N A N O E X P R E S S
Open Access
Pure electron-electron dephasing in percolative
aluminum ultrathin film grown by molecular
beam epitaxy
Shih-Wei Lin
1, Yue-Han Wu
2, Li Chang
2, Chi-Te Liang
3,4and Sheng-Di Lin
1*Abstract
We have successfully grown ultrathin continuous aluminum film by molecular beam epitaxy. This percolative aluminum film is single crystalline and strain free as characterized by transmission electron microscopy and atomic force microscopy. The weak anti-localization effect is observed in the temperature range of 1.4 to 10 K with this sample, and it reveals that, for the first time, the dephasing is purely caused by electron-electron inelastic scattering in aluminum.
Keywords: Weak anti-localization; Ultrathin metal films; Pure electron-electron dephasing Background
Weak localization (WL) is the quantum correction to the conductance which occurs in weakly disordered sys-tems due to coherent backscattering of electrons (or holes). As a results of spin-orbit coupling, weak anti-localization (WAL) may be observed in a weakly disor-dered electron (or hole) system [1]. The theoretic deriv-ation and experimental proof of WAL were extensively developed since 1980s, and the investigation on various materials in all dimensions has been a central topic in condensed matter physics for decades. In particular, a wide variety of experimental results of WAL were ob-tained in two-dimensional (2D) systems. 2D system is suitable for experimental study of WAL because of its stronger WAL contribution than three-dimensional (3D) ones and its easier sample fabrication than one-dimensional (1D) ones. Recently, due to its sensitivity to the electron dephasing and spin dephasing, WAL has been widely applied to studying the spin-orbit inter-action in new materials, such as graphene, topological insulator, magnetic-doped semiconductor, and narrow-gap semiconductor, to evaluate the potential for spintro-nics devices [2-7].
In addition, the interplay between superconducting ef-fect and WAL also attracts much attention and has been investigated extensively. Ebisawa et al. derived the rela-tionship between the superconducting pair-breaking par-ameterδ and the inelastic scattering rate τi−1by δ = (πℏ/ 8kBT)τi−1 [8]. Their results have been used to study ex-periments of WAL in aluminum thin films [9-11]. How-ever, it is a chanting challenge to grow an ideal 2D superconducting metallic sample. As the thickness of evaporated metal goes thinner, the discontinuity of the metal film un-avoidably appears due to large lattice mis-match between the template and metallic material as well as the surface non-uniformity of the bottom tem-plate. Previously, the reported metallic films were in the thickness of ten to a few tens of nanometers [9-14]. Al-though plenty of the theoretic works of WAL in two-dimensional systems have been published in the past few decades, the experimental proof toward WAL in an ideal two-dimensional metallic system is still lacking. In this work, we have used molecular beam epitaxy (MBE) sys-tem as the deposition technique to prepare ultrathin Al films. By using gallium-rich GaAs as the epi-template, we are able to successfully deposit ultrathin percolated Al film for studying WAL toward the 2D limit. Interest-ingly, we have observed a pure electron-electron dephas-ing in this sample over the whole temperature range that WAL effect exists. Note that all of our characteriza-tions including structural and electrical assessments
* Correspondence:[email protected]
1
Department of Electronics Engineering, National Chiao Tung University, 1001 University Road, Hsinchu 30010, Taiwan
Full list of author information is available at the end of the article
© 2015 Lin et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
were fulfilled ex situ, and the continuity of the Al film remains even after the post-processing for the Hall de-vice fabrication. Even though our sample is not thin enough to reach the ultimate two-dimensional limit such as monolayer graphene, our results still provide an ex-perimental proof that Nyquist scattering becomes the dominant inelastic scattering mechanism at all tempe-ratures when the system approaches an ideal two-dimensional one.
Theory
Magneto-resistance measurements are commonly used to study WAL. The theoretical calculation of 2D WAL in a perpendicular magnetic field was derived by Hikami, Larkin, and Nagaoka [1]. The difference of conductance induced by applied magnetic field can be expressed as:
ΔgWAL¼ g Bð Þ−g 0ð Þ ¼πhe2 3 2Y B2 B −1 2Y Bi B : ð1Þ
where B is the applied magnetic field. Y(x) represents Ψ(1/2 + x) − ln(x), B2= Bi+3/4Bso, Ψ(x) is the digamma function. Here, Bi and Bsorepresent the strength of elec-tron dephasing and spin-orbit interaction, respectively. For a superconducting material near its critical temperature Tc, superconducting fluctuations must be considered. Maki has calculated the effect of superconducting fluctuations in a 2D system (ΔgMT) [15]. Later, Thompson has modi-fied the model by introducing the superconductor pair-breaking parameter δ to avoid the unphysical divergence at temperature near Tc[16]. The Maki-Thompson correc-tion term was modified by Abrahams et al. that can be ap-plied to higher field [17]:
ΔgMT ¼ −β e2 πh Y Bi B −Y BT B ; ð2Þ
whereβ represents the interaction strength between the electron pair [18]. Accordingly, β ~ 1/[ln(T/Tc)− δ] at temperature close to Tc[17]. BT=2kBTln(T/Tc)/πDe and Dis the diffusion coefficient, D = 1/3vFl0. Here, vFis the Fermi velocity and l0is the mean free path.
Bergmann experimentally studied WAL in various me-tallic thin films [19] and confirmed that the Maki-Thompson correction gives the most contribution of superconducting fluctuations and cannot be neglected even for temperatures far above Tc. Therefore, in our fol-lowing analysis, both WAL effect and Maki-Thompson term will be taken into account.
Methods
Sample fabrication
The epitaxial aluminum thin film was fabricated on a semi-insulating gallium arsenide (SI-GaAs) substrate in
our Varian Gen II MBE system equipped with an arsenic cracking cell. First, a GaAs wafer was heated to 620°C for 20 min for de-oxidation under As flux, and then a 300-nm-thick undoped GaAs buffer layer was deposited at 590°C as the epitaxial template. The sample surface was turned into Ga-rich at 620°C in the absence of As flux. We kept the sample in high vacuum (approximately
3 × 10−10 Torr) until the background As vapor was
pumped out. The Al film with an intentional thickness of 3 nm was then deposited at room temperature with a growth rate of 0.366μm/h.
The 50-μm-wide Hall devices were processed using con-ventional lithography technique. Al was etched by 2% tetra-methylammonium hydroxide (TMAH) for 20 s to transfer the Hall bar pattern from photoresist to our Al film. Ti/Au (30/300 nm) was deposited using e-gun evaporation as the contact electrode. The finished devices were wire-bonded on a ceramic carrier and loaded into4He cryogenic system equipped with a superconducting magnet. A DC four-terminal electrical measurement has been performed in this work for studying WAL. We have used a Keithley 2602 multi-meter (Keithley Instruments Inc., Cleveland, OH, USA) as a constant current source as well as a voltage meter. Electrical current was set at 3 μA for clear signals and was low enough in order to avoid possible current heating effect. Voltage noise level was at about 5μV.
Results and discussion
We have used cross-sectional high-resolution transmission electron microscopy (TEM) and atomic force microscopy (AFM) for the investigation of film crystal quality and surface morphology. Figure 1 shows the TEM image of our epitaxial Al film taken with electron beam along the < 011>GaAszone axis. A clear interface between the GaAs template and deposited Al can be seen in the middle of the picture. At the bottom of Figure 1, we can see an amorphous layer which may be amorphous carbon from TEM specimen preparation. The thickness of Al layer is about 8 nm which is thicker than the deposited amount because our Al film is percolative as we shall see later in the AFM images. A detailed examination of the TEM image at the interface can show the existence of misfit dis-locations, implying that most of the strain in the Al film caused by lattice mismatch between Al and GaAs is re-leased. The inset of Figure 1 is the fast Fourier transform diffraction pattern taken around the interface between GaAs and Al. Clear diffraction spots can be seen. The inner hexagon is the diffraction spots of the bottom GaAs template, and the outer one is for epi-aluminum film, indicating that the epitaxial relationship of Al with GaAs is (100)GaAs// (111)Al, <011 >GaAs//< 211>Al, and < 011>GaAs // < 011>Al; the last axis is also the obser-vation plane of the TEM image. The axis arrangement is
different from the previous works [20-22] probably due to the different surface conditions used. Clear spots indicating good crystal quality and no deformation of the diffraction spots are observed even the strain accumulation due to lattice mismatch could occur in our sample. Figure 2 shows a 1 × 1 μm2 AFM image of our Al film. Obviously, a
percolating but continuous morphology has been seen. This kind of morphology was generally observed when metal films are deposited onto a semiconductor or dielectric tem-plate because of poor affinity between them [23]. Although the bi-polarized atomic bonding of the bottom GaAs tem-plate or inhomogeneous surface atomic deformation could
Figure 1 Cross-sectional TEM image of the Al sample. GaAs/Al interface in the middle and the native oxide of Al at the bottom are clearly spotted. The upper left inset is the diffraction pattern showing the GaAs (Al) diffracted spots in the inner (outer) hexagon.
also play a role here, we have used a smooth Ga-rich sur-face as the epitaxial template to minimize these two issues. The roughness of the film is about 4.9 nm which is thicker than the deposited thickness and is possibly caused by the Al oxidation after the exposure to the air.
We have measured the sheet resistance Rsof our sample in the temperature range of 1.4 to 10 K. At temperatures higher than 10 K, the signals became noisy and WAL is barely observable. We noted that the Hall resistance RH is small compared to Rs and remained unchanged even in-creasing the magnetic field to 1 T. Thus, we neglected the effect of RHin all the calculation and theoretical fitting. In Figure 3, the measured Rs is plotted against the applied magnetic field at various temperatures, together with the fit to the theoretical model. Around zero magnetic field, clear WAL in our Al film occurs as Rsincreases with increasing magnetic fields. In addition, Rsdecreases dramatically when the temperature goes below 4.5 K, indicating that the superconducting fluctuation plays an important role here. Therefore, when fitting our data to the theoretical model, we have considered the WAL theory stated in Equation 1 with the first Maki-Thompson term in Equation 2. The later term of Equation 2 vanishes because BT is always much larger than Biand Bso. The contribution of this term is much less. So, the used fitting formula is given by:
Δg ¼ e2 πh 3 2Y B2 B −1 2Y Bi B −βY Bi B : ð3Þ
In our fitting procedure, Bi and β were
temperature-dependent fitting parameters. Bso was chosen as a
temperature-invariant constant because it is commonly accepted that the electron configuration in the half-filled conduction bands is insensitive to the temperature, so the angular momentum and the spin-orbit interaction remains unchanged at all temperature in metals [10-13]. In Figure 3, the solid lines represent theoretic fits of our experimental data at three temperatures as an example. It is clear that the theory can only be applied to small magnetic field. At higher temperatures, the fits can be extended to higher magnetic fields. This is because the Maki-Thompson term is only valid at B≪ kB(T− Tc)/ 4eD [15-17].
With the fitted Biand Bso, the inelastic scattering timeτi and spin-orbit scattering timeτsocan be derived byτi= h/ (8πeDBi) and τso= h/(8πeDBso), as well as the phase co-herent length liand spin-orbit interaction length lsoby li= (h/8πeBi)1/2and lso= (h/8πeBso)1/2. We used Al bulk con-centration nAl= 1.81 × 1029 (m−3) to estimate the Fermi velocity (vF= 2.03 × 106m/s) and Fermi wavelength (λF= 0.36 nm) for our sample. Figure 4a shows the extracted parameter 1/τi as a function of temperature. The solid horizontal line indicates the level of spin orbit interaction rate 1/τso, and the dashed line represents the theoretic cal-culation of electron-electron scattering (Nyquist scatter-ing) rate based on the work of Altshuler et al. [24]:
τN−1¼ e2R s 2πℏ2kBTln πℏ e2R s : ð4Þ
In our 3-nm-thick aluminum sample, weak anti-localization changes into weak anti-localization when the
Figure 3 Measured and calculated sheet resistance. Measured (symbols) and calculated (solid lines) sheet resistance as a function of magnetic field of the Al film at 1.639, 2.248, and 4.650 K.
temperature goes above 10 K. In the temperature range for existing WAL (T < 10 K), we found that 1/τiis proportional to T, which agrees with the theoretic prediction of Nyquist scattering (dashed line) in the same order.
We would like to emphasize that our ultrathin Al film has a very high carrier concentration (n ~ 1029 m−3), so it is actually in mixed dimensions. That is, considering the density of states and diffusive motion, it is treated as a 3D system because the film thickness d is much larger than the Fermi wavelengthλF(d≫ λF~10−1nm) and its mean free path of electrons l0(d≫ l0~10−1 nm in our sample). However, in terms of electron dephasing, the
film thickness is much less than the dephasing length li(d≪ li~ few tens of nanometers) so the system is two-dimensional. Previous experimental studies on clean and dirty Al thin films demonstrated that, only for the temperature near Tc, Nyquist scattering was the major dephasing mechanism. When the temperatures were higher than about 4 K, electron-phonon scattering domi-nated the dephasing process [9-14]. In contrast, our re-sults show a pure electron-electron dephasing all the way up 10 K. For a type I superconducting material, there is only one experimental result on clean titanium film exhibiting a full T−1 dependence of dephasing rate
Figure 4 Temperature-dependent dephasing rate and interaction strength. (a) Temperature-dependent dephasing rate (1/τi) (symbol) of
the Al sample. The horizontal solid line represents the level of spin-orbit interaction rate (1/τso) and the dashed line represents the theoretic
[25]. However, our sample is in the dirty limit as the sheet resistivity of our sample is over 1.6 kΩ, at least one order higher than those all of their samples. On the other hand, for type II superconductors such as ZrRh and TaN, most experimental works did not observe the pure T−1 dependence of dephasing rate [26,27]. It is worth mentioning that the observation of Giannouri et al. [28] with their NbTa film is very similar to ours except that the sudden drop of dephasing rate at temperature approaching Tc has not been seen even at 1.4 K in our experiment.
Dephasing process which purely comes from electron-electron scattering observed here is unusual but is rela-tively easy to be obtained in semiconductor-based 2D systems, such as 2D electron or hole gases (2DEG and 2DHG), formed by modulation-doped heterostructures. The carriers (electrons or holes) are confined by the tri-angular potential well caused by remote ionized dopants and limited in the few monolayers next to the hetero-structure interface. The carrier concentration of 2DEG or 2DHG is controlled in the range of 1010~ 1012 cm−2, so the Fermi wavelengthλFeasily exceeds the size of the confinement potential well and the system is quantized in this dimension to become two-dimensional. However, for metallic materials even down to a few nanometers, the carrier concentration is still as high as 1015 cm−2, which makes the λFis much smaller than its thickness. Recently, the experimental results of WAL in various semiconductor heterostructures were published, including AlGaN/GaN, GaAs/InGaAs, InP/InGaAs, and AlGaAs/ GaAs [29-35]. Most of these works demonstrated a full T−1dependence of the dephasing rate up to 10 K and con-sidered as consistent with the theoretic calculation of Nyquist scattering because of their 2D nature. In our Al film, the observed pure 2D-like dephasing indicates that the superconducting metallic film can be an ideal 2D sys-tem in inelastic process.
We wish to address the issue of zero-temperature sat-uration rate (1/τi0) that is the dephasing rate in the zero temperature limit. The origin of the saturation rate is still under debate for the time being [36]. Some of the reported experimental works on superconducting metal-lic materials revealed a divergence dephasing rate when the temperature approached Tc [9,10,13,26,27]. In con-trast, such situations have not been observed in our sample that exhibits a full Nyquist scattering rate from liquid helium temperatures to 10 K, thus could be help-ful for developing the related theory.
In Figure 4b, the electron-electron interaction strength parameter β in the Maki-Thompson correction term is plotted as a function of temperature. It is clear that β has a trend to diverge at temperature approaching Tc, as expected with Larkin’s theory [18]. The value of our sample decreases dramatically with the increasing
temperature because the superconducting effect be-comes less significant. It is noted that β does not vanish but converge to about 0.9 instead. A similar result was observed in ZrRh films [26]. Further investigations on this non-zeroβ are certainly needed.
We note that the sheet resistivity of our sample is about 1.6 kΩ. The mean free path l0 actually is 0.082 nm estimated by the Drude model, which is slight shorter than λF. This appears to contradict to the basic assumption of the WAL theory that WAL occurs in a weakly disordered system (l0≫ λF). In a strongly disor-dered system, WAL should not be observed. We believe that, due to the percolative morphology of our sample, the actual electron path is much longer than the Hall bar size; therefore, the mean free path l0 has been underestimated.
Similar to the WAL effect, the universal conductance fluctuations (UCFs) due to the electron (or hole) inter-ference between two classical paths is closely related to the phase coherent length li[37-40]. We have also exam-ined the conductance fluctuations in our measured data, but no clear UCFs are observed, which could be due to the large size of our device and the limit of the measure-ment system noise (approximately 5μV).
Conclusions
We have presented the structural and electrical characte-rization of the ultrathin percolating aluminum film grown by MBE. The TEM results indicate a superior crystal qual-ity of the epitaxial aluminum film. WAL revealed by low-temperature magnetoresistance measurement showed its unusual dephasing mechanism. At all temperatures that WAL exists, a pure electron-electron scattering was ob-served, so the aluminum film behaves as an ideal two-dimensional system in this aspect. Based on this, we conclude that the MBE-grown aluminum films could achieve the two-dimensional limit of a superconducting metallic material.
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
S-WL carried out the sample growth, electrical measurements, and data analysis and drafted the manuscript. Y-HW and LC performed the TEM sample preparation and characterization. LC, C-TL, and S-DL participated in the data analysis and model discussion and helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
We thank the financial support from the NSC and ATU program of MOE in Taiwan. The equipment help from the Center for Nano Science and Technology at the National Chiao Tung University and National Device Laboratory is highly appreciated. C-TL acknowledges the support from the MOST, Taiwan (grant numbers: MOST 103-2918-I-002-028 and MOST 102-2119-M-002-016-MY3). Author details
1
Department of Electronics Engineering, National Chiao Tung University, 1001 University Road, Hsinchu 30010, Taiwan.2Department of Materials
Science and Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan.3Department of Physics, National Taiwan University, Taipei 10617, Taiwan.4Geballe Laboratory for Advanced Materials (GLAM), Stanford University, Stanford, CA 94305, USA.
Received: 12 December 2014 Accepted: 24 January 2015
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