Kinetic inductance

Kinetic inductance characterizes the contribution to the energy of electric current due to the kinetic energy of current carriers, in addition to the energy of the magnetic field (which is characterized by magnetic or geometric inductance ) ^[1]

F_{k}=\int n{\frac {mv^{2}}{2}}dV={\frac {L_{K}I^{2}}{2}},

{\ displaystyle F_ {k} = \ int n {\ frac {mv ^ {2}} {2}} dV = {\ frac {L_ {K} I ^ {2}} {2}},}

{\ displaystyle F_ {k} = \ int n {\ frac {mv ^ {2}} {2}} dV = {\ frac {L_ {K} I ^ {2}} {2}},}

,

where the integral is taken over the volume of the conductor, n , m , v - concentration, mass and speed of current carriers, I - total current in the conductor.

As a rule, kinetic inductance can be neglected compared to conventional, due to the small kinetic energy of electrons compared to electromagnetic energy. However, at optical frequencies and in the case of a superconductor, this is no longer the case. For example, for sufficiently thin superconducting wires, the kinetic inductance can make a noticeable or even decisive contribution to the inductance ^[2] .

Content

Superconductors

The kinetic inductance of the wire can be obtained by equating the kinetic energy of Cooper pairs and the equivalent inductive energy:

(mv^{2})(n_{s}lA)={\frac {1}{2}}L_{K}I^{2}

{\ displaystyle (mv ^ {2}) (n_ {s} lA) = {\ frac {1} {2}} L_ {K} I ^ {2}}

what gives ^[3]

L_{K}={\frac {ml}{2n_{s}e^{2}A}}

{\ displaystyle L_ {K} = {\ frac {ml} {2n_ {s} e ^ {2} A}}}

,

where A and l are the cross-sectional area of the wire and its length, n _s is the concentration of Cooper pairs, m and e are the electron mass and charge. This formula is valid for the case when the wire diameter is much less than the penetration depth (in ordinary superconductors - several hundred angstroms), that is, for superconducting nanowires with a diameter of ~ 10 nm ^[4] ^[5] ^[6] .

Since the concentration of Cooper pairs depends on temperature ( T ), in the framework of the Ginzburg – Landau theory, the kinetic inductance will depend on the temperature L _K ( T ) = L _K (0) (1- T / T _c ) ⁻¹ , where T _c - critical temperature of transition to normal state ^[3] .

Two-dimensional electron gas

The conductivity of a two-dimensional electron gas at a frequency ω in the Drude model is written as

\sigma ={\frac {\sigma _{0}}{1+j\omega \tau }},

{\ displaystyle \ sigma = {\ frac {\ sigma _ {0}} {1 + j \ omega \ tau}},}

where j is the imaginary unit, $\sigma _{0}=ne\mu$ ${\ displaystyle \ sigma _ {0} = ne \ mu}$ - low-frequency conductivity, τ - pulse relaxation time, n - 2DEG concentration, e - elementary electric charge , μ - current carrier mobility . When considering the impedance of a 2DEG with a width W and a length L

Z={\frac {L}{W\sigma }}=R+j\omega L_{K}.

{\ displaystyle Z = {\ frac {L} {W \ sigma}} = R + j \ omega L_ {K}.}

The coefficient in the imaginary part of the impedance at a frequency is called the kinetic inductance, by analogy with the magnetic part, which also comes in the form of a multiplier to the frequency. The kinetic inductance for 2DEG is

L_{K}={\frac {\tau }{\sigma _{0}}}{\frac {L}{W}}={\frac {m^{*}}{ne^{2}}}{\frac {L}{W}}

{\ displaystyle L_ {K} = {\ frac {\ tau} {\ sigma _ {0}}} {\ frac {L} {W}} = {\ frac {m ^ {*}} {ne ^ {2 }}} {\ frac {L} {W}}}

and depends on the concentration and effective mass ( m ^* ) of the carriers. It is assumed here that μ = e τ / m ^* . This part of the inductance is connected in series with the geometric inductance, therefore, at a sufficiently low concentration of electrons it can exceed the last. The equivalent circuit of the field effect transistor at high frequencies presented in the form of a transmission line with losses should take into account precisely this part of the inductance, which was demonstrated in an experiment on highly mobile 2DEGs ^[7] .

Notes

↑ Schmidt V.V. Introduction to the physics of superconductors. Moscow, Nauka, 1982. - 238 p., § 10. See also VV Schmidt, The Physics of Superconductors: Introduction to Fundamentals and Applications (Springer 1997)
↑ Annunziata AJ et. al. Variables of superconducting nanoinductance // Nanotechnology. - 2010 .-- T. 21 . - S. 445202 . - DOI : 10.1088 / 0957-4484 / 21/44/445202 . - arXiv : 1007.4187 .
↑ ¹ ² Annunziata, 2012.
↑ JT Peltonen et al., ArXiv: 1305.6692.
↑ OV Astafiev et al., Nature 484, 355–358 (2012).
↑ C. Schuck et al., Sci. Rep. 3, 1893 (2013).
↑ Burke, 2000 .

Literature

Champlin, KS, Armstrong DB, Gunderson PD Inertia of a charge carrier in semiconductors = Charge carrier inertia in semiconductors // Proceedings of the IEEE. - 1964. - Vol. 52 . - P. 677-685 . - DOI : 10.1109 / PROC.1964.3049 .
Burke PJ, Spielman IB, Eisenstein JP, Pfeiffer LN, West KW Conductivity of a high-mobility two-dimensional electron gas at high frequencies (English) = High frequency conductivity of the high-mobility two-dimensional electron gas // Appl. Phys. Lett .. - 2000. - Vol. 76 . - P. 745-747 . - DOI : 10.1063 / 1.125881 . - arXiv : http://core.kmi.open.ac.uk/display/4870668 . (inaccessible link)
S. Gordyunin. Ideal conductors and kinetic inductance // Quantum . - M .: Bureau "Quantum", 1996. - No. 4 . - S. 40–41 . (inaccessible link)

[1] Schmidt V.V. Introduction to the physics of superconductors. Moscow, Nauka, 1982. - 238 p., § 10. See also VV Schmidt, The Physics of Superconductors: Introduction to Fundamentals and Applications (Springer 1997)

[2] Annunziata AJ et. al. Variables of superconducting nanoinductance // Nanotechnology. - 2010 .-- T. 21 . - S. 445202 . - DOI : 10.1088 / 0957-4484 / 21/44/445202 . - arXiv : 1007.4187 .

[Annunziata,_2012.-3] ¹ ² Annunziata, 2012.

[4] JT Peltonen et al., ArXiv: 1305.6692.

[5] OV Astafiev et al., Nature 484, 355–358 (2012).

[6] C. Schuck et al., Sci. Rep. 3, 1893 (2013).

[_5c9c3d17df398bb4-7] Burke, 2000 .