E?ect of frequency and temperature on microwave-induced magnetoresistance oscillations in two-dimensional electron systems.
Jes? s I? arrea1,2 u n
Escuela Polit?cnica Superior,Universidad Carlos III,Leganes,Madrid,Spain and e 2 Unidad Asociada al Instituto de Ciencia de Materiales, CSIC, Cantoblanco,Madrid,28049,Spain. (Dated: August 11, 2008)
arXiv:0808.1489v1 [cond-mat.mes-hall] 11 Aug 2008
Experimental results on microwave-induced magnetoresistance oscillation in two-dimensional electron systems show a similar behavior of these systems regarding temperature and microwave frequency. It is found that these oscillations tend to quench when frequency or temperature increase, approaching magnetoresistance to the response of the dark system. In this work we show that this experimental behavior can be addressed on the same theoretical basis. Microwave radiation forces the electron orbits to move back and forth being damped by interaction with the lattice. We show that this damping depends dramatically on microwave frequency and also on temperature. An increase in frequency or temperature gives rise to an increase in the lattice damping producing eventually a quenching e?ect in the magnetoresistance oscillations.
The study of the e?ects that radiation can produce on nano-devices, such as two-dimensional electron systems (2DES), is attracting considerable attention since the last two decades both from theoretical and experimental sides1 . Important and unusual properties have been discovered when these systems are subjected to di?erent external potentials. In particular microwave-induced magentoresistance oscillations (MIRO) and Zero Resistance States (ZRS), are being a subject of intense investigation since its recent discovery2,3,4 . Since then, experimental results are being produced in a continuous basis which challenge the available theoretical models5,6,7,8,9,10,11,12,13 . Among all of those experimental outcomes we can cite for instance, temperature dependence of MIRO2,3,5 , absolute negative conductivity14,15,16 , bichromatic microwave (MW) exictation17,18 , polarization immunity of magnetoresistance (ρxx )19,20 , e?ect of in-plane magnetic ?elds21 . More recently another striking experimental result by Studenikin et al.,22 has joined the group. They demonstrate that at high MW frequencies (above 120 GHz) MIRO start to quench and for further increasing frequencies, MIRO practically disappear. They also show experimental results on temperature dependence of MIRO amplitude indicating that at increasing lattice temperature the oscillations amplitude dramatically decrease. This behavior was already obtained by the seminal experiments on MIRO and ZRS2,3,4 . For both cases, MW frequency and temperature, the system tends to recover the dark ρxx response. In this letter we present a common theoretical approach to explain the quanching e?ect of both, MW frequency and temperature, on MIRO. The starting point is the e?ect of MW radiation on 2DES, which is treated with the MW driving Larmor orbits model5,15 . According to it, MW radiation forces electronic orbits to move periodically back and forth at MW frequency. As a result, the whole two-dimensional electron gas (2DEG) moves harmonically in the current direction. In this pe-
24 12 0 24 12 0
24 12 0
24 12 0 24 12 0 24 12 0
FIG. 1: Calculated magnetoresistance di?erence between radiation and dark situations:?ρxx = ρM W ? ρDark as a funcxx xx tion of magnetic ?eld B for di?erent MW frequencies from 80 to 253 GHZ. TL = 1K.
riodic movement electrons in their orbits (quantum oscillators) interact with the lattice being damped and emiting acoustic phonons. If we increase the MW frequency leaving ?xed the lattice temperature (TL ), the average number of interactions between electrons and lattice ions will be larger. Thus, we obtain and increasing damping of the MW-driven oscillating motion. This is ?nally
2 allows us to obtain γ and its dependence on w and TL as follows. Following Ando and other authors24 , we propose the next expression for the electron-acoustic phonons scattering rate valid at low TL : 1 m? Ξ2 kB TL ac = 3 2 τac ρul < z >
20 15 10 5 0 -5 12
15 10 5 0 -5 8
0 -2 -4 -4
FIG. 2: (Calculated ρxx di?erence for di?erent frequencies plotted as a functions of w/wc . TL = 1K.
where Ξac is the acoustic deformation potential, ρ the mass density, ul the sound velocity and < z > is the effective layer thickness. However τ1 is not yet the ?nal ac expression for γ. First, we have to multiply τ1 by the ac average number of times that an electron in its orbit can interact e?ectively with the lattice ions. We can approximate this number dividing the length an electron runs in its Landau state by the lattice parameter of GaAs, (aGaAs ): 2πRc /aGaAs , Rc being the cyclotron radius for the corresponding orbit. Secondly, being γ a dissipative term, it is directly proportional to the MW energy and inversely proportional to the emitted acoustic phonon enw ergy. Thus, we have ?nally to multiply by the ratio: wac , wac being the average frequency of the emitted acoustic phonons25 . Finally we obtain an expression for γ: γ= 1 2πRc w × × τac aGaAs wac (3)
re?ected in the decreasing amplitude of MIRO. On the other hand if we increase the TL leaving ?xed the MW frequency w, the average electrons-lattice interactions will be larger producing more damping and smaller ρxx oscillations. Thus, the quenching of MIRO by high w or TL would be based on the same physical e?ect. We have recently15 proposed a model to explain the ρxx behavior of a 2DEG at low B and under MW radiation. According to it, due to the MW radiation, center position of electronic orbits are not ?xed, but they oscillate back and forth harmonically with w. The amplitude A for these harmonic oscillations is given by: A= m? eEo 2 (wc ? w2 )2 + γ 4 (1)
With the experimental parameters we have at hand2,3,4,25 and for an average magnetic ?eld it is straightforward to obtain a direct relation between γ and TL and w, resulting in a linear dependence: γ ∝ TL × w (4)
where e is the electron charge, wc the cyclotron frequency, E0 the intensity for the MW ?eld and γ is a damping parameter due to interaction with the lattice. Now we introduce the scattering su?ered by the electrons due to charged impurities randomly distributed in the sample5,15,23 . Firstly we calculate the electron-charged impurity scattering rate 1/τ , and secondly we ?nd the average e?ective distance advanced by the electron in every scattering jump: ?X MW = ?X 0 + A cos wτ , where ?X 0 is the e?ective distance advanced when there is no MW ?eld present. Finally the longitudinal conductivity MW σxx can be calculated: σxx ∝ dE ?Xτ (fi ? ff ), being fi and ff the corresponding distribution functions for the initial and ?nal Landau states respectively and E energy. xx To obtain ρxx we use the relation ρxx = σ2 σ+σ2 ? σxx , 2 σxy xx xy ni e where σxy ? B and σxx ? σxy . Therefor, ρxx is proportional to the MW-induced oscillation amplitude of the electronic orbits center: ρxx ∝ A cos wτ . At this point, we introduce a microscopic model which
Now is possible to go further and calculate the variation of ρxx with TL and w. In Fig. 1, we present calculated ρxx di?erence (between radiation and dark situations:?ρxx = ρMW ? ρDark ) as xx xx a function B for di?erent MW frequencies from 80 to 253 GHZ. From Fig.1a to 1f the ρxx evolution shows the quenching of MIRO being clearly visible from w = 110 GHz. At w = 253 GHz are hardly visible. The explanation can be readily obtained according to our theory. As we said above, ρxx is proportional to A: ρxx ∝ eEo m?
? w2 )2 + γ 4
From equations (4) and (5), we can deduce that higher values for w mean larger γ, ?nally re?ected in smaller ρxx . Eventually for large enough w, ρxx will be practically wiped out. In Fig. 2, we present calculated ?ρxx for di?erent frequencies plotted as a function of w/wc . Calculated results of Figs. 1 and 2 are in reasonable agreement with experiments22 . In Fig. 3 bottom panel, we show calculated ρxx as a function of B for di?erent TL from 1K to 3.5K. ρxx presents decreasing oscillations for increasing TL at a ?xed w = 80 GHz. In the top panel we represent TL
3 dependence of MIRO amplitude in arbitrary units determined by ?tting calculated ρxx results of the bottom panel. Calculated data have obtained from the largest ρxx peak (around B = 0.2 T). According to this ?tting MIRO amplitude depends roughly on TL as ?2 AMIRO ∝ TL . This behavior is explained similarly as in the w case following equations (4) and (5) including the AMIRO dependence on TL .
Calculated data Fitted curve
2.5 K 3.0 K 3.5 K
FIG. 3: (Color on line). Bottom panel: calculated ρxx as a function of B for di?erent TL from 1K to 3.5K. ρxx presents decreasing oscillations for increasing TL at a ?xed w = 80 GHz. TL = 1 K. Bottom panel: TL dependence of MIRO amplitude in arbitrary units determined by ?tting calculated ρxx results of the bottom panel. Calculated data have obtained from the largest ρxx peak (around B = 0.2 T
This work has been supported by the MCYT (Spain) under grant MAT2005-0644, by the Ram?n y Cajal o program and by the EU Human Potential Programme: HPRN-CT-2000-00144.
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