Resistivity saturation in a weakly interacting 2D Fermi liquid at intermediate temperatures
Xiaoqing Zhou, B. Schmidt, L. W. Engel, G. Gervais, L. N. Pfeiffer, K. W. West, S. Das Sarma
RResistivity saturation in a weakly interacting 2D Fermi liquid at intermediatetemperatures
Xiaoqing Zhou , B. Schmidt , L.W. Engel , G. Gervais , L.N. Pfeiffer and K.W. West , S. Das Sarma , Department of Physics, McGill University, Montreal, H3A 2T8, CANADA National High Magnetic Field Laboratory, Tallahassee, FL 32310, USA Department of Electrical Engineering, Princeton University, Princeton NJ 08544 USA and Condensed Matter Theory Center, Department of Physics,University of Maryland, College Park, MD 20742 USA (Dated: October 29, 2018)We report a highly unusual temperature dependence in the magnetoresistance of a weakly in-teracting high mobility 2D electron gas (2DEG) under a parallel magnetic field and in the currentconfiguration I ⊥ B . While the linear temperature dependence below 10 K and the exponentialtemperature dependence above 40 K agree with existing theory of electron-phonon scattering, afield induced resistivity saturation behaviour characterized by an almost complete suppression ofthe temperature dependence is observed from approximately 20 to 40 K, which is in sharp con-trast to the phenomenology observed in the configuration I (cid:107) B . Possible origins of this intriguingintermediate temperature phenomenon are discussed. PACS numbers: 73.43.Qt, 75.47.Gk
The dependence of resistance on temperature is a keyexperimentally observable feature of electronic systems,and provides an important testing ground for theoreti-cal models of the microscopic scattering mechanisms atplay. For the well understood Fermi liquid theory basedsystems, scattering is usually governed by the low energyexcitations near the Fermi surface, and resistance typi-cally decreases as the temperature is lowered since thereare fewer states to scatter from and into. Nevertheless,lowering the thermal energy can often allow other effectsto become more prominent, some of which can give riseto a negative or more generally non-monotonic temper-ature dependence. These include disorder-related effectssuch as Anderson localization , Coulomb interaction re-lated effects such as Mott physics , various spin-relatedphenomena and other interaction-driven quantum phasetransitions, all of which require working at the lowest pos-sible temperatures. In contrast, the intermediate tem-perature regime from ∼ ∼
77 K has been relativelyneglected. This is perhaps not surprising, since at a suffi-ciently high temperature the dominating electron-phononscattering typically results in a strong positive tempera-ture dependence, which has been well understood in con-ventional metals over a century ago, and in 2DEG over 20years ago . Therefore, it is highly unusual and counter-intuitive to observe a nearly flat temperature dependencein the intermediate temperature regime in a Fermi liquidsystem, which is what we report in this work.The application of a magnetic field brings another di-mension of possibility to the electrical transport prop-erties. While applying a magnetic field perpendicularto a 2D system leads to the extensively studied quan-tum Hall physics , here we focus on the relatively un-explored case when the field is applied parallel to the2D conducting plane. In a typical quantum well 2DEGsample, the parallel field induced quasiparticle cyclotronmotion is largely forbidden by the confinement potential, and it is often assumed that the most important effectof the parallel field is spin polarization . However,for a 2DEG with a large quantum well width, the cou-pling between the field and the ˆ z degree of freedom of thequasiparticle motion can be highly nontrivial when themagnetic length l (cid:107) ≡ (cid:113) ¯ hceB (cid:107) becomes comparable to thequasiparticle confinement width d z ≡ √ < z > , givingrise to mixing of 2D subbands. In a previous study ,in the low temperature limit ( < I (cid:107) B and I ⊥ B , the impactof the magneto-orbital coupling has been observed as acolossal magnetoresistance (CMR) with two changes inslope in the vicinity of 9 T, where the condition l (cid:107) ≈ d z ismet. This is in good agreement with the magneto-orbitalcoupling theoretical arguments, suggesting that the 2Dsystem gradually evolves into a novel quasi-3D phase inthis field range, as different 2D subbands become stronglycoupled.The magneto-orbital coupling has a profound and un-expected impact on the temperature dependence of the2DEG in the electron-phonon scattering regime. Naively,the I (cid:107) B case might be expected to be relatively sim-ple, since the Hamiltonian m ∗ (ˆ p − eA ) has no fieldterm along the current direction; we nevertheless findan apparent field-induced metal-insulator transition phe-nomenology characterized by a sign change in the tem-perature slope ∂ρ xx ∂T below ∼
10 K . The “transition”occurs within a narrow field window near ∼ . . Thenegative temperature slope is not related to a true “in-sulator” phase (in fact, the metallicity parameter k F λ remains large at high fields, λ being the mean free path),but an indication of a field-suppressed Fermi tempera-ture T F ( B ) (cid:28) T in the quasi-3D phase. We note that a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b this “classical” regime is accessible in conventional met-als ( T F ∼ K) only in the plasma phase. By contrast,in our case the electron-phonon scattering is still dom-inant at an intermediate temperature range ( >
10 K),signalling a novel regime for theoretical understandingand experimental exploration.In this work, we explore the temperature dependencein the I ⊥ B case on the same 2DEG sample. Surpris-ingly, the phenomenology here is distinctively differentfrom that in the I (cid:107) B case. It does not show the neg-ative temperature slope below ∼
10 K at all. Instead,the system remains largely “metallic” ( ∂ρ xx ∂T >
0) below ∼
20 K and above ∼
40 K, while in the intermediate rangethe temperature dependence of resistance is largely sup-pressed by a parallel magnetic field as low as 5 T. Thedistinct phenomenologies, occurring at field and temper-ature ranges that differ by about a factor of two, suggestthat the contrasting behaviours may result from entirelydifferent mechanisms, although both could ultimately bedriven by the same magneto-orbital coupling. The occur-rence of these contrasting and distinct phenomenologiesin a nearly ideal Fermi gas system poses an intriguingtheoretical challenge. The flat temperature dependentresistivity observed in the intermediate ( ∼
20 - 40 K) tem-perature range is somewhat reminiscent of the resistivitysaturation in metals for
T >
600 K typically .The sample used in this study is a typical Al-GaAs/GaAs/AlGaAs 2D system with an ultra-high mo-bility of µ (cid:39) cm /V · s . It is chosen with a chargecarrier density of n (cid:39) cm − so that the spin po-larization effect is very weak ( E z /E F less than 10 % at10 T). The electron-electron interaction energy E ee isalso comparable to the Fermi energy E F ( r s = E ee E F isabout 1.8 in 2D and 1 in quasi-3D), allowing the sys-tem to be described as a weakly interacting Fermi liquid.Given the very high mobility, disorder and localizationeffects should not play a significant role. In such a highmobility sample, we expect the resistivity behaviour in amagnetic field to be dominated by the modification of thefermi surface of the 2DEG, and its interaction with thephonon bath. The well width is chosen to be 40 nm sothat the magneto-orbital coupling effect is strong , andonly a single subband is occupied at zero field. The sam-ple has a rectangular shape with a long-to-short axis ratio ∼ R xx and the Hall resistance R xy were measured inthe rectangular Van der Pauw geometry using a standardlow frequency (13.5 Hz) lock-in technique, with 100 nAcurrent being applied along the long axis. Using an insitu rotation stage, the 2D electron plane of the samplewas carefully aligned to be parallel to the applied mag-netic field by nulling the Hall resistance. The qualitativebehavior of R xx ( B, T ) was confirmed to be insensitive topossible angular systematic errors. Its robustness againstpossible field and temperature errors was also verified byreproducing the data in three facilities with distinctivelydifferent field, temperature control and thermometry set-tings. Unless otherwise stated, the data we present here were taken in the NHMFL hybrid magnet facility, whichsupplied magnetic fields from 0 to 45 T.
Field ( T )
Field ( T ) R xx ( k Ω ) R xx ( B , T ) / R xx ( , T ) Note: scale the <10.5 T data down by subtracting a hall resistance Rxy~ [email protected] and [email protected]
Note: scale the <10.5 T data down by subtracting a hall resistance Rxy~ [email protected] and [email protected] I ⊥ B I ⊥ B FIG. 1. (Color online) Examples of longitudinal resistance R xx as a function of field at several temperatures in the config-uration I ⊥ B . Inset: the normalized resistance as a functionof field at several temperatures (matching those in the mainpanel) in the configuration I ⊥ B . The field dependence of R xx is shown in Fig. 1. At alltemperatures from 1.8 K to 43 K, the resistance increaseswith an increasing field. The low temperature ( < . < . With an increasing temperature,the qualitative shape of the magnetoresistance remainslargely unchanged.Our main finding, a resistivity saturation behaviourin the intermediate ( ∼
20 - 40 K) temperature regimebefore optical phonon scattering becomes important, ispresented in Fig. 2. Fig. 2a shows examples of the lowfield temperature sweep data taken at 0 T, 5 T, 8.5 Tand 10.5 T. At zero field, the resistance increases lin-early with temperature, while above ∼
40 K an exponen-tial increase becomes apparent. These temperature de-pendences are well described by an established theory as the signatures of the electron-acoustic-phonon scatter-ing and electron-optical-phonon scattering respectively.However, starting from 5 T, we observe the developmentof an apparent suppression of the temperature depen-dence in an intermediate temperature range from approx-imately 10 to 40 K. This suppression becomes strongeras the magnetic field increases, and at ∼ ∂ρ xx ∂T ∼ ∼ ∂ρ xx ∂T gradually becomes weakly negativewith an increasing field. Above 11.5 T, R xx ( T ) at differ-ent fields were extracted from the field sweep data takenat several fixed temperatures, but similar resistance satu-ration behaviours were still observed, as shown in Fig. 2c.In fact, the weakly negative slope of R xx ( T ) from 20 Kto 40 K persists to the highest field of 45 T, with bothits upper and lower boundaries showing only a weak fielddependence. On the other hand, the linear temperaturedependence at low temperature ( <
10 K) and the ex-ponential temperature dependence at high temperature( >
40 K) persist from zero field to high magnetic fields, assuggested by the normalized resistances shown in Fig. 2band Fig. 2d. In both the low field ( <
11 T) and high field( >
11 T) data, below ∼
10 K the normalized resistivitycurves all join a single trend, implying that the acousticphonon scattering remains robust against a strong mag-netic field, and that the resistance is simply scaled bya field-dependent factor. A similar argument can alsobe applied to the optical phonon scattering, as the ex-ponential temperature dependence beginning at >
40 Kremains apparent at all magnetic fields. If this exponen-tially increasing contribution of optical phonon scatteringis subtracted out from our measured resistivity, then theflat region is extended much further into the higher tem-perature regime. Therefore, the resistance saturation inthe intermediate temperature within the phonon scatter-ing regime is truly puzzling. R xx ( Ω ) R xx ( B , T ) / R xx ( B , . K ) ab T e m p e r a t u r e ( K ) cd T e m p e r a t u r e ( K ) FIG. 2. (Color online) a) Longitudinal resistance R xx ( B, T )and b) normalized resistance from 0 T to 10.5 T in the config-uration I ⊥ B . c) Longitudinal resistance R xx ( B, T ) and d)normalized resistance from 11.5 T to 45 T in the configuration I ⊥ B . In both the low temperature and high temperaturelimits the qualitative temperature dependence remains qual-itatively unsuppressed by the applied magnetic field, but inthe intermediate region a resistivity saturation is observed. To shine more light into this puzzling phenomenol-ogy, we compare it with that in the better understood I (cid:107) B case, as shown by the examples in Fig. 3. When T →
0, the qualitative field dependence of the mag-netoresistance was found to be relatively independent ofthe current configuration . However, the magnetoresis-tance at higher temperatures was found to be increas-ingly anisotropic. In the I (cid:107) B case, the resistivity sat-uration effect is absent, and below the 2D to quasi-3D transition( ∼ : R xx = a + a T + b e − b /T T , (1)where a + a T is the acoustic phonon term, and b =¯ hω/k B is the effective optical phonon energy. As shownin Fig. 3a (with data taken in the McGill facility), fittingparameter a ( B ) seems to be simply the colossal magne-toresistance R xx ( B,
0) observed in the low temperaturelimits. The fact that a and b do not show a strongfield dependence suggests that the phonon scattering islargely unaffected by the applied field. This is perhapsnot surprising, as the quasiparticles moving along thefield direction should not “see” the magnetic field, butonly the field-deformed Fermi surface. On the contrary,in the I ⊥ B case, below ∼
10 K the linear (acousticphonon scattering) temperature slope seems to be en-hanced by a field-dependent factor. One simple possi-bility is that the effective mass m ∗ increases with an in-creasing field, resulting in an enhanced Drude resistivity ρ ∼ m ∗ . However, a single field-dependent factor can-not explain the weakly temperature dependent “plateau”above ∼
20 K, which persists over a large field range, andthe lower bound of which ( < ∼ . ∼ I (cid:107) B case (withdata taken in the Toulouse pulse field facility), there is anemerging negative temperature slope below 10 K, remi-niscent of a metal-insulator transition . This behaviourhas been interpreted as the result of a field suppressedFermi energy T F ( B ), as in the classical T (cid:29) T F regimean upturn R xx ∝ T F /T would overcome the positivetemperature slope induced by electron-phonon scatter-ing. However, this upturn is clearly absent in the I ⊥ B case (with data taken in the NHFML hybrid magnet fa-cility). In fact, the R xx ( B, T ) curves in these two config-urations show contrasting temperature dependence andcross each other below ∼
10 K.To summarize, the details of the phenomenology, andin particular the relevant energy scales, are markedlydifferent in these two current configurations. The onlyresemblance between these two cases is that there isstill an evolution from positive to negative temperatureslope in the resistivity saturation regime. One possibleinterpretation could still be a cancellation between thequantum-classical crossover induced resistivity going as ρ ∼ /T and the acoustic phonon induced resistivitygoing as ρ ∼ T . But, given that these two effects have Temperature ( K ) R xx ( Ω ) a B = 9 T, I || BB = 9 T, I ⊥ B b Temperature ( K ) B = 15 T, I ⊥ BB = 15 T, I || BB = 0 T FIG. 3. (Color online) a) The resistance anisotropy data at0 T (black square); 9 T with I (cid:107) B (red circle); 9 T with I ⊥ B (blue square), extracted from the temperature sweepdata at fixed fields. The resistivity saturation effect is onlyapparent when I ⊥ B . b) The resistance anisotropy data at15 T with I (cid:107) B (red circles); 15 T with I ⊥ B (blue square),extracted from the field sweep data at fixed temperatures.The 0 T and the 9 T I (cid:107) B data can be fit to Eq.1 with a = 0 Ω, a = 2 .
57 ΩK, b = 3 . × ΩK and b = 453K(black solid line) and a = 66 .
82 Ω, a = 2 .
92 ΩK, b =3 . × ΩK and b = 453K (red solid line). The others(9 T I ⊥ B data, 15 T I ⊥ B data and 15 T I (cid:107) B data)cannot be fit to this functional form, and the dashed lines areguides-to-the-eye. completely independent physical origins, it is difficult tosee such a perfect cancellation over a large temperaturerange. Furthermore, the interpretation of the anisotropyand the apparent energy scale differences is of great the-oretical challenge. We also note that common mecha-nisms that could lead to a negative temperature slopesuch as the spin degree-of-freedom or Coulomb interac-tions, can most likely be ruled out in such a high den-sity and high mobility sample. The cutoff temperatureas high as ∼
20 K also poses a challenge to common in-terpretations. Even the magneto-orbital coupling effect,which has been successful in the I (cid:107) B case, has difficul-ties explaining this phenomenology. Although in the thequantum Hall physics case (where I ⊥ B is always sat-isfied) field induced change in the functional power lawof the electron-phonon scattering term has been reportedbefore , to the best of our knowledge, a complete sup-pression of the overall temperature dependence has neverbeen reported.It is worth mentioning that certain aspects of the re- sistivity saturation phenomenology might have been ob-served in other systems under very different conditions.In particular, a somewhat similar phenomenology was ob-served in the work by Gao et al. below 1 K, with a muchnarrower temperature range and at a lower field range.Similarly, an evolution of a non-monotonic temperaturedependence with the carrier density was reported below1 K . There are also a few zero field studies with fixedcarrier density that perhaps show certain phenomenolog-ical resemblances to the resistivity saturation behaviourat intermediate temperature ranges (Fig. 2 in ref. ). Weemphasize, however, that none of these studies reporteda flat temperature dependence in resistivity over a largetemperature range as we have discovered. Furthermore,our system behaves like a conventional 2D metal below ∼ ∼
20 to ∼
40 K, driven by a parallel magnetic field from aslow as 5 T to as high as 45 T. The strong suppression ofthe temperature dependence of resistivity in a well estab-lished metallic system, in a wide and high temperaturerange by a relatively small field, is unprecedented, unex-pected and might not be readily understood within thecurrent theoretical framework. We believe that we havediscovered a new transport phenomenon in an interme-diate temperature regime.This work has been supported by NSERC, CIFAR,FQRNT and Microsoft Station-Q. The work at Prince-ton was partially funded by the Gordon and Betty MooreFoundation as well as the National Science FoundationMRSEC Program through the Princeton Center for Com-plex Materials (DMR-0819860). A portion of this workwas performed at the National High Magnetic Field Lab-oratory, which is supported by NSF Cooperative Agree-ment No. DMR-0084173, by the State of Florida, and bythe DOE. We also thank T. Murphy, E. Palm, R. Talbot,R. Gagnon and J. Smeros for technical assistance. P. W. Anderson, Phys. Rev. , 1492 (1958). E. Abrahams et al. , Phys. Rev. Lett. , 673 (1979). N. F. Mott, Proc. Phys. Soc. A , 416 (1949). T. Kawamura and S. Das Sarma, Phys. Rev. B , 3725(1990). T. Kawamura and S. Das Sarma, Phys. Rev. B , 3612(1992). K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev.Lett. , 494 (1980). D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev.Lett. , 1559 (1982). V. M. Pudalov et al. , JETP Lett. , 932 (1997). D. Simonian et al. , Phys. Rev. Lett. , 2304 (1997). J. Yoon et al. , Phys. Rev. Lett. , 4421 (2000). Xiaoqing Zhou et al. , Phys. Rev. Lett. , 216801 (2010). Xiaoqing Zhou et al. , Phys. Rev. Lett. , 086804 (2011). S. Das Sarma and E. H. Hwang, Phys. Rev. Lett. , 164(1999). E. H. Hwang and S. Das Sarma, Phys. Rev. Lett. , 5596(2000). O. Gunnarsson, M. Calandra and J. E. Han, Rev. Mod.Phys. , 1085 (2003) L. Smrˇcka and T. Jungwirth, J. Phys. Cond. Matt. , 55(1994) S. Das Sarma and E. H. Hwang, Phys. Rev. B , 205303 (2005); , 035311 (2005), and references therein. S. Das Sarma and E. H. Hwang, Phys. Rev. B , R7838(2000). H. L. Stormer et al. , Phys. Rev. B , 1278 (1990). Xuan P.A. Gao et al. , Phys. Rev. Lett. , 166803 (2002). A. P. Millis et al. , Phys. Rev. Lett. , 2805 (1999). B. Spivak et al. , Rev. Mod. Phys.82