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

N1: 1=0.93x106 cm2/Vs; n1~12x1011 cm-2 N2: 2=0.82x106 cm2/Vs; n2~8.5x1011 cm-2 GaAs QW B J

1 2 3 4 6 7

50m×250m Hall bar

EH

At strong magnetic field current density J is almost perpendicular to electric field E=(EXX , EH ): EXX << EH

Typical samples:

Experimental setup

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: continuous spectrum Quantum conductors: quantized spectrum

• 4
• 2

2 4 1 2

meV

DOS

hC

• 4 -3 -2 -1 0 1 2 3 4

1

meV

DOS; 2D; B=0T

F

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)

2 1

0.0 0.1 0.2 40 42 44 46 48 50

Classical Joule Heating Quantal Heating

Resistance (Ohm)

Magnetic field (T) (B)

Heating by T bath

T~ 2K

Difference between dc heating (red curve) and regular heating (black curve). Arrow marks the magnetic field above which the quantization

• f electron spectrum (Landau levels) appears.

1

2

T=2K ; I=0 µA T=2K; I=6 µA

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

• n 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

Strong “overheating” inside Landau levels and “overcooling” between them

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

• f f corresponds to a “cooling”

Such peculiar electron distribution results in extraordinary decrease of electron conductivity with dc bias

Tbath=8 K

• 20

20 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

DOS Electron energy





Din=



(eE)

2(R 2 c/)in= 2

=h/(q); q-electron lifetime

Conditions for Quantal Heating

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 VMI=VZDR ?

RECENT DEVELOPMENT

• 80
• 40

40 80 1.0 1.5 2.0

Idc [A] B [T]

1 2 3 20 40 60 80 0.31 0.32 0.33

RXX,  B [T] J [A/m] +A +B +C T=2.1 K

• 4
• 2

2 4 0.0 0.2 0.4 0.6

j=-1 +C +B +A

• A
• B

j=1 j=2

B[T] J[A/m]

j=3

• C
• 50
• 25

25 50

• 40

40 80 120

Idc (A) rxx ()

(b)

ZDRS T = 1.6 K B = 0.841 T T =4.2 K Ith Hall Bar

• 600
• 300

300 600

• 1

1 2 3

g12 (mS) Edc (V/m) B = 0.847 T T = 4.2 K T = 1.6 K

(a)

Eth ZDCS Corbino Disc

Competing nonlinear mechanism for separated levels: One band populated -SdH

• 80
• 40

40 80 1.0 1.5 2.0

Idc [A] B [T]

0.01350 0.02331 0.03313 0.04294 0.04784 0.05275 0.06256 0.06747 0.07238 0.08219 0.09200 0.1018 0.1067 0.1116 0.1214 0.1313 0.1411 0.1509 0.1705

• 80
• 40

40 80 1.0 1.5 2.0

Idc [A] B [T]

0.003000 0.01416 0.02531 0.03089 0.03647 0.04763 0.05878 0.06994 0.08109 0.09225 0.1034 0.1146 0.1257 0.1369 0.1480 0.1536 0.1815

𝜐𝑟=1 ps 𝜐𝑟=4 ps

Scott Dietrich, Sean Byrnes, Sergey Vitkalov, D. V. Dmitriev, and A. A. Bykov, Phys. Rev. B 85, 155307 (2012)

0.0 0.5 1.0 1.5 2.0 0.0 0.1 0.2

• 80
• 40

40 80 0.00 0.05 0.10 0.15 0.0 0.5 1.0 1.5 2.0 0.0 0.1 0.2

• 80
• 40

40 80 0.00 0.05 0.10 0.15

• 80
• 40

40 80 0.00 0.05 0.10

• 80
• 40

40 80 0.00 0.05 0.10 0.15

• 80
• 40

40 80 0.00 0.05 0.10 0.15

• 80
• 40

40 80 0.00 0.05 0.10

• 80
• 40

40 80 1.0 1.5 2.0 B [T] Idc [A] B [T]

• 80
• 40

40 80 1.0 1.5 2.0 Idc [A]

0.01350 0.03313 0.04784 0.06256 0.07238 0.09200 0.1067 0.1214 0.1411 0.1705

b1)

RXX [k]

T=4.20 K d1)

RXX [k] Idc [A] B=2.0T

c) b)

B [T] B [T] RXX [k]

T=4.77 K d)

RXX [k] Idc [A] B=2.0T B=0.6T B=0.6T

e)

RXX [k] B=1.87 T

e1)

RXX [k] B=1.87 T

c1) a1)

Idc [A] Idc [A] RXX [k] Idc [A] RXX [k] Idc [A]

a)

• 20
• 15
• 10
• 5

5 10 15 20

UX X GaAs

-Si

Z [nm] Density

-Si

AlAs X



U EF U[a.u.]

yo

SL

• - -

(b)

 V

SL H

V

2D H

y

(a)

• - - SL

2DEG

n(y)

EH

deff

+++ +++ +++

• - -

• 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).

Boundary condition: 𝐾𝑧=0 Quantum oscillations Potential in the capacitor

For 𝐽0=35 𝜈𝐵 𝑒𝑓𝑔𝑔=36 nm which is comparable with the width of the screening layers 27 and 76 nm Test #1

Dependence of the resistance on magnetic field with no dc bias applied. Sample N1.

0.0 0.2 0.4 60 70 80 90

M

2

B [T] RXX [] T=4.35 K

1 2 1

P

hC hC/2

Two populated subbands: Magneto-Inter-Subband Oscillations: MISO .

• 4
• 2

2 4 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 (a)

3 2

• 2

2

• 1

j=0 1

• 4
• 2

2 4 60 70

B [T] Rxx []

(b)

1

• 1
• 2
• 3

j=0

B=0.12T J [A/m]

Landau –Zener transitions inside lowest subband

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

1 2 3 20 40 60 80 0.31 0.32 0.33

RXX,  B [T] J [A/m] +A +B +C T=2.1 K

• 4
• 2

2 4 0.34 0.36 0.38 0.40 0.42 0.44 (a)

Rxx [] B [T]

O +C +B

• C
• B

+A

• A
• 4
• 2

2 4 20 40 60 80

(b)

J [A/m] B=0.408T B=0.418T

Dependence of resistance on magnetic field B and current density J , indicating correlation

• f features ±A and ±C with MISO minima and features ±B with MISO maxima.

• 2

2 0.0 0.1 0.2 0.3 0.4 0.5 0.6

j=2 j=1 B=0.532 T j=-2 j=-1

80 160

J=3.03 A/m RXX[] j=2 BC J=0 A/m

80 160

_ _ _ + + +C +B +A

• A
• B
• C

RXX[] B[T] J[A/m] B=0.548 T + _

• 4
• 2

2 4 0.0 0.2 0.4 0.6

j=-1 +C +B +A

• A
• B

j=1 j=2

B[T] J[A/m]

j=3

• C

Scott Dietrich, Sean Byrnes, Sergey Vitkalov, A. V. Goran, and A. A. Bykov

• Phys. Rev. B 86, 075471 (2012)

Zero-differential conductance state

0.2 0.4 0.6 0.8 1.0 4 8 12

g12 (mS) B (T) T = 1.6 K Edc = 0 Edc = 250 V/m I12 = Iac + Idc Vac Vdc 2 1

~

• 50
• 25

25 50

• 40

40 80 120

Idc (A) rxx ()

(b)

ZDRS T = 1.6 K B = 0.841 T T =4.2 K Ith Hall Bar

• 600
• 300

300 600

• 1

1 2 3

g12 (mS) Edc (V/m) B = 0.847 T T = 4.2 K T = 1.6 K

(a)

Eth ZDCS Corbino Disc 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

Conclusions

• In conducting systems with discrete spectrum

Joule heating results in a spectacular decrease of electron resistance and a transition of the electron systems into state in which voltage V does not depend on current I – the state with zero differential resistance dV/dI=0 (ZDR).

• The quantal heating leads to an apparent dc

driven metal-insulator transition, which correlates with the transition into the ZDR state. The correlation is remarkable and is not understood.