The limits of rechargeable spin battery A. V. Yanovsky1and P. V. Pyshkin2y 1B. Verkin Institute for Low Temperature Physics and Engineering of the National

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The limits of rechargeable spin battery

A. V. Yanovsky1, ∗and P. V. Pyshkin2, †

1B. Verkin Institute for Low Temperature Physics and Engineering of the National

Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkov 61103, Ukraine

2Department of Physical Chemistry, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain

We discuss how the ideal rechargeable energy accumulator can be made, and what are the limits

for solid state energy storage. We show that in theory the spin batteries based on heavy fermions

can surpass the chemical ones by energy capacitance. The absence of chemical reactions in spin

batteries makes them more stable, also they don’t need to be heated in cold conditions. We study

how carriers statistics and density of states aﬀect energy capacity of the battery. Also, we discuss

hypothetical spin batteries based on neutron stars.

I. INTRODUCTION

Rechargeable electric batteries are one of the most im-

portant devices of modern civilization. It is obvious that

their role will only increase in the future. Unfortunately,

the existing chemical rechargeable batteries (based on

reversible electrochemical reactions) are far from ideal.

This manifests itself, for example, in their inevitable ir-

reversible degradation, slow charging, relatively low en-

ergy capacitance per unit mass, the need for heating

when the temperature drops, etc. Of course, the progress

doesn’t stop, however the most eﬀorts now are applied to

chemistry and physical chemistry properties of batteries

(see1,2). Relatively new physical idea of quantum battery

explores quantum state and entanglement properties3–6.

The idea of using spin degree of freedom to store energy

attracts a lot of attention last years7–10. Particularly,

in recent article11 authors propose a spin battery (SB)

which is half-metal spin valve with suppressed spin ﬂips

of conducting electrons. This solution would allow us to

store reversibly the electric energy without any chemical

reactions at the charging process using nonequilibrium

states of quasiparticles in a conductor instead.

Hence, the following question appears: is it possible for

SB to surpass chemical battery, and what are the prop-

erties of ideal SB? In this paper we theoretically “test”

possible limits of solid state for SB-s and more exotic

matter of neutron stars as well.

II. CHEMICAL POTENTIAL

The energy in SB is stored in spin carriers’ density de-

viation from its equilibrium value under condition that

spin relaxation is suppressed in this conductor. In other

words, such a battery is just a certain volume ﬁlled with

spin particles, and energy accumulation appears due to

a development of the non-equilibrium spin state. Such

spin particles can be electrons in conductor11, whereas

being charge carriers, and their spin direction ±can

be determined by the external magnetic ﬁeld or mag-

netic contacts. In order to charge such a battery con-

taining charged spin carriers or to transfer accumulated

non-equilibrium spin concentration into charge current

H H

FIG. 1. Schematic illustration of spin battery (denoted by

“SB” in picture) which is a conductor between half-metal

electrodes (denoted by “H” in picture) with opposite spin

polarization.

one can use for instance antiparallel magnetized half-

metal12–14 electrodes which passes only + or −spin cor-

respondingly11, see Fig. 1.

In this situation the charging potential diﬀerence δϕ

induces variations of chemical potentials of ±compo-

nents (after the charging time when the equilibrium es-

tablished):

µ±=µ0,±+η±,

where µ0,±are equilibrium electrochemical potentials of

the discharged battery determined by density of corre-

spondent ±components. Chemical potentials η±are

induced by charging process, and their values could be

found from the conditions δϕ =η+/q+−η−/q−(where

q±are electric charges of corresponded carriers), together

with spatial electroneutrality which follows from Poisson

equation ∆ϕ=−4π{q+[ρ+(µ+)−ρ+(µ0)]+q−[ρ−(µ−)−

ρ−(µ0)]}(with RHS equal to zero):

q+[ρ+(µ+)−ρ+(µ0)] + q−[ρ−(µ−)−ρ−(µ0)] = 0 (1)

where ϕis electrical potential, ρ±are the densities, q±

are charges of correspondent spin ±carriers. Here, for

simplicity, we assume µ0,+=µ0,−=µ0, as it is obvious

that the equilibrium value of electrochemical potential

does not aﬀect the basic principles of energy storage.

For one-band conductor we have q+=q−=e, where

eis the electron charge. Moreover, we can consider the

arXiv:2210.14029v1 [cond-mat.mes-hall] 25 Oct 2022

FIG. 2. (a) One-band “purely-electronic” spin battery. (b)

Two-band “electron-holes” spin battery. Filled levels are

drawn in gray color. The number of carriers of one spin de-

creases when the number of carriers of another spin increases

for one-band battery, and polarity is not important. We chose

polarity of two-band battery in such a way to have increasing

of both types of carriers during charging process.

usage of two-band conductors which contain not only the

electrons but also “holes” with opposite charge. What is

more, the carriers are polarized in such a way as to spin ±

connected to charge ±. In this situation the charges q±

have the opposite sign: q+=−q−=e. This is possible

when interaction between carriers from diﬀerent bands

is weak15, or by using electron-hole pairing methods16–18

for the spin-ﬂip suppressing mentioned in11. When the

one-band battery is being charged it follows to increas-

ing the number of certain spin carriers, and it necessarily

leads to decreasing the number of opposite spin carriers in

order to preserve electroneutrality (1). In two-band bat-

tery we always chose polarity of charging voltage in such

a way to have increasing of the number of carriers of both

spins (η+>0). It means that we connect positive-guided

contact to the electrode with “+”-polarity (correspond-

ing to “holes” with charge q+), and negative-guided con-

tact to the electrode with “−”-polarity (corresponding to

electrons with charge q−). Non-equilibrium states caused

by deviations of spin densities in one-band and two-band

SB-s are shown in Fig. 2. Here we show schematically

the ﬁlling energy levels εas functions of corresponded

densities of states (DOS) D±for spins ±.

As can be seen, two-band battery has polarity, and

such SB is equal to chemical battery but with the follow-

ing diﬀerence. In chemical battery we have the concen-

trations changing, and correspondingly the changing of

chemical potentials with respect to electrodes, while q±

corresponds to the one ion charge: −q+=q−=−zF/Na,

where zis positive valency, Fis Faraday constant, and

Nais Avogadro number (for deﬁniteness we chose the

sign the same as elementary charge has, we assume the

same absolute value of valency of all ions). In the ab-

sence of charging potential diﬀerence in the circuit, the

presence of non-equilibrium chemical potentials leads to

appearance of diﬀusion forces, which pull-in or push-out

charges into the circuit. It happens on electrodes of the

opposite “aﬃnity” (such an aﬃnity is related to chem-

ical reactions in usual chemical battery, or it is related

to the presence of conduction band only for certain spin

on ±electrodes in the SB). Asymmetry of charge mov-

ing during relaxation into thermodynamic equilibrium

state causes electric current in the full circuit11. Also,

we can consider SB with spin/charge carriers are not be-

ing usual conductive electrons but quasi-particles. Such

quasi-particles even can have zero electric charge, but in

this situation the movement of such quasi-particles do

not cause electric current, and thus the energy extrac-

tion from this battery is diﬃcult. SB doesn’t require

chemical reactions, and therefore it does not suﬀer from

chemical degradation. As we show below, SB doesn’t re-

quire heating in the case it consists of degenerate gas of

charge/spin carriers. Obviously, SB can be a source not

only of charge but also of spin current19–2122. At last, SB

can be charged without electrodes with using polarized

electromagnetic radiation23.

III. GENERAL FORMULAS FOR THE ENERGY

OF CHARGED SB

Let us denote E±(µ±) as the total internal energy of

carriers of ±components for given value of electrochem-

ical potential µ±. The energy stored in the battery is a

diﬀerence between the internal energies of charged and

discharged states

δE =E+(µ0+η+)+E−(µ0+η−)−E+(µ0)−E−(µ0).(2)

At microscopic level the value of Ein SB as well in chem-

ical battery is determined by equilibrium energy distri-

bution of carriers n±(ε, µ), DOS D±(ε) and by the vol-

ume Ω:

E±(µ±)=Ω

∞

dεεD±(ε)n±(ε, µ±).(3)

Also, we can write the following expression for ρ±in order

to substitute it in (1)

ρ±(µ±) =

∞

dεD±(ε)n±(ε, µ±).(4)

As can be seen from (2) and (3), the energy distribution

determines which parts of the DOS dependencies D±(ε)

give the main contribution. Of course, the energy dis-

tribution can be classical (Boltzmann), Bose, Fermi or,

even, so-called fractional statistics.

Note, usually, εD(ε) increases with ε, and on the other

hand Bose distributions collect particles in states with

lower energies where DOS has its minimal values. Of

course, DOS of Bose distributions can have certain sin-

gularities (see24), thus some low dimensional systems re-

quire special investigation. However, in general case, if

we do not consider non-physical DOS which diverge at

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ThelimitsofrechargeablespinbatteryA.V.Yanovsky1,andP.V.Pyshkin2,y1B.VerkinInstituteforLowTemperaturePhysicsandEngineeringoftheNationalAcademyofSciencesofUkraine,47NaukyAve.,Kharkov61103,Ukraine2DepartmentofPhysicalChemistry,UniversityoftheBasqueCountryUPV/EHU,48080Bilbao,SpainWediscusshowtheidealre...

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The limits of rechargeable spin battery A. V. Yanovsky1and P. V. Pyshkin2y 1B. Verkin Institute for Low Temperature Physics and Engineering of the National

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