KT The solution of the above differential equation is: no x  constant  e

But what is that “constant” in the above equation ???

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

Thermal Equilibrium - II

qx KT We have: no x  constant  e

Note: one can only measure potential differences and not the absolute values of potentials Convention: The potential of pure intrinsic Silicon is used as the reference value and assumed to be equal to zero. q x KT So for intrinsic Silicon, no x  constant  e  constant

But we already know that in intrinsic Silicon, no x  ni

So it must be that, constant  ni qx KT And we get the final answer, no x  ni e

Consider Holes Now: qx  KT One can repeat the above analysis for holes and obtain: po x  ni e

2 Check: no x po x  ni

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

10 Potential of Doped Semiconductors What are the values of potentials in N-doped and P-doped semiconductors ??

N-doped Semiconductors (doping density is Nd ): The potential in n-doped semiconductors is denoted by: n Example: no x  Nd Suppose, qn x KT 17 -3 10 -3  Nd  ni e Nd  10 cm and ni  10 cm KT N     log d KT Nd  n q  n   n  log    0.4 Volts  i  q  ni 

P-doped Semiconductors (doping density is Na ): The potential in p-doped semiconductors is denoted by: p p x  N Example: o a Suppose, qp x  KT 17 -3 10 -3  Na  ni e Na  10 cm and ni  10 cm KT N  KT N      log a a p    p   log    0.4 Volts q  ni  q  ni 

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

ECE 315 – Spring 2005 – Farhan Rana – Cornell University

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