Negative magnetoresistance in Poiseuille flow Negative magnetoresistance in Poiseuille flow

of two-dimensional electronsof two-dimensional electrons

P. S. Alekseev 1 and M. I. Dyakonov 2

1 A. F. Ioffe Physico-Technical Institute, St. Petersburg, Russia

2 Université Montpellier 2, CNRS, France

Outline:

• Experiments

• Viscous flow of electronic fluid

• Decrease of viscosity in magnetic field

• Interpretation of experimental magnetoresistance data

• Prediction: temperature and magnetic field dependent Hall resistance

• Unresolved problems

Goal

Recently, several groups reported strong negative

magnetoresistance in 2D electron gas at low temperatures and

moderate magnetic fields.

« huge » « colossal » « giant »

So far, there is no explanation of these results

We propose a new mechanism, which might be responsible

(However, we still have some problems)

Brief review of experimental results

Giant negative magnetoresistance in high-mobility 2D electron systems A.T. Hatke, M.A. Zudov, J.L. Reno, L.N. Pfeiffer, K.W. West

Phys. Rev. B 85, 081304(R) (2012)

Size-dependent giant magnetoresistance in millimeter scale GaAs/AlGaAs 2D electron devices. R. G. Mani, A. Kriisa, and W. Wegscheider (2013)

L. Bockhorn, A. Hodaei, D. Schuh, W. Wegscheider, R. J. Haug

HMF-20, Journal of Physics: Conference Series 456 (2013) 012003

« We observe for each sample geometry a strong negative magnetoresistance

around zero magnetic field which consists of a peak around zero magnetic field and of

a huge magnetoresistance at larger fields».

Colossal negative magnetoresistance in a 2D electron gas

Q. Shi, P.D. Martin, Q.A. Ebner, M.A. Zudov, L.N. Pfeiffer, K.W. West (2014)

Announcing our main ideas

1) The resistance might be due to the viscosity of the electronic fluid

Then resistivity is proportional to viscosity

eec2) The viscosity decreases in magnetic field on the scale defined by

As a consequence, negative magnetoresistance appears

3) There should be a corresponding correction to the Hall resistance

Electronic viscosity

Fv - Fermi velocity, ee - electron-electron collision time

The idea of a viscous flow of electronic fluid was put forward by Gurzhi more than 50 years ago:

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

R. N. Gurzhi and S. I. Shevchenko, Sov. Phys. JETP 27, 1019 (1968)

R. N. Gurzhi, Sov. Phys. Uspekhi 94, 657 (1968)

eeFv 221

For degenerate electrons at low temperatures 2

1

Tee

Viscosity is relevant when the mean free path lee = vFτee is << sample width w

L. W. Molenkamp and M. J. M. de Jong, Phys. Rev. B 49, 5038 (1994)

R. N. Gurzhi, A. N. Kalinenko, and A. I. Kopeliovich, Phys. Rev. Lett., 72, 3872 (1995)

H. Buhmann et al, Low Temp. Phys. 24, 737 (1998)

H. Predel et al, Phys. Rev. B 62, 2057 (2000)

Z. Qian and G. Vignale, Phys. Rev. B 71, 075112 (2005)

A. Tomadin, G. Vignale, and M. Polini, Phys. Rev. Lett. 113, 235901 (2014)

More recently, this idea was discussed in connection with 2D transport

lee

lph

Calculated e-e and e-ph mean free paths as functions of temperature

Viscous flow of electronic fluid in 2D

wx

y

E

Em

e

y

v

t

v

2

2

Boundary condition:2

at 0)(w

yyv

(Poiseuille parabolic profile)

Jean Léonard Marie Poiseuille (1797 – 1869)

Steady state solution:

2

2

42y

w

m

eEyv

(total current ~ w3)

*ne

m

2

12

2w*

Pure viscous resistivity

22

11

T,

w Unusual temperature dependence!

These results are modified if the momentum relaxation time τ due to interaction

with phonons and static defects is comparable to τ*. In this case, the usual friction

term −v/ τ should be added to the right-hand side of the Navier-Stocks equation.

,

Taking in account electron viscosity

AND

scattering by phonons and defects [Gurzhi-Shevchenko (1968)]

12

2w*

)tanh(

1

12

ne

m

ll

w

ee

*

2

3

Interestingly, this formula can be replaced (with an accuracy better than 12%) by:

,ne

m

11

*2

Which means that the effect of viscosity can be considered as

a parrallel channel of electron momentum relaxation !

(Here l is the mean free path for scattering by phonons and defects)

Calculated resistivity at B=0 as a function of temperature

Poiseuille flow regime – below the minimum at ~ 8K

Main point: decrease of viscosity in magnetic field

Like other kinetic coeeficients, e.g. conductivity, in magnetic field the

viscosity becomes a tensor with B-dependent components

20

)2(1 eecyyxx

2

0

)2(1

)2(

eec

eecyxxy

THE VISCOSITY OF A PLASMA IN A STRONG MAGNETIC FIELD

Yu. M. Aliev

Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 19-26, 1965

In magnetic field electrons carry their momentum to adjacent layers on a

smaller distance. Thus the internal friction (viscosity) must diminish

Physical reason for the decrease of viscosity in magnetic field

Equations for viscous electronic liquid in electric and magnetic fields

xy

xyx

xxycx E

m

e

y

v

y

vv

t

v

2

2

2

2

yx

xyy

xxxcy E

m

e

y

v

y

vv

t

v

2

2

2

2

02

2

xx

xx Em

e

y

v

02

2

yxcx

xy Em

ev

y

v

Under stationary conditions and in the absence of Hall

The first equation says that the resistance is given by previous formulas,

where η is replaced by ηxx (which decreases with magnetic field!!).

The second equation serves for finding the Hall field Ey

2at 0)(

wyyvx current vy = 0 for all y, while

,RRc

*ee

HH

2

ee

)0(

)2(1

121

nec

BRH )0(

12

2w*

New prediction: correction to Hall resistance

(depending on sample width, magnetic field, and temperature)

This is for pure viscous flow!

(Terms – v/τ are ignored)

Calculated resistivity as function of magnetic field for different temperatures,

assuming 1/τee ~ T2 down to zero temperature + phonon scattering

Calculated resistivity as function of magnetic field for different temperatures assuming 1/τee = aT2+b (b is a fitting parameter) + phonon scattering

T = 1, 5, 9, 12, 15, 18, 21, 24, 27, 30 K

Comparison of our calculations with

the experimental results of Shi et al

experimental

« theoretical »

Hall resistance calculated with 1/τee =aT^2+b

Problems

1. To fit the experimental data reasonably well we need to assume that

τee remains finite in the limit T 0

2. We also need to assume that electron-phonon scattering time τph

behaves as 1/T down to very low temperatures

(this was already noted by Q. Shi et al (2014))

Conclusions: our theory in a nutshell

2

2

)2(1

1 12

2

eec

ee*ee

w

l

, 2

21

eeFv ,weec

* 22

2112

xy*ee

cx*x E

m

evv

dt

dv

2

111

yx*ee

cy*y E

m

evv

dt

dv

2

111

Simplified Drude-like equations:

2

200

)2(1

1 121

21

eec

eeH*

eeHH w

lRRR

, 11

2

*ne

m

nec

BRH 0

Results

That’s the end

Thank you!