Pis’ma v ZhETF, vol. 111, iss. 3, pp. 173 - 174

February 10

Joule-Thomson cooling in graphene

K.Zarembo¹⁾

Nordita, Kungliga Tekniska Högskolan Royal Institute of Technology and Stockholm University, SE-106 91 Stockholm, Sweden

Niels Bohr Institute, Copenhagen University, 2100 Copenhagen, Denmark

Institute of Theoretical and Experimental Physics, 117218 Moscow, Russia

Submitted 31 August 2019

Resubmitted 20 December 2019

Accepted 22

December 2019

DOI: 10.31857/S0370274X20030066

Collective effects of electron interactions may pre-

lation ǫ + P = µn + T s, where s is the entropy density,

vail over impurity scattering in a very clean sample.

then yields

Electrons then flow like a viscous fluid [1], as observed

A+T^∂ŝ∂µ

α=-

(2)

in graphene in a certain range of temperatures [2-7].

Aŝ+ T^∂ŝ

∂T

Spectacular consequences of hydrodynamic electronic

where ŝ = s/n is specific entropy and A = 1. The ra-

transport include negative local resistivity [2, 3, 4, 7],

tionale to introduce a special notation for a constant

violation of ballistic bound on conductance [2, 6, 8, 9]

equal to one is that a more accurate calculation gives

(Gurzhi effect [1]), breakdown of Wiedemann-Franz law

A ≈ 2/3, while the overall functional form of the cooling

[5, 9, 10], and negative magnetoresistance [9, 11]. An-

coefficient remains the same.

other possible manifestation of hydrodynamic mode of

Electrons in graphene form a 2d Fermi gas with lin-

transport, discussed here, is cooling of electron fluid that

ear dispersion relation, and their pressure is given by

passes through a narrow constriction. In the solid state

∫

∑ (

)

setting electric current normally generates heat, and

d²p

qµ-vF |p|

P = 4T

1+ e

(3)

cooling of electron flow may look counterintuitive, but in

(2πℏ)²

q=±

fluid mechanics this phenomenon, the Joule-Thomson

(JT) effect [12], is well known and underlies a widely

where v_F is the Fermi velocity, q labels particles/holes,

used method of gas refrigeration.

and the overall factor of four takes into account valley

Consider two strips of graphene connected by a nar-

and spin degeneracy. We tacitly assume that holes and

row bridge and subject to a constant voltage δU. As-

electrons are in thermodynamic equilibrium, which is

suming that one strip is kept at temperature T₁ and

not a good approximation at the neutrality point, where

denoting electron temperature in the other by T₂, the

our derivation is not applicable.

cooling/heating effect can be characterized by the tem-

The rest of thermodynamic quantities can be calcu-

perature drop δT = T1 - T2 relative to the potential

lated from dP = ndµ + sdT . When applied to (2) the

difference δµ = µ₁ - µ₂ = eδU:

standard thermodynamic machinery gives

3AFF′

δT = αδµ,

(1)

- ξ,

ξ=

(4)

A+2-3FF′′

F^′2

where

where µ₁ and µ₂ are chemical potentials on the two sides

F (ξ) = Li₃(- e^ξ) + Li₃(- e^-ξ).

(5)

of the bridge. The dimensionless coefficient α can take

either sign and is defined such that α > 0 corresponds

In the two limiting cases we get:

to cooling.

3Aµ

Textbook derivations of the JT effect start with the

^µ≫T≃

^µ≪T≃ -

(6)

2(1 + A)π²T

(1 + A)µ

enthalpy conservation: δ[(ǫ + P )/n] = 0, where ǫ and P

are the energy density and pressure of the electron fluid

The JT effect results in cooling in the Fermi liquid

and n is the charge carrier density. Thermodynamic re-

regime (µ ≫ T) and in heating in the Dirac fluid case

(µ ≪ T ). The sign of the effect changes at an inversion

point, which for the physical case of A = 2/3 lies at

¹⁾e-mail: zarembo@nordita.org

µ_inv = 3.32T.

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2020

173

174

K.Zarembo

The thermodynamic derivation of the JT effect rests

where L is the total size of the system.

on enthalpy conservation, but enthalpy production in

In conclusion, hydrodynamic nature of electron flow

the moving electron fluid cannot be neglected. The

in biased graphene leads to JT cooling when the cur-

viscous heating turns out to be the prime source of

rent is forced through a narrow constriction. Cooling

enthalpy production and leads to order one effects,

occurs in the Fermi liquid regime, for sufficiently large

while Ohmic resistivity only gives small corrections. The

charge imbalance or at sufficiently low temperatures.

derivation is based on the hydrodynamic theory of elec-

For lower chemical potentials the JT effect results in

tron transport in the commonly used Stockes approxi-

heating, which is most pronounced in the Dirac liq-

mation:

uid regime at µ ≪ T . Although similar to conventional

Joule heating the mechanics behind this effect is quite

η∂²v_i -

v_i = ∂_iP,

∂_ivⁱ = 0,

(7)

λ²

different, in particular the temperature increment is lin-

where v_i is the electron velocity, η is the shear viscosity

ear in applied voltage and not quadratic. It is necessary

to keep in mind however that our derivation becomes

of the electron fluid and λ is the momentum-relaxation

invalid close to the neutrality point.

length.

The solution is known explicitly when λ = ∞ [8]:

Full text of the paper is published in JETP Letters

4ηu

journal. DOI: 10.1134/S0021364020030030

P =P₀ -

√

a²

-z²

2uy

v_x =

√

R. Gurzhi, Sov. Phys. JETP 17, 521(1963).

a²

-z²

I. Torre, A. Tomadin, A. K. Geim and M. Polini, Phys.

√

Rev. B 92, 165433 (2015); 1508.00363.

2uy

v_y =

-z² -

√

(8)

L. Levitov and G. Falkovich, Nat. Phys. 12, 672 (2016);

a²

-z²

1508.00836.

D. A. Bandurin, I. Torre, R. K. Kumar, M. Ben Shalom,

Here z = x+iy, a is the width of the bridge, and u is the

A. Tomadin, A. Principi, G. H. Auton, E. Khestanova,

maximal velocity attained by the fluid. The square root

K. S. Novoselov, I. V. Grigorieva, L. A. Ponomarenko,

is analytic on the complex plane with a semi-infinite cut

A. K. Geim, and M. Polini, Science 351, 1055 (2016);

representing the constriction.

1509.04165.

The flow is sustained by the pressure drop:

J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim,

8ηu

A. Lucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watan-

δP =

(9)

abe, T. A. Ohki, and K. C. Fong, Science 351, 1058

(2016); 1509.04713.

The entropy production rate in the moving fluid is cal-

culated according to

R. K. Kumar, D. A. Bandurin, F. M. D. Pellegrino

)

et al. (Collaboration), Nat. Phys. 13,

1182

(2017);

Π²

vⁱ∂_iŝ =

Π_ij = ∂_iv_j + ∂_jv_i.

(10)

1703.06672.

nT λ²

D. A. Bandurin, A. V. Shytov, L. S. Levitov, R. K. Ku-

Integrating the rate along the midflow (and setting

mar, A. I. Berdyugin, M. Ben Shalom, I. V. Grigorieva,

A. K. Geim, and G. Falkovich, Nat. Commun. 9, 4533

λ = ∞) gives:

(2018); 1806.03231.

∫

2ηua

dy y²

16ηu

H. Guo, E. Ilseven, G. Falkovich, and L. S. Levi-

-δŝ =

δP.

(11)

(a2

tov, Proc. Natl. Acad. Sci. USA 114, 3068 (2017);

3anT

3nT

+y²

-∞

1607.07269.

The relation δP = nδµ + sδT then results in the same

J. Gooth, F. Menges, N. Kumar, V. Sü, C. Shekhar,

formula (2) for the JT coefficient, but with A = 2/3.

Y. Sun, U. Drechsler, R. Zierold, C. Felser, and B. Gots-

mann, Nat. Commun. 9, 4093 (2018); 1706.05925.

When momentum relaxation is taken into account,

the coefficient A starts to depend on the dimensionless

10.

A. Lucas, J. Crossno, K. C. Fong, P. Kim, and

ratio a/λ. Assuming that this ratio is small, a ≪ λ, cor-

S. Sachdev, Phys. Rev. B 93, 075426 (2016); 1510.01738.

rections due to momentum relaxation and Ohmic heat-

11.

A. Lucas, R. A. Davison, and S. Sachdev, Proc. Natl.

ing appear to be quadratic in the small parameter a/λ,

Acad. Sci. USA 113, 9463 (2016); 1604.08598.

but are logarithmically enhanced:

12.

W. Thomson and J. P. Joule, Philos. Trans. R. Soc. Lon-

(

)

don 143, 357 (1853).

a²

(12)

32λ²

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2020