Dynamics of Quantal Heating in Electron Systems with Discrete Spectra

William Mayer1,2, S. Dietrich1,2, S. Vitkalov1, A. A. Bykov3,4

1. City College of City University of New York, New York 10031, USA2. Graduate Center of City University of New York, New York 10016, USA3. A. V. Rzhanov Institute of Semiconductor Physics, Novosibirsk 630090, Russia4. Novosibirsk State University, Novosibirsk 630090, Russia

Thursday, May 28, 2015

Quantum transport in 2D systemsMay 23 - 30, 2015, Luchon, France

Strong nonlinear responses in 2DEG

•Due to MW pumping

•Due to DC bias

J.Q. Zhang, S. Vitkalov, A.A. Bykov Phys. Rev. B 80, 045310 (2009)

M. A. Zudov, R. R. Du, L. N. Pfeiffer and K. W. West, Phys. Rev.Lett. 90, 046807 (2003)

I. A. Dmitriev, M. G. Vavilov, I. L.Aleiner, A. D. Mirlin, and D.G. Polyakov, Phys. Rev. B 71, 115316 (2005)

S. I. Dorozhkin, JETP Lett, 77, 577 (2003)

Quantal Heating is effect of quantum mechanics on Joule Heating

• decreases conductivity

• occurs in electron systems with quantized spectrum

• does not exist in classical electron systems

J.Q. Zhang, S. Vitkalov, A.A. Bykov , Phys. Rev. B 80, 045310 (2009)

Quantal Heating is… Apply bias , E

Spatial Spectral Diffusion

�⃗�

Selective Flattening of

−𝜕 𝑓𝜕𝑡

+𝐸2𝜎 𝐷

❑

𝜈 (𝜖 )𝜕𝜖 [ 𝜈 (𝜖 )2

𝜈02 𝜕𝜖 𝑓 (𝜖 )]= 𝑓 (𝜖 )− 𝑓 𝑇 (𝜖)

𝜏 𝑖𝑛

Lower longitudinal conductivity

𝜎 𝑛𝑙=∫𝜎 (𝜖 )(− 𝜕 𝑓𝜕𝜖 )𝑑𝜖

𝜖𝑡𝑜𝑡𝑎𝑙=𝜖 𝐾 (𝑡)+𝑒𝐸 x (t)

𝑥

I. A. Dmitriev, M. G. Vavilov, I. L.Aleiner, A. D. Mirlin, and D.G. Polyakov, Phys. Rev. B 71, 115316 (2005)

Quantal Heating in the dc-domain

−𝜕 𝑓𝜕𝑡

+𝐸2𝜎 𝐷𝐶

❑

𝜈 (𝜖 )𝜕𝜖 [𝜈 (𝜖 )2

𝜈02 𝜕 𝑓 (𝜖 )]= 𝑓 (𝜖 )− 𝑓 𝑇 (𝜖 )

𝜏 𝑖𝑛

• Gaussian DOS for electrons• Conductivity:• No electron spatial

redistribution from dc bias

𝜎 𝑛𝑙=∫𝜎 (𝜖 )(− 𝜕 𝑓𝜕𝜖 )𝑑𝜖

Inelastic rate

Why dynamics?• There is a difficulty with the inelastic

mechanism in MW domain: the polarization dependence seems does not agree with experiment.

• There is a nonlinearity related to spatial electron redistribution due to applied bias. The nonlinearity is comparable with quantal heating in SdH regime.

• SdH method indicates inelastic rate proportional to temperature TM.G. Blyumina, A. G. Denisov, T. A.

Polyanskaya, I. G. Savel’ev, A. P. Senichkin, and Yu. V. Schmartsev, JETP Lett., 44,257 (1986)

Scott Dietrich, S. A. Vitkalov, D. V. Dmitriev and A. A. Bykov, Phys. Rev. B 85, 115312 (2012).

J. H. Smet, et al Phys. Rev. Lett. 95, 116804 (2005).

Samples

r2=1mm

r1=0.9mm

• MBE grown• Selectively doped single

GaAs quantum wells• GaAs/AlAs superlattice

barriers

Corbino geometry provides well determined radial field distribution. Important for nonlinear measurements

• High electron density decreases e-e scattering

• High mobility strong variations in the density of states

GaAs/AlAs

GaAs QW 13nm

Si

𝜔1𝜔2

𝜔=𝜔1−𝜔2

Dynamics of Quantal Heating:Difference Frequency Method

Total field:

where

Analyzer measures signal

Analyzer:

SRC: SRC:

LPF Bias-Tee

Lockin

Scott Dietrich, William Mayer, Sergey Vitkalov, A. A. Bykov, cond-mat > arXiv:1410.2618, Phys. Rev. B 91, 205439 (2015).

Dynamics of Quantal Heating

𝑗 𝜔=𝐸0∫𝜎 𝜖 [−𝜕𝜖 ( 𝛿 𝑓 𝜔 ) ] 𝑑 𝜖=2𝐸0𝐸1𝐸2exp (𝑖𝜔𝑡 )

𝑖 𝜔+1/𝜏 𝑖𝑛Σ(𝐵)

−𝜕 𝑓𝜕𝑡

+𝐸2𝜎 𝐷

❑

𝜈 (𝜖 )𝜕𝜖 [ 𝜈 (𝜖 )2

𝜈02 𝜕𝜖 𝑓 (𝜖 )]= 𝑓 (𝜖 )− 𝑓 𝑇 (𝜖)

𝜏 𝑖𝑛

Heating (excitation)

Now time dependent & modulated by beating of two sources.

Cooling (relaxation)

𝑓 = 𝑓 𝑇+𝛿 𝑓 𝜔&

We measure -signal∝𝐸0𝐸1𝐸2

√𝜔 2+1/𝜏 𝑖𝑛2

Magnetic Field Dependence

0.0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

5

0.0

0.2

0.4

0.6

R(k

)

-s

igna

l (V

)

B (T)

f=1.0 MHz

Dc Bias Dependence-signal ∝

𝑬 𝟎 𝐸1𝐸2√𝜔 2+1/𝜏 𝑖𝑛2

-30 -20 -10 0 10 20 300.0

0.1

0.2

0.3

0.4

0.2

0.3

0.4

0.5

-s

ign

al (m

V)

Vdc

(mV)

1 MHz

r(k

)

1 GHz

1.5 GHz

T=4.8 KB=0.333 T

Power Dependence-signal ∝𝐸0𝑬 𝟏 𝑬 𝟐

√𝜔 2+1/𝜏 𝑖𝑛2

-25 -15 -5 50.01

0.1

1

10

0 2 4 6 8 10 12 14 16185

190

195

200

205

210

215

220

225

20log(E1E

2) (dBm)

-s

ignal (m

V)

1 MHz 1 GHz 1.5GHz

T=4.8KB=0.33T

1.5 GHz

1 MHz

R (

Oh

m)

P1+P

2 (mW)

1 GHz

Dynamics of Quantal Heating

-signal ∝𝐸0𝐸1𝐸2

√𝝎 2+1/𝜏 𝑖𝑛2/

Dynamics of Quantal Heating

electron-phonon interactions

∝𝑇 3

electron-electron interactions

∝𝑇 2

∝𝑇 2

e-e interaction dominates

Dynamics of Quantal Heating

CV

𝑘𝑇 ≫ћ𝜔 𝑐

e-phonon interaction dominates

∝𝑇 3

𝑘𝑇 ≈ћ𝜔𝑐

J.Q. Zhang, S. Vitkalov, A.A. Bykov , Phys. Rev. B 80, 045310 (2009)

Comparison of two methods

•Order of magnitude agreement

• ω-signal is direct measurement

•dc-domain may experience electron spatial redistribution

Conclusions• direct

measurements of inelastic relaxation

• Observe T2 dependence for

• Approaches T3 dependence for

Scott Dietrich, William Mayer, Sergey Vitkalov, A. A. Bykov, cond-mat > arXiv:1410.2618, Phys. Rev. B 91, 205439 (2015)

Acknowledgements

https://sites.google.com/site/ccnymw

NSF DMR 1104503 & RFBR #14-02-01158

Thank You