Effective mass

Effective mass is a quantity that has a mass dimension and is used to conveniently describe the motion of a particle in the periodic potential of a crystal . It can be shown that electrons and holes in a crystal react to an electric field as if they were moving freely in vacuum , but with a certain effective mass, which is usually determined in units of the electron mass m _e (9.11 × 10 ⁻³¹ kg ). The effective mass of an electron in a crystal, generally speaking, is different from the mass of an electron in a vacuum and can be either positive or negative ^[1] .

Definition

The velocity of an electron in a crystal is equal to the group velocity of electron waves and is determined by the formula

v_{g}={\frac {d\omega }{dk}}={\frac {1}{\hbar }}{\frac {dE}{dk}}

{\ displaystyle v_ {g} = {\ frac {d \ omega} {dk}} = {\ frac {1} {\ hbar}} {\ frac {dE} {dk}}}

.

Here $\omega$ ${\ displaystyle \ omega}$ - frequency $k$ ${\ displaystyle k}$ - wave vector $E$ ${\ displaystyle E}$ - electron energy. During $d t$ ${\ displaystyle dt}$ external force $F$ ${\ displaystyle F}$ does the work of moving an electron equal to

dE=v_{g}dtF={\frac {F}{\hbar }}{\frac {dE}{dk}}dt

{\ displaystyle dE = v_ {g} dtF = {\ frac {F} {\ hbar}} {\ frac {dE} {dk}} dt}

.

From here we find $F=\hbar {\frac {dk}{dt}}$ ${\ displaystyle F = \ hbar {\ frac {dk} {dt}}}$ . Differentiating $v_{g}$ ${\ displaystyle v_ {g}}$ in time, we determine the electron acceleration

a={\frac {dv_{g}}{dt}}={\frac {1}{\hbar }}{\frac {d^{2}E}{dk^{2}}}{\frac {dk}{dt}}

{\ displaystyle a = {\ frac {dv_ {g}} {dt}} = {\ frac {1} {\ hbar}} {\ frac {d ^ {2} E} {dk ^ {2}}} { \ frac {dk} {dt}}}

.

Substituting here ${\frac {dk}{dt}}$ ${\ displaystyle {\ frac {dk} {dt}}}$ from the formula $F=\hbar {\frac {dk}{dt}}$ ${\ displaystyle F = \ hbar {\ frac {dk} {dt}}}$ we get

a={\frac {1}{\hbar ^{2}}}{\frac {d^{2}E}{dk^{2}}}F

{\ displaystyle a = {\ frac {1} {\ hbar ^ {2}}} {\ frac {d ^ {2} E} {dk ^ {2}}} F}

.

This formula expresses Newton’s second law. $a={\frac {F}{m^{*}}}$ ${\ displaystyle a = {\ frac {F} {m ^ {*}}}}$ . Here $m^{*}$ ${\ displaystyle m ^ {*}}$ - effective mass. Comparing these two formulas, we obtain ^[2] :

m^{*}=\hbar ^{2}\cdot \left[{{d^{2}E} \over {dk^{2}}}\right]^{-1}.

{\ displaystyle m ^ {*} = \ hbar ^ {2} \ cdot \ left [{{d ^ {2} E} \ over {dk ^ {2}}} \ right] ^ {- 1}.}

For a free particle, the dispersion law is quadratic, and thus the effective mass is constant and equal to the rest mass. In a crystal, the situation is more complicated and the dispersion law is different from quadratic. In this case, the concept of mass can be used only near the extrema of the curve of the dispersion law, where this function can be approximated by a parabola and, therefore, the effective mass is independent of energy.

The effective mass depends on the direction in the crystal and is generally a tensor.

There are other approaches for calculating the effective mass of an electron in a crystal ^[3] .

Effective mass tensor is a term in solid state physics that characterizes the complex nature of the effective mass of a quasiparticle ( electron , hole ) in a solid. The tensor nature of the effective mass is illustrated by the fact that the electron in the crystal lattice does not move as a particle with a rest mass , but as a quasiparticle, in which the mass depends on the direction of motion relative to the crystallographic axes of the crystal. The effective mass is introduced when there is a parabolic dispersion law , otherwise the mass begins to depend on energy. In this regard, a negative effective mass is possible.

By definition, the effective mass is found from the dispersion law ^[4] $\varepsilon =\varepsilon ({\vec {k}})$ ${\ displaystyle \ varepsilon = \ varepsilon ({\ vec {k}})}$

m_{ij}^{-1}={\frac {1}{\hbar ^{2}k}}{\frac {\partial \varepsilon }{\partial k}}\delta _{ij}+{\frac {1}{\hbar ^{2}}}\left({\frac {\partial ^{2}\varepsilon }{\partial k^{2}}}-{\frac {1}{k}}{\frac {\partial \varepsilon }{\partial k}}\right){\frac {k_{i}k_{j}}{k^{2}}},\qquad (1)

{\ displaystyle m_ {ij} ^ {- 1} = {\ frac {1} {\ hbar ^ {2} k}} {\ frac {\ partial \ varepsilon} {\ partial k}} \ delta _ {ij} + {\ frac {1} {\ hbar ^ {2}}} \ left ({\ frac {\ partial ^ {2} \ varepsilon} {\ partial k ^ {2}}} - {\ frac {1} { k}} {\ frac {\ partial \ varepsilon} {\ partial k}} \ right) {\ frac {k_ {i} k_ {j}} {k ^ {2}}}, \ qquad (1)}

Where ${\vec {k}}$ ${\ displaystyle {\ vec {k}}}$ - wave vector $\delta _{ij}$ ${\ displaystyle \ delta _ {ij}}$ - Kronecker symbol , $\hbar$ ${\ displaystyle \ hbar}$ - Planck's constant .

Effective mass for some semiconductors

The table below shows ^[5] ^{[6] the} effective mass of electrons and holes for semiconductors - simple substances of group IV and binary compounds A ^III B ^V and A ^II B ^VI .


Material	Effective mass of electrons, m _e	Effective hole mass, m _h
Group IV
Si (4.2 K)	1,08	0.56
Ge	0.55	0.37
A ^III B ^V
Gaas	0,067	0.45
Insb	0.013	0.6
A ^II B ^VI
Znse	0.17	1.44
Zno	0.19	1.44

This site provides the temperature dependence of the effective mass for silicon.

Experimental Definition

Traditionally, effective carrier masses were measured by the cyclotron resonance method, which measures the absorption of a semiconductor in the microwave range of the spectrum as a function of the magnetic field. When the microwave frequency equals the cyclotron frequency $\omega _{c}={\frac {eB}{m^{*}c}},$ ${\ displaystyle \ omega _ {c} = {\ frac {eB} {m ^ {*} c}},}$ a sharp peak is observed in the spectrum. In recent years, effective masses have usually been determined from measuring the band structure using methods such as angular resolution photoemission (ARPES), or the more direct method based on the de Haas - van Alphen effect .

The effective masses can also be estimated using the coefficient γ from the linear term of the low-temperature electron contribution to the heat capacity at a constant volume $c_{v}.$ ${\ displaystyle c_ {v}.}$ The heat capacity depends on the effective mass through the density of states at the Fermi level .

Importance

As the table shows, semiconductor compounds A ^III B ^V , such as GaAs and InSb, have much lower effective masses than semiconductors from the fourth group of the periodic system — silicon and germanium. In the simplest theory of electron transport, Drude, the drift velocity of carriers is inversely proportional to the effective mass: ${\vec {v}}={\begin{Vmatrix}\mu \end{Vmatrix}}\cdot {\vec {E}},$ ${\ displaystyle {\ vec {v}} = {\ begin {Vmatrix} \ mu \ end {Vmatrix}} \ cdot {\ vec {E}},}$ Where ${\begin{Vmatrix}\mu \end{Vmatrix}}={\frac {e\tau }{\begin{Vmatrix}m^{*}\end{Vmatrix}}}$ ${\ displaystyle {\ begin {Vmatrix} \ mu \ end {Vmatrix}} = {\ frac {e \ tau} {\ begin {Vmatrix} m ^ {*} \ end {Vmatrix}}}}$ and $e$ ${\ displaystyle e}$ Is the charge of an electron . The speed of integrated circuits depends on the speed of the carriers, and thus, a small effective mass is one of the reasons that GaAs and other semiconductors of group A ^III B ^V are used instead of silicon in applications where a wide bandwidth is required.

Links

NSM archive
Pastori Parravicini, G. Electronic States and Optical Transitions in Solids. - Pergamon Press, 1975. - ISBN ISBN 0-08-016846-9 . The book contains a comprehensive but accessible discussion of the topic with an extensive comparison between theory and experiment.

Notes

↑ Epifanov, 1971 , p. 137.
↑ Epifanov, 1971 , p. 136.
↑ Pekar S.I. Conduction electrons in crystals // Problems of Theoretical Physics. Collection dedicated to Nikolai Nikolaevich Bogolyubov in connection with his sixtieth birthday. - M., Nauka , 1969. - Circulation 4000 copies. - c. 349—355
↑ Askerov, BM Electron Transport Phenomena in Semiconductors, 5th ed. - Singapore: World Scientific, 1994. - P. 416. - ISBN ISBN 981-02-1283-6 .
↑ Sze SM Physics of Semiconductor Devices . - John Wiley & Sons, 1981. - (Wiley-Interscience publication). - ISBN 9780471056614 .
↑ Harrison WA Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond . - Dover Publications, 1989. - (Dover Books on Physics). - ISBN 9780486660219 .

Literature

Epifanov G. I. Physical foundations of microelectronics. - M .: Soviet Radio, 1971. - 376 p.

[_1be5d21e92403e26-1] Epifanov, 1971 , p. 137.

[_1be5d21e92403e27-2] Epifanov, 1971 , p. 136.

[3] Pekar S.I. Conduction electrons in crystals // Problems of Theoretical Physics. Collection dedicated to Nikolai Nikolaevich Bogolyubov in connection with his sixtieth birthday. - M., Nauka , 1969. - Circulation 4000 copies. - c. 349—355

[4] Askerov, BM Electron Transport Phenomena in Semiconductors, 5th ed. - Singapore: World Scientific, 1994. - P. 416. - ISBN ISBN 981-02-1283-6 .

[5] Sze SM Physics of Semiconductor Devices . - John Wiley & Sons, 1981. - (Wiley-Interscience publication). - ISBN 9780471056614 .

[6] Harrison WA Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond . - Dover Publications, 1989. - (Dover Books on Physics). - ISBN 9780486660219 .