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Quantal Joule Heating
Joule heating is a remarkable physical phenomenon which transforms electric energy into heat. Recent experiments on 2D electrons, placed in quantizing magnetic fields, have identified a new kind of Joule heating, which occurs exclusively in conducting quantum systems. The effect is absent in classical electron systems. Quantal Joule heating produces peculiar spectral distribution of electrons, which deviates significantly from the Fermi-Dirac form. Experiments show that even weak quantal heating induces an exceptionally strong violation of Ohm's Law, changing radically the electron transport. Overheated electrons undergo a transition into a state in which voltage does not depend on current, called the zero differential resistance (ZDR) state. These intriguing phenomena as well as some properties of the quantal heating are not understood and contradictory. Ongoing studies in our lab probe the fundamental properties of quantal heating, the ZDR state. A variety of experiments are being done on a set of high mobility 2D electron systems with high electron density. Recent research is focused on dynamic of quantal heating and the dynamical response of ZDR state. The research is supported by National Science Foundation.
Figure A presents heating effects observed in magnetoresistance of 2D electrons in a GaAs quantum well. In strong magnetic fields the magnetoresistance at T=2.16K (green curve) shows quantum resistance oscillations due to the oscillations of the density of states induced by the quantization of the electron spectrum in magnetic fields . Presented in the figure blue arrow marks the magnetic field above which the density of states begins to oscillate due to the spectrum quantization.
The black curve shows the effect of the usual thermal bath heating of the sample with no dc bias applied (no Joule heating). The thermal bath keeps the electron and phonon systems in the thermodynamic equilibrium at a temperature T. As expected the increase of the temperature T from 2.16 K to 4.2 K decreases the amplitude of quantum resistance oscillations due to averaging of the oscillations of the density of states inside the energy interval kT near Fermi energy. The averaging effect is more profound at the higher temperature T=4.2K due to the larger energy interval kT for the average. Figure B is an enlargement of Fig. A at small magnetic fields, revealing another effect of the thermal bath heating: the increase of the resistance due to the enhancement of the electron-phonon scattering at the higher temperature.
In figures A and B the red curve demonstrates the effect of Joule heating induced by applied dc bias of 6 µA at bath temperature T=2.04 K. At small magnetic fields below 0.1 T (left from the blue arrow) the electron spectrum is not quantized and Joule heating affects very weakly the electron transport: the green (dc bias is OFF) and red (dc bias is ON) curves are essentially coincide. Drastically different behavior is observed in the magnetic fields quantizing the electron spectrum (right from the blue arrow). Quantal Joule heating leads to outstanding decrease of the sample resistance. This effect is much stronger than the effect of Joule heating at small magnetic fields at the same dc bias.
To understand the difference between these two regimes of Joule heating one need to see how the Joule heating occurs in metals. At thermodynamic equilibrium the electron distribution is described by Fermi-Dirac function, fe. At temperature T=0K the distribution is a step like function: electron states with energy below the Fermi energy, EF , are fully occupied by electrons while the electron states above the Fermi energy are empty. At a finite temperature T>0 the transition from the fully occupied states to the empty ones occurs in energy interval kT in the vicinity of Fermi energy, where k is Boltzmann constant. This property is called as broadening of the electron distribution by the temperature. In the kT interval near EF the quantum states are partially occupied by electrons. The electron transport occurs due to the electric field induced quantum transitions between these partially occupied electron states. Fully occupies states do not contribute to the electron transport since an electron (a Fermi particle) cannot make a transition to a fully occupied quantum state. Such transitions are prohibited by the Pauli principle.
At low temperatures within the kT interval an electron performs a chaotic motion mostly due to scattering on static impurities. In such motion the electron position R(t) is a stochastic function of the time t. In addition, since the impurity scattering is an elastic process, the total electron energy is conserved: Etot=P+K=const. With no electric field applied the potential energy of the electron, P, does not depend on the stochastic electron position R(t) and, thus, is a constant. Therefore the kinetic energy of the electron, K=Etot -P, is also a constant. An application of the electric field, E, leads to stochastic fluctuations of the electron potential energy: P(t)=eER(t) and, thus, to stochastic fluctuations of the kinetic energy K(t)=Etot -P(t). The fluctuations of the electron kinetic energy lead to a spectral diffusion of electrons in the energy space and generate a spectral electron flow along the gradient of the electron distribution. The spectral flow is proportional to the product of the coefficient of spectral diffusion and the gradient of the electron distribution. The spectral diffusion tends to reduce this energy gradient of the electron distribution. In result in the electric field the electron distribution is determined by an interplay between the coefficient of the spectral diffusion and the gradient of the electron distribution. The resultant electron distribution is stabilized by an inelastic dissipation of electron energy by phonon system. For a stationary electron distribution and an uniform rate of the inelastic dissipation the electric field induced spectral flow should be approximately the same at different energies within the kT interval.
For 2D electron systems with no spectrum quantization (called here as classical 2D electron systems) the coefficient of the spectral diffusion is quite uniformed in the energy since the density of electron quantum states is a constant for 2D electrons. This property together with the uniform spectral flow lead to a quite uniform reduction of the gradient of electron distribution for all energies resulting in the increase of the overall broadening of the electron distribution. In figure C green curve presents numerical computations of the electron distribution under effect of Joule heating for the classical 2D electron systems at lattice temperature T1. The computed distribution has considerably larger broadening and is well described by Fermi -Dirac distribution with an elevated temperature T2>T1.
Fig.(C) Blue curve presents the electron distribution at thermodynamic equilibrium (Fermi-Dirac distribution) at bath temperature T1. Green curve presents numerical computations of the electron distribution under Joule heating for classical 2D electron system at bath temperature T1. The computed curve is well approximated by Fermi-Dirac distribution with an elevated temperature T2 . The red curve presents numerical computation of the electron distribution in the regime of quantal heating for 2D electrons with quantized spectrum at bath temperature T1. The grey curve presents the quantized electron spectrum used in the numerical computations.
In figure C the red curve presents numerical computations of Joule heating of 2D electrons placed in a strong magnetic field. Due to the wave nature of electrons, the magnetic field quantizes the circular electron orbits leading to Landau levels - infinite set of quantum states, separated by the cyclotron energy (the cyclotron gap). The electron-impurity scattering makes the quantum lifetime of an electron in these levels to be finite, leading to the quantum broadening of these levels. In Fig.C the grey oscillating curve presents a typical example of the quantum electron spectrum (so-called density of states (DOS)) in strong magnetic field. The DOS maxima corresponds to the energy of Landau levels. The levels are broaden by the scattering and separated from each other by the cyclotron gap. (The energy interval between two nearest maximums of the DOS is the cyclotron gap.)
The presented DOS oscillates strongly between a large value at the maxima and zero at the minima (in the middle of the cyclotron gap). For such DOS the coefficient of the spectral diffusion exhibits strong oscillations. In the DOS maxima the spectral diffusion is strong since there are many quantum levels available for the scattering promoting the spectral diffusion. At the DOS minima the coefficient of the spectral diffusion reduces since less quantum levels are available for the scattering. In fact for the presented spectrum the coefficient of the spectral diffusion is zero in the DOS minima, since no quantum states are available there. Thus the coefficient of the spectral diffusion strongly oscillates with the energy, that leads to a significant modification of the electron distribution by Joule heating.
In Fig. C the red curve presents results of quantal Joule heating. Quantal Joule heating does not increase the broadening but, instead, leads to a quantization of the electron distribution. Under quantal Joule heating the equilibrium Fermi-Dirac distribution (the blue curve) evolves into the step-like function (the red curve) containing as many steps as the number of quantum (Landau) levels located within the energy interval kT1.The resultant dc-biased non-equilibrium distribution function is clearly different from the Fermi-Dirac form and, thus, cannot be described by a temperature T.
The presented properties of quantal heating are result of the quantization of the electron spectrum. Due to electric charge conservation in a stationary state the spectral flow (which is a product of the coefficient of spectral diffusion and the gradient of the distribution function) is a constant. At a small electric field and a large cyclotron gap the coefficient of the spectral diffusion inside the cyclotron gap is exponentially suppressed. To keep the constant spectral flow in this region the gradient of the distribution function is strongly enhanced. For the same reason inside a Landau level the strong spectral diffusion reduces significantly the gradient of the electron distribution making the distribution to be almost flat in this region. It leads to the step like shape of the electron distribution shown in Fig. C by the red line.
The finite quantum gap between Landau levels reduces the overall transfer of electrons from one Landau level to another reducing the strength of the global spectral flow. In fact for a large spectral gap the global spectral electron flow is exponentially suppressed preserving the overall broadening of the electron distribution as shown in Fig.C. Indeed the overall broadening of the red (quantal heating) and blue (thermal equilibrium) curves are essentially the same. Thus quantal heating keeps fixed the number of quantum states participating in the electron transport in contrast to the classical Joule heating. On the other hand Fig.C demonstrates that in quantal heating regime the actual overheating of electrons inside a Landau level is enormous due to this quantum “insulation” of the level. Indeed, in Fig.C the flat parts of the red curve indicate an infinitely large "local temperature" within Landau levels.
Due to the strong overheating and the conservation of the number of quantum states contributing to the conductivity quantal heating provides outstanding impact on the electron transport. Fig.D demonstrates V-I characteristics of 2D electrons in GaAs quantum wells placed in magnetic field B=0.784T at different temperatures as labeled. At low temperatures the figure shows a Zero Differential Resistance (ZDR) state. In this state the voltage V=VZDR does not depend on the applied current IDC . Surprisingly this voltage corresponds also to an apparent dynamical Metal-Insulator (M-I) transition in the 2D electron system.
Fig.(D) . V-I characteristics of 2D electrons in GaAs quantum wells placed in magnetic field B=0.784T at different temperatures as labeled. V-I characteristics are sublinear for all temperatures. Below T=2K at IDC > 6 uA the 2D system undergoes to a state with Zero Differential Resistance at which dV/dI=0. In a broad range of dc bias the ZDR state exhibits an independence of the voltage on the applied current.
More on our research and properties of quantal Joule heating is in:
A. A. Bykov, Jing-qiao Zhang, Sergey Vitkalov, A. K. Kalagin, and A. K. Bakarov ”Effect of dc and ac excitations on the longitudinal resistance of a two-dimensional electron gas in highly doped GaAs quantum wells” Phys. Rev. B 72, 245307 (2005)
Jing-qiao Zhang, Sergey Vitkalov, A.A. Bykov, A.K. Kalagin, A.K. Bakarov "Effect of DC electric field on longitudinal resistance of two dimensional electrons in a magnetic field" Phys. Rev. B 75, 081305(R) (2007).
A.A. Bykov, Jing-Qiao Zhang, Sergey Vitkalov, A.K. Kalagin, A.K. Bakarov "Zero differential resistance state of two dimensional electron systems in strong magnetic fields" Phys. Rev. Lett. 99, 116801 (2007).
N.Romero Kalmanovitz, A. A. Bykov, S.A. Vitkalov, A. I. Toropov "Warming in systems with discrete spectrum: spectral diffusion of two dimensional electrons in magnetic field" Phys. Rev. B 78, 085306 (2008).
N. Romero, S. Mchugh, M. P. Sarachik,S. A. Vitkalov,A. A. Bykov "Effect of parallel magnetic field on the Zero Differential Resistance State" Phys. Rev. B 78, 153311 (2008) .
Jing Qiao Zhang, Sergey Vitkalov, and A. A. Bykov, "Nonlinear resistance of 2D electrons in crossed electric and magnetic fields" Phys. Rev. B 80, 045310 (2009).
Sergey Vitkalov (review article) "Nonlinear transport of 2D electrons in crossed electric and quantizing magnetic fields" International Journal of Modern Physics B v.23 No.23, 4727-4753 (2009).
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