Outline: Experiments Viscous flow of electronic fluid Decrease of viscosity in magnetic field

1. Negative magnetoresistance in Poiseuille flow of two-dimensional electronsP. S. Alekseev 1 and M. I. Dyakonov 21A. F. Ioffe Physico-Technical Institute, St. Petersburg, Russia2Université 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

2. So far, there is no explanation of these results We propose a new mechanism, which might be responsible (However, we still have some problems) Goal Recently, several groups reported strong negative magnetoresistance in 2D electron gas at low temperatures and moderate magnetic fields. « colossal » « giant » « huge »

3. Brief review of experimental results

4. 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)

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

6. 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».

7. 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)

8. 2) The viscosity decreases in magnetic field on the scale defined by As a consequence, negative magnetoresistance appears Announcing our main ideas • The resistance might be due to the viscosity of the electronic fluid • Then resistivity is proportional to viscosity 3) There should be a corresponding correction to the Hall resistance

9. - Fermi velocity, - electron-electron collision time For degenerate electrons at low temperatures Electronic viscosity Viscosity is relevant when the mean free path lee = vFτee is << sample width w 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)

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

11. lph lee Calculated e-e and e-ph mean free paths as functions of temperature

12. y w x Jean Léonard Marie Poiseuille (1797 – 1869) E Boundary condition: (total current ~ w3) Steady state solution: (Poiseuille parabolic profile) Viscous flow of electronic fluid in 2D

13. Pure viscous resistivity Unusual temperature dependence! These results are modified if the momentum relaxation time τdue to interaction with phonons and static defects is comparable to τ*. In this case, the usual friction term −v/τshould be added to the right-hand side of the Navier-Stocks equation.

14. Interestingly, this formula can be replaced (with an accuracy better than 12%) by: Which means that the effect of viscosity can be considered as a parrallel channel of electron momentum relaxation ! Taking in account electron viscosity AND scattering by phonons and defects [Gurzhi-Shevchenko (1968)] , (Herelis the mean free path for scattering by phonons and defects)

15. Calculated resistivity at B=0 as a function of temperature Poiseuille flow regime – below the minimum at ~ 8K

16. 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

17. 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

19. Physical reason for the decrease of viscosity in magnetic field In magnetic field electrons carry their momentum to adjacent layers on a smaller distance. Thus the internal friction (viscosity) must diminish

20. Equations for viscous electronic liquid in electric and magnetic fields

21. current vy = 0 for all y, while Under stationary conditions and in the absence of Hall The first equation says that the resistance is given by previous formulas, whereη is replaced by ηxx (which decreases with magnetic field!!). The second equation serves for finding the Hall fieldEy

22. 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)

23. Calculated resistivity as function of magnetic field for different temperatures, assuming1/τee ~ T2down to zero temperature + phonon scattering

24. T = 1, 5, 9, 12, 15, 18, 21, 24, 27, 30 K Calculated resistivity as function of magnetic field for different temperatures assuming1/τee = aT2+b(b is a fitting parameter) + phonon scattering

25. Comparison of our calculations with the experimental results of Shi et al experimental « theoretical »

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

27. 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))

28. Simplified Drude-like equations: Results Conclusions: our theory in a nutshell

29. That’s the end Thank you!