Abstract
Magnetotransport measurements are performed on an aluminum thin film grown on a GaAs substrate. A crossover from electron to holedominant transport can be inferred from both longitudinal resistivity and Hall resistivity with increasing the perpendicular magnetic field B. Also, phenomena of localization effects can be seen at low B. By analyzing the zerofield resistivity as a function of temperature T, we show the importance of surface scattering in such a nanoscale film.
Introduction
Aluminum has found a wide variety of applications in heat sinks for electronic appliances such as transistors and central processing units, electrical transmission lines for power distribution, and so forth. As a result, it is highly desirable to prepare highquality aluminum materials for practical device applications. In particular, the epitaxial growth of Al thin films on GaAs substrates has attracted much interest because of its relevance to the field of electronic interconnects [1,2]. Fundamental limitations on the speed of interconnects are the various scattering processes [3,4] occurring in lowdimensional systems. In order to fully utilize it in the integrated circuits consisting of GaAsbased high electron mobility transistors, investigations of the scattering mechanism on an Al thin film grown on a GaAs substrate are necessary.
One of the most important issues regarding the power dissipation and the speed of the device is the inelastic process such as electronphonon scattering and electronelectron scattering. It is also important for the illustrations of quantum interference phenomena [512], one of which is weak localization [WL]. In the WL regime, phasecoherent loops formed by the paths of electrons undergoing multiple scattering events and the timereversed ones lead to constructive interference at the original position of electrons at zero magnetic field under the assumption that the inelastic scattering time is much larger than the elastic one. However, phase coherence would be destroyed under a perpendicular B and lead to the negative magnetoresistance [NMR]. Positive magnetoresistance [PMR] can also be observed in the WL regime if the spinorbit scattering [6,8,12] is strong enough.
Here, we review the temperature dependences of resistivity for various scattering mechanisms [13,14] that are generally observed in bulk materials. At low temperatures, T (lower than the Debye temperature), electronphonon scattering is usually the dominant one, which is expected to give a BlochGruneisen T^{5 }contribution to the resistivity. However, for the materials with complex Fermi surfaces or are suffering from interband scattering, Umklapp process [1315] should be taken into account, leading to the T^{3 }dependence instead. Umklapp process means that the crystal momentum is not conserved after an electronphonon scattering event. A reciprocal lattice vector is added after this process, possibly leading to a largeangle scattering [1517]. That is, the resistivity would not decrease as rapidly as T^{5}, which introduces an additional factor of T^{2 }for the lowangle phonon scattering at low T. Also, the T^{2 }term expected for electronelectron scattering may possibly appear at low T [13,15], while at extremely high T (much larger than the Debye temperature), the resistivity follows AT [15], where A is a constant depending on the properties of the system.
It is well known that electronic transport is significantly affected by surface scattering [1820], in addition to electronelectron scattering and electronphonon scattering, as the thickness of a system is reduced to become comparable to the electron mean free path. There are several theories dealing with surface scattering.
As proposed by Olsen [21], neglecting the Umklapp process, lowangle scattering of electrons by phonons is important in a thin film where electrons are deflected by lowenergy phonons to the surface [22,23] more easily than that in the bulk sample. That is, surface scattering occurs frequently in a thin film. A more careful treatment for the size effects considering the surface conditions is proposed by Soffer [24]. Here, we use Soffer's theory as the beginning of our analyses for the zerofield resistivity.
An Al thin film is investigated in our experiments especially for its special properties. With increasing B, a crossover from electron to holedominant transport occurs as a result of its nonsimple Fermi surface [2528]. Also, it is a good material for the investigations of quantum phenomena in lowdimensional systems ascribed to its long inelastic scattering time [7].
Experimental details
The sample used in this study was grown by molecular beam epitaxy [MBE]. The following layer sequence is grown on a semiinsulating GaAs (100) substrate: 200nm undoped GaAs and 60nm Al film. All the processes were performed in the ultrahighvacuum MBE chamber to prevent unnecessary defects. The Al thin film investigated here is a single crystalline, which can be checked by the Xray shown in Figure 1a. Figure 1b shows an atomic force microscopy [AFM] image of the Al thin film. Fourterminal magnetotransport measurements were performed in a toploading He^{3} system equipped with a superconducting magnet over the temperature range from T = 4 K to T = 78 K using standard ac phasesensitive lockin techniques. The magnetic field is applied perpendicular to the plane of the Al thin film. It is necessary to mention that all the resistivity results have been divided by the thickness (60 nm).
Figure 1. Xray and AFM of the Al thin film. (a) The φ scanning of Al(111) peak of the sample. (b) An AFM 5 × 5μm^{2 }image of a 60nmthick Al thin film.
Result and discussion
Longitudinal resistivity and Hall resistivity (ρ_{xx }and ρ_{xy}) as a function of magnetic field B at various temperatures T are shown in Figure 2a,b, respectively. PMR [7,9] can be observed at all T. It is generally believed that PMR is proportional to the quadratic B in the lowfield region followed by a linear dependence on B with increasing B for noncompensated (the numbers of electrons and holes are different) metals [14,26], such as aluminum investigated here. A classical PMR based on the twoband model [14,15,29] results in this B^{2 }dependence in the lowfield regime where the Fermi surface is spherical. With increasing B, the number of electrons undergoing Bragg reflection at the cusps in the second Brillouin zone increases, leading to the linear dependence on B for ρ_{xx }[26,27]. Another phenomenon regarding the crossover from electron to holedominant transport is the reverse of the sign of the Hall resistivity [28] with increasing B, as presented in Figure 2b. Such a bipolar phenomenon with increasing B can also be understood by the Bragg reflection occurring at the cusps, leading to the holelike orbit.
Figure 2. Resistivity at various temperatures T. (a) Longitudinal resistivity, ρ_{xx}. (b) Hall resistivity, ρ_{xy}, as a function of magnetic field B at various temperatures T.
While deviations from the B^{2 }dependence in the lowfield regime at various T can be observed in Figure 3a, it is beyond the classical mechanism. Thus, we know that quantum interferenceinduced corrections are needed to be taken into account for the exact illustration of our results. The contribution of weak localization [6,10] is usually dominant for T ≧ 20 K. At high B, ρ_{xx }shows a trend toward a linear dependence on B, shown in Figure 3b, representing that the holelike transport becomes dominant indeed. It is worth mentioning that the PMR can still be observed at T ≧ 20 K, without turning into the NMR [6]. Most of the measurements on Al [610] show that the PMR is almost diminished at T > 10 K due to its weak spinorbit scattering. As suggested by Bergmann et al. [7], PMR almost diminishes at T ≧ 9.4 K for Al in the lowfield regime. In order to study the scattering mechanisms in different T ranges, we analyzed the zerofield ρ_{xx }as a function of T in the next section.
Figure 3. Deviations from the B^{2 }dependence in the lowfield regime at various T. ρ_{xx }as function of B^{2 }(a) and B (b). The dotted lines in blue represent linear parts of the data.
As shown in Figure 4a, for 4.8 K ≦ T ≦ 78 K, the metallic behavior can be observed without a transition to the insulator, as is the case for a pure metal [11]. The mean free path for the bulk Al is approximately equal to 17.5 μm [23], substantially larger than the thickness of the thin film studied here (60 nm). It prevails that surface scattering is important instead of the grain boundary scattering in such a thin film. For a polycrystalline material, grain boundary scattering needs to be considered, while for the single crystal, it is a minor effect. In accordance with Soffer's model [24] of surface scattering and the extensive work of Sambles et al. [19,20], the resistivity takes the form
Figure 4. Resistivity and metallic behavior. (a) Zerofield resistivity as a function of T ranging from T = 4.8 K to T = 78 K. The red solid line corresponds to a fit to Eq. (1). The best fit is limited at T > 30 K, as shown in the inset. (b), (c) ρ_{xx }(B = 0) as functions of T^{2 }and T^{3}, respectively. The red dashed lines are a guide to the eye.
where A and B are systemdependent constants. The first term represents the residual resistivity. The second and the third terms are due to electronelectron scattering and BlochGruneisen electronphonon scattering, respectively. The fittings of Eq. (1) to the resistivity over the whole temperature range and above T = 30 K are shown in Figure 4a and its inset, respectively. It can be seen that the good fitting is limited to the temperature above 30 K. The obtained coefficient of T^{2 }dependence is approximately equal to 600 fΩmK^{2}. However, Soffer's theory cannot produce such a large T^{2 }term over such a wide temperature range 30 K < T < 78 K. Also, electronelectron scattering would not exist at such high T. It is believed that the violation of Soffer's theory in aluminum is due to its complex Fermi surface. As suggested by Sambles et al. [30], T^{2 }dependence can exist alone without a T^{5 }term, which is derived by considering the Umklapp scattering process occurring at the surface for materials with a disconnected Fermi surface [31]. Figure 4b shows that ρ_{xx }follows the T^{2 }dependence as T > 30 K, indeed consistent with the model of surface Umklapp scattering. On the other hand, it shows a trend toward a T^{3 }dependence with decreasing T below 30 K, as shown in Figure 4c, which can be ascribed to the electronphonon scattering introducing the Umklapp process, usually observed in the bulk material [13]. Even though we know that the Umklapp process is likely to be important in our system, the crossover from T^{2 }to T^{3 }dependence with decreasing T can still be explained by Olsen's argument for lowangle scattering qualitatively. At relatively low T, the magnitude of the momentum of phonons is too small to induce the size effect such that the Umklapp scattering process occurring in the interior may possibly be dominant over that occurring at the interface. Thus, the crossover from the T^{2 }dependence to T^{3 }dependence of resistivity with decreasing T below 30 K can be predicted. A similar T^{2 }term can be observed for 46 K < T < 90 K performed in a subsequent cooldown in a closed cycle system, as shown in Figure 5. A deviation from this dependence at T > 90 K is ascribed to the mean free path shortening with decreasing T. Thus, the size effect becomes less important, also consistent with Olsen's argument. At T > 105 K, ρ_{xx }shows a tendency toward a linear dependence on T, as shown in the inset of Figure 5. A classical model has predicted such a linear term at high T (much larger than the Debye temperature, about 394 K for aluminum). However, our result is not in this case. The onset of this linear dependence with increasing T and how the size effects modulate the magnetoresistance requires further investigations.
Figure 5. ρ_{xx }as a function of T^{2 }performed in a subsequent cooldown in a closed cycle system ranging from T = 46 K to T = 298 K. Inset: ρ_{xx }as a function of T, where the red dashed line represents the linear fit at T > 105 K.
Here, it is worth mentioning that the electronphonon impurity interference also leads to the T^{2 }contribution to the resistivity [3234], which should be smaller than the residual resistivity. However, in our results, the difference between ρ(T = 78 K) and ρ(T = 30 K) is approximately equal to 0.059 Ω, which is larger than ρ(T = 4.8 K) = 0.025 Ω, taken as the residual resistivity, inconsistent with the requirement for the correction term. Also, there are several experimental results indicating that such a mechanism is not the dominant one for a relatively pure metal. Therefore, we can safely neglect the influence of the electronphonon impurity interference in our Al thin film.
Conclusions
In conclusion, we have performed magnetotransport measurements on an aluminum thin film grown on a GaAs substrate. A crossover from electron to holedominant transport can be inferred from both longitudinal resistivity and Hall resistivity with increasing B, characteristic of the complex Fermi surface of aluminum. The existence of positive magnetoresistance at T ≧ 20 K indicates that the spinorbit scattering should be taken into account for the exact treatment of localization effects. The observed surface caused T^{2 }term for ρ_{xx }demonstrates that surface Umklapp scattering is important. With decreasing T, a tendency toward a T^{3 }dependence suggests that an Umklapp process occurring in the interior is more important than that occurring at the surface. Such a crossover is consistent with Olsen's argument for lowangle electronphonon scattering qualitatively. All these experimental results show that the nature of the interface between the Al thin film and the GaAs substrate would significantly affect the electrical properties of such a nanoscale film.
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
The authors declare that they have no competing interests. This work was funded by the NSC, Taiwan.
STL and CC performed the lowtemperature experiments on the Al film and drafted the manuscript. KYC and MRY performed the lowtemperature experiments on the Al film. SDL and CTL conceived of the study. JYW fabricated the Al samples. SWL prepared the Al samples and performed the AFM and XRay measurements. All authors read and approved the final manuscript.
References

Liu HF, Chua Sh, Xiang N: Growthtemperature and thermalannealinduced crystalline reorientation of aluminum on GaAs (100) grown by molecular beam epitaxy.
J Appl Phys 2007, 101:053510. Publisher Full Text

Feiginov MN, Kotel'nikov IN: Evidence for attainability of negative differential conductance in tunnel Schottky structures with twodimensional channels.
Appl Phys Lett 2007, 91:083510. Publisher Full Text

Davis JA, Venkatesan R, Kaloyeros A, Beylansky M, Souri SJ, Banerjee K, Saraswat KC, Rahman A, Reif R, Meindl JD: Interconnect limits on gigascale integration (GSI) in the 21st century.
Proc IEEE 2001, 89:305. Publisher Full Text

Havemann RH, Hutchby JA: Highperformance interconnects: an integration overview.
Proc IEEE 2001, 89:586. Publisher Full Text

Santhanam P, Prober DE: Inelastic electron scattering mechanisms in clean aluminum films.
Phys Rev B 1984, 29:3733. Publisher Full Text

Bergmann G: Quantum corrections to the resistance in twodimensional disordered superconductors above Tc: Al, Sn, and amorphous Bi_{0.9}Tl_{0.1 }films.
Phys Rev B 1984, 29:6114. Publisher Full Text

Santhanam P, Wind S, Prober DE: Localization, superconducting fluctuations, and superconductivity in thin films and narrow wires of aluminum.
Phys Rev B 1987, 35:3188. Publisher Full Text

Lin JJ, Bird JP: Recent experimental studies of electron dephasing in metal and semiconductor mesoscopic structures.
J Phys Condens Matter 2002, 14:R501. Publisher Full Text

Chui T, Lindenfeld P, McLean WL, Mui K: Localization and ElectronInteraction Effects in the Magnetoresistance of Granular Aluminum.
Phys Rev Lett 1981, 47:1617. Publisher Full Text

Mui KC, Lindenfeld P, McLean WL: Localization and electroninteraction contributions to the magnetoresistance in threedimensional metallic granular aluminum.
Phys Rev B 1984, 30:2951. Publisher Full Text

Berengue OM, Lanfredi AJC, Pozzi LP, Rey JFQ, Leite ER, Chiquito AJ: Magnetoresistance in SnDoped In_{2}O_{3 }Nanowires.
Nanoscale Res Lett 2009, 4:921. PubMed Abstract  Publisher Full Text  PubMed Central Full Text

Hikami S, Larkin AI, Nagaoka Y: SpinOrbit Interaction and Magnetoresistance in the Two Dimensional Random System.
Prog Theor Phys 1980, 63:707. Publisher Full Text

Fickett FR: A review of resistive mechanisms in aluminum.
Cryogenics 1971, 11:349. Publisher Full Text

Krevet B, Schauer W: Transverse magnetoresistance and its temperature dependence for highpurity polycrystalline aluminum.
J Appl Phys 1976, 47:3656. Publisher Full Text

Ashcroft NW, Mermin ND: Solid State Physics. Chicago: Holt, Rinehart, and Winston; 1976.

Ekin JW, Maxfield BW: Umklapp Processes and the LowTemperature (T < 7°K) Electrical Resistivity of Aluminum.
Phys Rev B 1970, 2:4805. Publisher Full Text

Holwech I, Jeppesen J: Temperature dependence of the electrical resistivity of aluminium films.
Phil Mag 1967, 15:217. Publisher Full Text

Garland JC, van Harlingen DJ: Lowtemperature electrical and thermal transport properties of pure aluminium.
J Phys F 1978, 8:117. Publisher Full Text

Sambles JR, Elsom KC, SharpDent G: The effect of sample thickness on the resistivity of aluminium.
J Phys F 1981, 11:1075. Publisher Full Text

Sambles JR, Mundy JN: A reanalysis of resistive size effects in tungsten.
J Phys F 1983, 13:2281. Publisher Full Text

Blatt FJ, Burmester A, LaRoy B: Resistance and Magnetoresistance of Thin Indium Wires.
Phys Rev 1967, 155:611. Publisher Full Text

von Bassewitz A, Mitchell EN: Resistivity Studies of SingleCrystal and Polycrystal Films of Aluminum.
Phys Rev 1969, 182:712. Publisher Full Text

Soffer SB: Statistical Model for the Size Effect in Electrical Conduction.
J Appl Phys 1967, 38:1710. Publisher Full Text

Ashcroft NW: The Fermi surface of aluminium.
Phil Mag 1963, 8:2055. Publisher Full Text

Balcombe RJ: The MagnetoResistance of Aluminium.
Proc Roy Soc 1963, 275:113. Publisher Full Text

Feder J, Lothe J: Magnetoresistance and Hall effect due to Bragg reflection of free electrons in aluminium.
Philos Mag 1965, 12:107. Publisher Full Text

Banik NC: Overhauser AW. Hall coefficient of a holelike Fermi surface.
Phys Rev B 1978, 18:1521. Publisher Full Text

Stamenov P, Venkatesan M, Dorneles LS, Maude D, Coey JMD: Magnetoresistance of Codoped ZnO thin films.
J Appl Phys 2006, 99:08M124. Publisher Full Text

Sambles JR, Elsom KC: Thickness effects and the T^{2 }dependence of the resistivity of aluminium.
J Phys F 1985, 15:161. Publisher Full Text

Tsoi VS, Razgonov II: Reflection of conductivity electrons by surface of a tungsten sample.

Lin JF, Bird JP, Rotkina L, Sergeev A, Mitin V: Large effects due to electronphononimpurity interference in the resistivity of Pt/CGa composite nanowires.
Appl Phys Lett 2004, 84:3828. Publisher Full Text

Ptitsina NG, Chulkova GM, Il'in KS, Sergeev AV, Pochinkov FS, Gershenzon EM, Gershenson ME: Electronphonon interaction in disordered metal films: The resistivity and electron dephasing rate.
Phys Rev B 1997, 56:10089. Publisher Full Text

Il'in KS, Ptitsina NG, Sergeev AV, Gol'tsman GN, Gershenzon EM, Karasik BS, Pechen EV, Krasnosvobodtsev SI: Interrelation of resistivity and inelastic electronphonon scattering rate in impure NbC films.