General case Effective mass (solid-state physics)

1 general case

1.1 inertial effective mass tensor
1.2 cyclotron effective mass
1.3 density of states effective masses (lightly doped semiconductors)

general case

in general dispersion relation cannot approximated parabolic, , in such cases effective mass should precisely defined if used @ all. here commonly stated definition of effective mass inertial effective mass tensor defined below; however, in general matrix-valued function of wavevector, , more complex band structure itself. other effective masses more relevant directly measurable phenomena.

inertial effective mass tensor

a classical particle under influence of force accelerates according newton s second law, = mf. intuitive principle appears identically in semiclassical approximations derived band structure. however, each of symbols has modified meaning; acceleration becomes rate of change in group velocity:

a

=


d

d
⁡
t






v


g


=


d

d
⁡
t




(

∇

k



ω

(

k

)

)

=

∇

k





d
⁡
ω

(

k

)



d
⁡
t



=

∇

k



(



d
⁡

k



d
⁡
t



⋅

∇

k



ω
(

k

)
)

,


{\displaystyle \mathbf {a} ={\frac {\operatorname {d} }{\operatorname {d} t}}\,\mathbf {v} _{\text{g}}={\frac {\operatorname {d} }{\operatorname {d} t}}\left(\nabla _{k}\,\omega \left(\mathbf {k} \right)\right)=\nabla _{k}{\frac {\operatorname {d} \omega \left(\mathbf {k} \right)}{\operatorname {d} t}}=\nabla _{k}\left({\frac {\operatorname {d} \mathbf {k} }{\operatorname {d} t}}\cdot \nabla _{k}\,\omega (\mathbf {k} )\right),}

where ∇k del operator in reciprocal space, , force gives rate of change in crystal momentum pcrystal:

f

=



d
⁡


p


crystal




d
⁡
t



=
ℏ



d
⁡

k



d
⁡
t



,


{\displaystyle \mathbf {f} ={\frac {\operatorname {d} \mathbf {p} _{\text{crystal}}}{\operatorname {d} t}}=\hbar {\frac {\operatorname {d} \mathbf {k} }{\operatorname {d} t}},}

where ħ = h/2π reduced planck constant. combining these 2 equations yields

a

=

∇

k



(



f

ℏ


⋅

∇

k



ω
(

k

)
)

.


{\displaystyle \mathbf {a} =\nabla _{k}\left({\frac {\mathbf {f} }{\hbar }}\cdot \nabla _{k}\,\omega (\mathbf {k} )\right).}

extracting ith element both sides gives

a

i


=

(


1
ℏ







∂

2


ω

(

k

)



∂

k

i


∂

k

j





)



f

j


=

(


1

ℏ

2









∂

2


e

(

k

)



∂

k

i


∂

k

j





)



f

j


,


{\displaystyle a_{i}=\left({\frac {1}{\hbar }}\,{\frac {\partial ^{2}\omega \left(\mathbf {k} \right)}{\partial k_{i}\partial k_{j}}}\right)\!f_{j}=\left({\frac {1}{\hbar ^{2}}}\,{\frac {\partial ^{2}e\left(\mathbf {k} \right)}{\partial k_{i}\partial k_{j}}}\right)\!f_{j},}

where ai ith element of a, fj jth element of f, ki , kj ith , jth elements of k, respectively, , e total energy of particle according planck–einstein relation. index j contracted use of einstein notation (there implicit summation on j). since newton s second law uses inertial mass (not gravitational mass), can identify inverse of mass in equation above tensor

[

m

inert


−
1


]


i
j


=


1

ℏ

2








∂

2


e


∂

k

i


∂

k

j






.


{\displaystyle \left[m_{\text{inert}}^{-1}\right]_{ij}={\frac {1}{\hbar ^{2}}}{\frac {\partial ^{2}e}{\partial k_{i}\partial k_{j}}}\,.}

this tensor expresses change in group velocity due change in crystal momentum. inverse, minert, known effective mass tensor.

the inertial expression effective mass commonly used, note properties can counter-intuitive:

the effective mass tensor varies depending on k, meaning mass of particle changes after subject impulse. cases in remains constant of parabolic bands, described above.
the effective mass tensor diverges (becomes infinite) linear dispersion relations, such photons or electrons in graphene. (these particles said massless, refers having 0 rest mass; rest mass distinct concept effective mass.)

cyclotron effective mass

classically, charged particle in magnetic field moves in helix along magnetic field axis. period t of motion depends on mass m , charge e,

t
=

|



2
π
m


e
b



|



{\displaystyle t=\left\vert {\frac {2\pi m}{eb}}\right\vert }

where b magnetic flux density.

for particles in asymmetrical band structures, particle no longer moves in helix, motion transverse magnetic field still moves in closed loop (not circle). moreover, time complete 1 of these loops still varies inversely magnetic field, , possible define cyclotron effective mass measured period, using above equation.

the semiclassical motion of particle can described closed loop in k-space. throughout loop, particle maintains constant energy, constant momentum along magnetic field axis. defining k-space area enclosed loop (this area depends on energy e, direction of magnetic field, , on-axis wavevector kb), can shown cyclotron effective mass depends on band structure via derivative of area in energy:

m

∗



(
e
,



b
^



,

k



b
^




)

=



ℏ

2



2
π



⋅


∂

∂
e



a

(
e
,



b
^



,

k



b
^




)



{\displaystyle m^{*}\left(e,{\hat {b}},k_{\hat {b}}\right)={\frac {\hbar ^{2}}{2\pi }}\cdot {\frac {\partial }{\partial e}}a\left(e,{\hat {b}},k_{\hat {b}}\right)}

typically, experiments measure cyclotron motion (cyclotron resonance, de haas–van alphen effect, etc.) restricted probe motion energies near fermi level.

in two-dimensional electron gases, cyclotron effective mass defined 1 magnetic field direction (perpendicular) , out-of-plane wavevector drops out. cyclotron effective mass therefore function of energy, , turns out related density of states @ energy via relation




g
(
e
)

=





g

v



m

∗




π

ℏ

2








{\displaystyle \scriptstyle g(e)\;=\;{\frac {g_{v}m^{*}}{\pi \hbar ^{2}}}}

, gv valley degeneracy. such simple relationship not apply in three-dimensional materials.

density of states effective masses (lightly doped semiconductors)

in semiconductors low levels of doping, electron concentration in conduction band in general given by

n

e


=

n

c


exp
⁡

(
−




e

c


−

e

f




k
t



)



{\displaystyle n_{e}=n_{c}\exp \left(-{\frac {e_{\text{c}}-e_{\text{f}}}{kt}}\right)}

where ef fermi level, ec minimum energy of conduction band, , nc concentration coefficient depends on temperature. above relationship ne can shown apply conduction band shape (including non-parabolic, asymmetric bands), provided doping weak (ec-ef >> kt); consequence of fermi–dirac statistics limiting towards maxwell–boltzmann statistics.

the concept of effective mass useful model temperature dependence of nc, thereby allowing above relationship used on range of temperatures. in idealized three-dimensional material parabolic band, concentration coefficient given by

n

c


=
2


(



2
π

m

e


∗


k
t


h

2




)



3
2





{\displaystyle \quad n_{c}=2\left({\frac {2\pi m_{e}^{*}kt}{h^{2}}}\right)^{\frac {3}{2}}}

in semiconductors non-simple band structures, relationship used define effective mass, known density of states effective mass of electrons. name density of states effective mass used since above expression nc derived via density of states parabolic band.

in practice, effective mass extracted in way not quite constant in temperature (nc not vary t). in silicon, example, effective mass varies few percent between absolute 0 , room temperature because band structure changes in shape. these band structure distortions result of changes in electron-phonon interaction energies, lattice s thermal expansion playing minor role.

similarly, number of holes in valence band, , density of states effective mass of holes defined by:

n

h


=

n

v


exp
⁡

(
−




e

f


−

e

v




k
t



)

,


n

v


=
2


(



2
π

m

h


∗


k
t


h

2




)



3
2





{\displaystyle n_{h}=n_{v}\exp \left(-{\frac {e_{\text{f}}-e_{\text{v}}}{kt}}\right),\quad n_{v}=2\left({\frac {2\pi m_{h}^{*}kt}{h^{2}}}\right)^{\frac {3}{2}}}

where ev maximum energy of valence band. practically, effective mass tends vary between absolute 0 , room temperature in many materials (e.g., factor of 2 in silicon), there multiple valence bands distinct , non-parabolic character, peaking near same energy.