Topic 10-2 effective mass and holes - Solid State...
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Colorado School of Mines Solid State Physics in a Nutshell solidstate.mines.edu Topic 11-2: Effective Mass and Introduction to Holes Summary: In this video we aim to see how an intrinsic semiconductor acts in an electric field. We begin by deriving an expression for the force on an electron within an intrinsic semiconductor in an electric field. Next we introduce and explore a new term called the effective mass. Finally, we get an expression for electrical conductivity of both the conduction and valence bands. We use these to introduce holes and compare the conduction and valence bands. • Goal: see how intrinsic semiconductors behave in an electric field • Previously we showed that a free electron will be shifted by dk in an electric field and will thus be accelerated o ! = ħ ! !! ! ! • Now we need to look in the context of the semiconductor dispersion instead of a parabolic dispersion • Build up the force for a semiconductor F = ma [1] F= -e ε [2] o F= force on electron, e=charge, ε= external field • From before we have an expression for !" !" from an external electric field o Work= force times distance o In an external field: !" = −!" ∙ ! ! !" [3] ! ! = !" !" [4] ! = ħ! [5] ! ! = ! ħ !" !" [6] !" = !!" ħ !" !" !" [7] ħ !" !" = −!" [8] ! = ħ !" !" [9]
Transcript of Topic 10-2 effective mass and holes - Solid State...
Colorado School of Mines Solid State Physics in a Nutshell solidstate.mines.edu
Topic 11-2: Effective Mass and Introduction to Holes
Summary: In this video we aim to see how an intrinsic semiconductor acts in an electric field.
We begin by deriving an expression for the force on an electron within an intrinsic
semiconductor in an electric field. Next we introduce and explore a new term called the effective
mass. Finally, we get an expression for electrical conductivity of both the conduction and
valence bands. We use these to introduce holes and compare the conduction and valence bands.
• Goal: see how intrinsic semiconductors behave in an electric field
• Previously we showed that a free electron will be shifted by dk in an electric field and
will thus be accelerated
o ! = ħ!
!!!!
• Now we need to look in the context of the semiconductor dispersion instead of a
parabolic dispersion
• Build up the force for a semiconductor
F = ma [1]
F= -e ε [2]
o F= force on electron, e=charge, ε= external field
• From before we have an expression for !"!"
from an external electric field
o Work= force times distance
o In an external field:
!" = −!" ∙ !!!" [3]
!! =!"!"
[4]
! = ħ! [5]
!! =!ħ!"!"
[6]
!" = !!"ħ
!"!"!" [7]
ħ !"!"= −!" [8]
! = ħ !"!"
[9]
Colorado School of Mines Solid State Physics in a Nutshell solidstate.mines.edu
• Should be a mass involved????
• Define effective mass: not a real mass but it describes how an electrons group velocity
changes in an electric field
!!"∙ !! =
!ħ!!!!"!#
∙ !"!"
[10]
! = !ħ!!!!!!
∙ !"!"
[11]
! = !ħ!
!!!!!!
[12]
!!∗ =
!ħ!
!!!!!!
[13]
where m* is the effective mass
!!∗ is a convention, since we treated k as our independent variable people
tend to expect to see k on the bottom of the derivative
• Back to the free electron model
• Check this approach by seeing if it gives that the effective mass is the rest mass of an
electron within the free electron model
o It should
! = ħ!
!!!! [14]
!!!!!!
= ħ!
!! [15]
!!∗ =
!ħ!
!!!!!!
= !ħ!( ħ
!
!!) [16]
o m* = me
• m* depends on ! and band index
• Near the bottom of the conduction band of a semiconductor we can fit a parabola
Colorado School of Mines Solid State Physics in a Nutshell solidstate.mines.edu
• Define a conduction band m* that assumes the electrons are just at the bottom of the
conduction band
• Example: GaAs
o m* is about 0.06 me for the conduction band edge (if you fit the band edge with a
parabolic) which is very light!
o For comparison and electron in a vacuum is about 1 me
• How can an electron in a crystal accelerate faster than and electron in a vacuum
o Wave particle duality
o Not talking about a physical electron moving, we are actually talking about the
group velocity of a wave moving
• Need an expression for conductivity from the conduction band
o Recall from the free electron model ! = !!!!!!
o Since m* defines how an electron behaves in an electric field !! =!!!!!!!!∗
o !! is the conduction band conductivity
o ne is the electron concentration in the conduction band
o !! is the electron relaxation time
• Now deal with the conduction band m*
• Still assume a parabola but this one is pointed down so the double derivative is negative
• For this band the electrons have a net momentum parallel to the electric field
o In the conduction band the momentum was antiparallel
• In other words in the valence band vg is with the electric field
Colorado School of Mines Solid State Physics in a Nutshell solidstate.mines.edu
• Can also treat this as the holes have a positive charge and therefore they move with the
electric field as expected
• In summary the valence band electrons accelerate in the wrong direction for negatively
charged particles in an electric field as if the mass is negative in F= ma
• Valence band contributes to conductivity as much as the conduction band
• Conduction band
o m* : positive
o acceleration: positive
o charge: negative
• Valence band (holes)
o m* : positive
o acceleration: negative
o charge: positive
• Net current density
o je- (conduction band) to the left
o jh+ (valence band) to the left
o have a net current density to the left as expected
Colorado School of Mines Solid State Physics in a Nutshell solidstate.mines.edu
Questions to Ponder
1. Compare and contrast the shape of bands with heavy and light effective mass.
2. If we increase our Fourier coefficients what will happen to the effective mass of the conduction and valance bands?
3. Sketch the k space for the conduction and valance bands at some finite temperature for a 2D intrinsic semiconductor with a square lattice. Then apply an electric field pointing to the left and describe how these states shift over time.
