Microwave-induced zero-resistance states on 2D electron gas: theoretical explanation and temperature dependence

Microelectronics Journal 36 (2005) 334–337 www.elsevier.com/locate/mejo

Microwave-induced zero-resistance states on 2D electron gas: theoretical explanation and temperature dependence Jesu´s In˜arreaa,b, Gloria Plateroa,* a

Instituto de Ciencia de Materiales (CSIC), Cantoblanco, Madrid, Spain Escuela Polite´cnica Superior, Universidad Carlos III, Leganes, Madrid

b

Available online 12 April 2005

Abstract A new theoretical model is presented in which the radiation-induced zero-resistance states and resistivity oscillations are analyzed. The basis of our model is an exact solution for the harmonic oscillator wave function in the presence of radiation and a perturbation treatment for elastic scattering due to randomly distributed charged impurities. Following this model most experimental results are reproduced: zeroresistance states, resistivity oscillations and the dependence of resistivity with temperature, where a physical model of interaction with acoustic phonon is presented, and an activation energy is calculated. q 2005 Elsevier Ltd. All rights reserved. Keywords: Two-dimensional electron gas; Resistivity oscillations; Zero-resistance states; Temperature dependence; Microwave

1. Introduction The discovery of zero-resistance states (ZRS) and resistivity oscillations in a two-dimensional electron gas (2DEG) subjected to a static magnetic filed B and microwave (MW) radiation [1–5] has led to a great deal of theoretical activity in order to describe the physics which drives and is behind this effect. Some like Girvin et al. [6,7], argue that this striking effect has to do with a multiphotonassisted impurity scattering or alternatively acoustic phonon scattering [8]. Others [9,10] relate the ZRS observed with a new structure of the DOS of the system in the presence of light. Andreev’s [11] approach consists in the existence of an inhomogeneous current flowing through the sample due to the presence of a domain structure in it. To date there is no consensus about the real origin. In this Letter we develop a semi-classical model that is based on the exact solution of the electronic wave function in the presence of a static magnetic field interacting with MW-radiation, i.e. a quantum forced harmonic oscillator, * Corresponding author. Address: Departamento de Teoria de la Materia Condensada, Instituto de Ciencia de Materiales (CSIC), Cantoblanco, Madrid 28049, Spain. Tel.: C34 91 334 9046; fax: C34 91 372 0623. E-mail addresses: [email protected] (J. In˜arrea), gplatero@ icmm.csic.es (G. Platero).

0026-2692/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2005.02.096

and a perturbation treatment for elastic scattering from randomly distributed charged impurities. We explain and reproduce most experimental features, and clarify the physical origin of ZRS. We explain the dependence of rxx on temperature (T) by means of the electron interaction with acoustic phonons which acts as a damping factor for the forced quantum oscillators. We proved that in the resistivity minima the variation with temperature has to do with a thermally activated transport and we calculated the corresponding activation energy.

2. Model We first obtain an exact expression of the electronic wave vector for a 2DEG in a perpendicular B, a DC electric field and MW radiation which is considered semi-classically. The total hamiltonian H HZ

P2x 1 1 E2 C m * w2c ðx K XÞ2 K eEdc X C m * dc * 2 2 2m B2 K eE0 cos wtðx K XÞ K eE0 cos wtX

where X is the center of the orbit for the electron spiral motion XZ

Zky eE K dc2 eB m * wc

J. In˜arrea, G. Platero / Microelectronics Journal 36 (2005) 334–337

w is the MW frequency, wc the cyclotron frequency, E0 the intensity for the MW field and Edc the DC electric field in the x direction. H can be solved exactly [12,13], to obtain Jðx;tÞZfn ðxKXKxcl ðtÞ;tÞ 

 ð m* dxcl ðtÞ i t 0 ½xKxcl ðtÞC !exp i Ldt Z 0 Z dt " !# N X eE0 1 w C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eimwt ! X Jm w Z ðw2c Kw2 Þ2 Cg4 mZKN (1) where g is a phenomenologically introduced damping factor for the electronic interaction with acoustic phonons, fn is the solution for the Schro¨dinger equation of the unforced quantum harmonic oscillator and xcl(t) is the classical solution of a forced harmonic oscillator [13] eE0 xcl Z pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi coswt 2 * m ðwc Kw2 Þ2 Cg4 L is the classical lagrangian, and Jm are Bessel functions. Apart from phase factors, the wave function for H is the same as the standard harmonic oscillator, where the center is displaced by xcl(t). Now we introduce the impurity scattering suffered by the electrons in our model [14] applying Fermi’s Golden Rule to calculate the transition rate from an initial state Jn(x,t), to a final state Jm(x,t): Wn,m [15]. From the point of view of the scattering process in the presence of MW radiation, we have to consider two important factors: first, when an electron suffers a scattering process with a probability given by Wn,m, it takes a time tZ1/Wn,m for that electron to jump from an orbit to another. Secondly, in the jump, as in any other scattering event, the electron loses memory and phase with reference to the previous situation, and thus when it arrives at the final state the oscillation condition is going to be different from the starting point. If we consider that the oscillation is at its midpoint when the electron jumps from the initial state, and that it takes a time t to get to the final one, then we can write for the average coordinate change in the x direction with the presence of MW-radiation ! eE0 MW 0 DX Z DX C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos wt m * ðw2c K w2 Þ2 C g4 where DX0 is the average distance advanced in the absence of radiation. When MW field is on, the orbits are not fixed, and instead move back and forth through xcl. Three cases can be distinguished. When the orbits are moving backwards during the electron jump, on average, due to scattering processes, the electron advances a larger distance than in the no MW case, DXMWODX0. This corresponds to an increasing conductivity. In the second case, the orbits are

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moving forwards and the electron advances a shorter distance, DXMW!DX0. This corresponds to a decrease in the conductivity. If we increase the MW intensity, we will eventually reach the situation where orbits are moving forwards, but their amplitude is larger than the electronic jump. In that case the electronic jump is blocked by the Pauli exclusion principle because the final state is occupied. This is the physical origin of ZRS. If the average value DXMW is different from zero over all the scattering processes, the electron possesses an average drift velocity vn,m in the x direction [14]. This drift velocity can be calculated readily by introducing the term DXMW into the integrand of the transition rate, and finally the longitudinal conductivity sxx can be written as ð 2e rðEn Þvn;m ½f ðEn Þ K f ðEm Þ dEn sxx Z Edc and finally sxx ¼

  e 7 n i B2 S X G 16p5 32 Z3 Edc n;m ½Zwc ðnKmÞ2 þG2   ð G ! dEn ½f ðEn ÞKf ðEm Þ ½En KZwc ðnþ 12 Þ2 þG2  ð qm ð  qðqcosqR2 þAcoswtÞ ! dq dq ðqþqs Þ2 0     n ! 1 2 2 1 2 2 n1 Kn2 n1 Kn2 1 2 2 2 q R q R ! 1 eK2 q R Ln2 n2 ! 2 2 !J02 ðAm ÞJ02 ðAn Þ

ð2Þ

where G is the Landau level broadening, ni is the impurity density, R is the magnetic characteristic length R2 ZZ=eB, Lnn12Kn2 are the associated Laguerre polynomials, n1Zmax(n,m) and n2Zmin(n,m) and AZ

m*

eE0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 ðwc Kw2 Þ2 Cg4

To obtain rxx we use the relation s s rxx Z 2 xx 2 x xx sxx Csxy s2xy where sxy xni e=B and sxx /sxy . 3. Results All our results have been based on experimental parameters corresponding to the experiments of Mani et al. [1]. In Fig. 1 we show the magnetoresistivity rxx obtained using our model, as a function of B for a MW frequency w/2pZnZ103.5 GHz. The darkness case is also presented. The minima positions as a function of B are indicated with arrows, corresponding to w=wc Z jC 1=4. In the minima corresponding to jZ1, ZRS are found. Using our model, calculated rxx as a function of B for different MW field intensities and frequencies, can be similarly

J. In˜arrea, G. Platero / Microelectronics Journal 36 (2005) 334–337

336

1.0 × 10–4

ν

ρxx(Ω)

8.0×10–4 6.0×10–4 4.0×10–4 2.0×10–4 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 B(T) Fig. 1. Calculated magnetoresistivity rxx as a function of magnetic field B, for a MW intensity of 70 V/m, and a frequency of nZ103.5 GHz. The darkness case is also presented.

obtained [16]. Although the qualitative behavior of rxx as a function of B is very similar to the experimental one, the absolute value is smaller. It could be due to the smaller carrier density, or the simplified model for the electronic scattering with impurities that we have considered. The calculated temperature variation of resistivity rxx has been obtained at nZ103.5 GHz (see Fig. 2) and it is remarkable that the strong temperature dependence: as T increases, the MW-induced rxx response is damped and eventually it almost disappears. The explanation for such a variation can be readily obtained according to our model. As said above, we have phenomenologically introduced a damping parameter g in the expression for the total wave function. The origin for this parameter corresponds to the following: when the electronic orbits are ‘dancing’ harmonically due to the time dependence external force, interact with the lattice ions yielding acoustic phonons. From a physical standpoint this interaction is acting as an important damping factor for the orbits movement. As the temperature is increased, the interaction

lattice-orbits, is going to be stronger and the damping effect upon orbits ‘dancing movement’ will be stronger as well. As a consequence, we expect a damping in the MW-induced rxx response. Following Studenikin et al. [3] we have introduced a linear dependence between the damping factor g and temperature. In Fig. 2 we show the natural logarithm of the resistivity versus the inverse temperature at the first minima positions for the five different temperatures considered. Accordingly, we conclude that rxx vanishes exponentially with decreasing temperature,

following a general expression like: rxx f exp KET =kb T . This indicates thermally activated transport being ET the activation energy and kb the Boltzman constant. The actual value we obtain ETZ3.14 K, is small compared to the experimental evidence [1,2,5], questioning the linear model we have used between g and temperature. In order to get a more realistic value for the activation energy, a microscopic model for the interaction between electrons and acoustic phonons at low temperatures has to be introduced. This is the object for a future work. Using our model, now is possible to give some light about the peculiar shape for the rxx oscillations with the magnetic field and minima positions. From formula (3) we can extract the direct dependence of rxx with B: w rxx f B cosðwtÞf B cos (3) B From a mathematical point of view considering the magnetic field B as the independent variable, the expression above corresponds to a mathematical function (see Fig. 3) with similar shape to the experimental results, i.e. experimental behavior follows approximately a function like aCx cos(c/x) being a and c constants, except by the minima which do not show up due to Pauli exclusion principle. To try to understand minima positions, we examine cosine argument, i.e. cos(wt). Now it is easy to understand that if we change MW frequency w, minima positions will change as well. Regarding tZ1/Wn,m we can say that the scattering transition rate Wn,m is mostly

f(x)=a+x cos(c/x)

–9.5

E/ k b = 3. 14K

–10.5

f(x)

In (ρ x x ( Ω) )

–10.0

–11.0

0.0

–11.5 –12.0 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

T –1 ( K –1 ) Fig. 2. Natural logarithm of rxx versus the inverse of temperature at the first minima positions for the five different temperatures considered.

0.0

0.1

0.2

0.3

0.4

0.5

X Fig. 3. Mathematical function which explains experimental behavior of MW-induced rxx.

J. In˜arrea, G. Platero / Microelectronics Journal 36 (2005) 334–337

dependent on sample and scattering variables and that specific minima positions will be function of those variables too. In this way we expect that for significantly different samples minima positions and other features of rxx oscillations will not match, like in experimental data available [1,2]. 4. Conclusions As a summary, in this paper we have presented a new theoretical model whose main foundations are in one hand an exact solution for the quantum harmonic oscillator in the presence of microwave radiation considered as a external driven force. On the other hand we apply a perturbation treatment for elastic scattering due to randomly distributed charged impurities. Accordingly, we are able to reproduced and explain most experimental information and regarding temperature dependence of rxx, a physical model of interactions with acoustic phonons is introduced, showing thermally activated transport at minima positions and obtaining a calculated value for the activation energy.

Acknowledgements This work was supported by the MCYT (Spain) grant MAT2002-02465, the ‘Ramon y Cajal’ program (J.I.)

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and the EU Human Potential Programme HPRN-CT-200000144.

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