S. A. Vitkalov Department of Physics, City College of the City - PowerPoint PPT Presentation

Quantal Heating in Electron Systems with Discrete Spectrum. S. A. Vitkalov Department of Physics, City College of the City University of New York 10031, USA e-mail: vitkalov@gmail.com New York City College of Technology. 11/7/2013

CCNY Microwave Lab Graduate students: Scott Dietrich, Sean Byrnes, William Mayer PhDs: J. Q. Zhang , Natalia Romero Samples • High electron mobility GaAs quantum wells • LaSrCuO high temperature superconductors Facilities • Liquid Helium system with liquefier • He3 Probe (0.3K) • Nanofabrication Lab

Experimental setup 50  m × 250  m Hall bar 6 GaAs QW J 7 1 2 E H 3 4 B At strong magnetic field current density J is almost perpendicular to electric field E=(E XX , E H ): E XX << E H Typical samples: N1:  1=0.93x10 6 cm 2 /Vs; n1~12x10 11 cm -2 N2:  2=0.82x10 6 cm 2 /Vs; n2~8.5x10 11 cm -2

Quantal heating We have observed a particular kind of Joule heating, which occurs in conducting quantum systems. The quantal heating has extraordinary properties and provides extreme violation of the Ohm’s Law in normal metals. The heating may not increase the electron “temperature” and is qualitatively different from the heating in classical systems. It results in nontrivial spectral distribution of electrons, radical change of the electron transport, and transition of the electrons into a state, in which voltage (current) does not depend on current (voltage). The phenomena were recently observed in a conductivity of two dimensional electrons placed in quantizing magnetic fields. Supported by National Science Foundation: DMR 1104503

Electron spectrum Classical conductors: Quantum conductors: quantized spectrum continuous spectrum 1 2 h  C DOS; 2D; B=0T DOS  F 1 2 1 0 0 -4 -3 -2 -1 0 1 2 3 4 -4 -2 0 2 4  meV  meV

1 2 Difference between dc heating (red curve) and regular heating (black curve). Arrow marks the magnetic field above which a quantization of electron spectrum (Landau levels) appears. J. Q. Zhang, S. A. Vitkalov, and A. A. Bykov, Phys. Rev. B 80 , 045310 (2009)

50 2 1 Heating by T bath Resistance (Ohm) 48 T=2K ; I=0 µA 46  T~ 2K 44 Classical Joule Heating T=2K; I=6 µA 42 Quantal Heating 40 0.0 0.1 0.2 (B) Magnetic field (T) Difference between dc heating (red curve) and regular heating (black curve). Arrow marks the magnetic field above which the quantization of electron spectrum (Landau levels) appears.

The nonlinearity is so strong that the 2D electron system forms a state with zero differential resistance (ZDR) A. A. Bykov, J. Q. Zhang, S. Vitkalov, A. K. Kalagin and A. K. Bakarov, Phys. Rev. Lett. 99 , 116801 (2007).

In presence of dc electric field E stochastic elastic electron scattering on impurities induces stochastic variations of electron kinetic energy  K (t) - spectral diffusion . Quantal heating is result of the diffusion through the quantized spectrum.

Spectral Diffusion : equation for distribution function f in electric field E is Drude conductivity in magnetic field is dimensionless density of states Nonlinear conductivity is where I. A. Dmitriev, M. G. Vavilov, I. L.Aleiner, A. D. Mirlin, and D.G. Polyakov, Phys. Rev. B 71, 115316 (2005)

Quantal Heating: stratification of electron distribution T bath =8 K The electron distribution f is divided on regions with fast ( between Landau levels) and slow ( inside the levels) variations with energy. Slow variation of f indicates a “heating” whereas fast variation of f corresponds to a “cooling” Such peculiar electron distribution results in extraordinary decrease of electron conductivity with dc bias Strong “overheating” inside Landau levels and “overcooling” between them

Conditions for Quantal Heating    D   in =   2.0 1.8 c / ) in =  2 (R 2 2 1.6 (eE) 1.4 1.2  =h/(  q ); 1.0 DOS 0.8  q - electron lifetime 0.6 0.4  0.2 0.0 -20 0 20 Electron energy

Property , which is understood: variation of the resistance with dc bias Phys. Rev. B 80 , 045310 (2009)

Property , which is not understood: apparent “ M- I” transition Phys. Rev. B 80 , 045310 (2009)

Property, which is not understood: relation between apparent “ M- I” transition and ZDR state: why V MI =V ZDR ?

RECENT DEVELOPMENT T=2.1 K +A 0.1815 +B 0.1536 0.1480 0.1369 2.0 0.1257 0.1146 0.1034 R XX ,  0.09225 0.08109 0.06994 0.05878 0.04763 0.03647 B [T] 0.03089 0.02531 80 +C 0.01416 0.003000 1.5 60 40 20 0 0 0 0.33 1.0 1 -80 -40 0 40 80 J [A/m] Idc [  A] 0.32 B [T] 2 3 (a) 3 0.31 B = 0.847 T Corbino Disc 2 -C -B +A -A +B +C g 12 (mS) 0.6 1 T = 4.2 K 0 j=-1 j=1 T = 1.6 K E th ZDCS 0.4 B[T] -1 -600 -300 0 300 600 E dc (V/m) 120 j=2 (b) 0.2 80 r xx (  ) B = 0.841 T Hall Bar j=3 40 T =4.2 K 0.0 0 -4 -2 0 2 4 T = 1.6 K ZDRS I th J[A/m] -40 -50 -25 0 25 50 I dc (  A)

Competing nonlinear mechanism for separated levels: One band populated -SdH 𝜐 𝑟 =4 ps 𝜐 𝑟 =1 ps 0.1705 0.1509 0.1815 0.1411 0.1536 0.1313 0.1480 0.1214 2.0 0.1369 2.0 0.1116 0.1257 0.1067 0.1146 0.1018 0.1034 0.09200 0.09225 0.08219 0.08109 0.07238 0.06747 0.06994 0.06256 0.05878 0.05275 0.04763 0.04784 0.03647 B [T] 0.04294 0.03089 B [T] 0.03313 0.02531 0.02331 0.01416 0.01350 0.003000 1.5 1.5 1.0 1.0 -80 -40 0 40 80 -80 -40 0 40 80 Idc [  A] Idc [  A] Scott Dietrich, Sean Byrnes, Sergey Vitkalov, D. V. Dmitriev, and A. A. Bykov, Phys. Rev. B 85, 155307 (2012)

0.1705 2.0 2.0 0.1411 0.1214 B [T] 0.1067 B [T] 0.09200 0.07238 1.5 1.5 0.06256 0.04784 0.03313 a1) a) 0.01350 1.0 1.0 -80 -40 0 40 80 -80 -40 0 40 80 Idc [  A] Idc [  A] R XX [k  ] R XX [k  ] 0.2 0.2 T=4.77 K T=4.20 K b) b1) 0.1 0.1 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 B [T] B [T] 0.10 0.10 R XX [k  ] R XX [k  ] B=0.6T B=0.6T c1) 0.05 0.05 c) 0.00 0.00 -80 -40 0 40 80 -80 -40 0 40 80 Idc [  A] Idc [  A] R XX [k  ] R XX [k  ] 0.15 0.15 B=2.0T B=2.0T 0.10 d) 0.10 d1) 0.05 0.05 0.00 0.00 -80 -40 0 40 80 -80 -40 0 40 80 Idc [  A] Idc [  A] R XX [k  ] R XX [k  ] 0.15 0.15 B=1.87 T B=1.87 T 0.10 0.10 e1) e) 0.05 0.05 0.00 0.00 -80 -40 0 40 80 -80 -40 0 40 80 Idc [  A] Idc [  A]

 (a)  -Si  -Si AlAs GaAs 2D V H SL V H U  U[a.u.] y y o E F (b) U X - - - SL +++ Density E H  n(y) d eff  - - - X X +++ 2DEG - - - SL +++ -20 -15 -10 -5 0 5 10 15 20 Z [nm]

Boundary condition: 𝐾 𝑧 =0 Quantum oscillations Potential in the capacitor A. A. Shashkin, V. T. Dolgopolov, and S. I. Dorozhkin, Sov. Phys. JETP 64 , 1124 (1986). M. I. Dyakonov, Solid State Commun. 78 , 817 (1991).

Test #1 For 𝐽 0 =35 𝜈𝐵 𝑒 𝑓𝑔𝑔 =36 nm which is comparable with the width of the screening layers 27 and 76 nm

Two populated subbands: Magneto-Inter-Subband Oscillations: MISO . 1 2 h  C P 90 T=4.35 K 80 R XX [  ] 70 h  C / 2 60 M 1 2 0.0 0.2 0.4 B [T] Dependence of the resistance on magnetic field with no dc bias applied. Sample N1.

Landau – Zener transitions inside lowest subband j=0 -2 -1 2 1 0.16 (a) 0.14 0.12 B [T] 0.10 0.08 0.06 0.04 0.02 -4 -2 0 2 4 3 -3 2 -2 70 R xx [  ] 1 -1 j=0 (b) B=0.12T 60 -4 -2 0 2 4 J [A/m]

T=2.1 K +A +B R XX ,  80 +C 60 40 20 0 0 0 0.33 1 J [A/m] 0.32 B [T] 2 3 0.31 Stronger magnetic fields . Dependence of resistance on magnetic field and current density J . Labels +A, +B and +C indicate different maxima induced by dc bias

-C -B -A O +A +B +C 0.44 (a) 0.42 0.40 B [T] 0.38 0.36 0.34 -4 -2 0 2 4 80 R xx [  ] 60 B=0.418T (b) 40 B=0.408T 20 -4 -2 0 2 4 J [A/m] Dependence of resistance on magnetic field B and current density J , indicating correlation of features ±A and ±C with MISO minima and features ±B with MISO maxima.

B=0.532 T R XX [  ] B=0.548 T +C 160 -C + _ +B _ +A -B -A 80 _ _ + + J=0 A/m J=3.03 A/m J[A/m] -2 0 2 B[T] 0.6 0.5 j=-1 0.4 B C 0.3 j=-2 j=1 0.2 j=2 0.1 j=2 R XX [  ] 0.0 80 160

-C -B +A -A +B +C 0.6 j=-1 j=1 0.4 B[T] j=2 0.2 j=3 0.0 -4 -2 0 2 4 J[A/m] Scott Dietrich, Sean Byrnes, Sergey Vitkalov, A. V. Goran, and A. A. Bykov Phys. Rev. B 86 , 075471 (2012)

Zero-differential conductance state 12 I 12 = I ac + I dc 3 (a) ~ V ac Corbino Disc B = 0.847 T 2 8 1 g 12 (mS) g 12 (mS) V dc 1 2 T = 4.2 K 4 0 E dc = 0 T = 1.6 K ZDCS E th -1 -600 -300 0 300 600 0 E dc (V/m) T = 1.6 K E dc = 250 V/m 120 (b) 0.2 0.4 0.6 0.8 1.0 B (T) 80 r xx (  ) B = 0.841 T Hall Bar 40 T =4.2 K 0 T = 1.6 K ZDRS I th -40 -50 -25 0 25 50 I dc (  A) PHYSICAL REVIEW B 87 , 081409(R) (2013) Zero-differential conductance of two-dimensional electrons in crossed electric and magnetic fields A. A. Bykov,* Sean Byrnes, † Scott Dietrich, † and Sergey Vitkalov ‡ I. V. Marchishin and D. V. Dmitriev