Electron-electron scatttering in Sn-doped indium oxide thick films
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Electron-electron scatttering in Sn-doped indium oxide thick films
Yu-Jie Zhang and Zhi-Qing Li ∗ Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology,Department of Physics, Tianjin University, Tianjin 300072, China
Juhn-Jong Lin † NCTU-RIKEN Joint Research Laboratory and Institute of Physics,National Chiao Tung University, Hsinchu 30010, Taiwanand Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan (Dated: November 11, 2018)We have measured the low-field magnetoresistances (MRs) of a series of Sn-doped indium oxidethick films in the temperature T range 4–35 K. The electron dephasing rate 1 /τ ϕ as a function of T for each film was extracted by comparing the MR data with the three-dimensional (3D) weak-localization theoretical predictions. We found that the extracted 1 /τ ϕ varies linearly with T / .Furthermore, at a given T , 1 /τ ϕ varies linearly with k − / F l − / , where k F is the Fermi wavenumber,and l is the electron elastic mean free path. These features are well explained in terms of thesmall-energy-transfer electron-electron scattering time in 3D disordered conductors. This electrondephasing mechanism dominates over the electron-phonon ( e -ph) scattering process because thecarrier concentrations in our films are ∼ e -ph relaxation rate. PACS numbers: 72.15.Rn, 73.20.Fz, 73.23.-b, 72.15.Qm
I. INTRODUCTION
The problem of the electron dephasing processes in dis-ordered conductors has long been of fundamental impor-tance and great interest. In general, the responsible elec-tron dephasing mechanisms are determined by the sys-tem dimensionality, the level of disorder, and the mea-surement temperature T . In three-dimensional (3D)weakly disordered metals, electron-phonon ( e -ph) scat-tering is often the dominant dephasing mechanism. While in lower dimensions, electron-electron ( e - e ) scat-tering is the major dephasing process, which gives riseto a phase relaxation rate of 1 /τ ee ∝ T in two dimen-sions (2D) and 1 /τ ee ∝ T / in one dimension. In1974, Schmid had already theoretically investigated thepossible influence of disorder on the e - e scattering in 3Dconductors. He found that the diffusive electron dynam-ics should enhance the scattering strength, and thus thetotal e - e scattering rate could be written as τ ee = π k B T ) ~ E F + √ ~ √ E F (cid:18) k B Tk F l (cid:19) / , (1)where k B is Boltzmann constant, E F ( k F ) is the Fermienergy (wavenumber), ~ is the Planck constant dividedby 2 π , and l is the electron elastic mean free path. Asimilar expression has also been derived by Altshuler andAronov. The first term on the right-hand side of Eq. (1)is the e - e scattering rate in a perfect, periodic poten-tial, while the second term is the enhanced contributiondue to the introduction of imperfections (defects, impu-rities, etc.). Microscopically, the second term stands forthe small-energy-transfer e - e scattering process and is re-sponsible at k B T < ~ /τ e , while the first term representsthe large-energy-transfer process and would dominate at k B T > ~ /τ e , where τ e is the electron elastic mean-freetime. As mentioned, the scattering rate predicatedin Eq. (1) is much weaker than the e -ph scattering ratein typical 3D disordered metals. Thus, the predictionof Eq. (1) has not been fully experimentally tested thusfar, although a qualitative (but not quantitative) T / temperature dependence of 1 /τ ee has occasionally beenreported in literature. To provide a convincing testof the validity of the second term of Eq. (1), in additionto the temperature dependence 1 /τ ee ∝ T / , the varia-tion on carrier concentration ( k F ) and disorder ( l ), i.e.,1 /τ ee ∝ k − / F l − / , is of critical importance.Tin-doped indium oxide (ITO) is the most widely usedtransparent conducting oxide in current optoelectronicdevices. In terms of the advantages for fundamental stud-ies, the ITO material possesses a unique free-electron-like energy bandstructure. The electron concentra-tions, n , are ∼ Recently, Wu and coworkers have stud-ied the electron dephasing processes in 2D ITO thinfilms. They found that the e - e scattering dominated thedephasing in a wide T range from liquid-helium temper-atures up to nearly 100 K. In 3D disordered conductors,we notice that the e -ph relaxation rate, 1 /τ e -ph , scalesessentially linearly with n (Refs. 4, 22 and 23), whilethe second term in Eq. (1) predicts 1 /τ ee ∝ E − / F k − / F .Approximately, we may write 1 /τ ee ∝ k − / F ∝ /n andestimate the relative dephasing strength to vary roughlyas (1 /τ ee ) / (1 /τ ep ) ∝ /n . In other words, one can ex-pect the e - e scattering to dominate the total dephasingin those 3D systems having sufficiently low values of n .Indeed, as to be shown in this paper, ITO thick films ful-fill this criterion and can be used to manifest the small-energy-transfer e - e scattering rate predicted in Eq. (1).Moreover, since ITO possesses free-electron-like charac-teristics as aforementioned, the relevant parameters ( E F , k F , l , etc.) can be faithfully evaluated through electrical-transport measurements. The theoretical evaluation ofEq. (1) can thus be known to a high degree of accuracy.In this paper, we report our experimental confirmationfor the numerical prediction of Eq. (1), by studying a se-ries of ITO thick films in a wide T range 4–35 K. OurITO thick films have values of n ∼ × cm − (seeTable I), as compared with n ∼ × cm − in typicalmetals. II. EXPERIMENTAL METHOD
Our ITO thick films were deposited on glass substratesby the standard rf sputtering method. A commercial Sn-doped In O target (99.99% purity, the atomic ratio of Snto In being 1:9) was used as the sputtering source. Thebase pressure of the vacuum chamber was 8 × − Pa,and the sputtering deposition was carried out in a mix-ture of argon and oxygen (99.999% purity) atmospherewith a pressure of 0.6 Pa. During the deposition process,the percentage of oxygen O p , together with the substratetemperature T s , was varied to “tune” the carrier (elec-tron) concentration and the amount of disorder. Twoseries of ITO thick films were fabricated. In the first se-ries, T s was fixed at 630, 650, 670, 690, or 720 K; andfor each T s , O p was kept at 0.8%, 1.0%, or 1.2%. In thesecond series, T s was fixed at 670 K, and O p was keptat 1.4%, 1.8%, 2.0%, or 4.0%. Altogether, 19 sampleswith different amounts of disorder and carrier concen-trations were deposited. Hall-bar-shaped samples (1-mmwide and 1-cm long) were defined by using mechanicalmasks and used for electrical-transport measurements.The thicknesses of the films were determined by a sur-face profiler (Dektak, 6M). The films were all at least 1 µ m thick to ensure that they were 3D with respect to theweak-localization (WL) and e - e interaction effects. Thestructures of the films were determined by a Rigaku x-raydiffractometer (D/max-2500v/pc) with Cu K α radiation.The measurements indicated that the films were singlephased with a cubic bixbyite structure characteristic tothat of the undoped In O . The observed strongest andsecond strongest diffraction peaks corresponded to the(400) and the (800) planes, respectively. The intensitiesof other diffraction peaks were less than one thirtieth thatof the (400) peak. Thus, the preferred growth orienta-tion was along the [100] direction. The low-magnetic-fieldmagnetoresistance (MRs) were measured on a physicalproperty measurement system (PPMS-6000, QuantumDesign). The magnetic fields were always applied per-pendicular to the film plane. Hall effect measurementswere also performed to determine the values of n . III. RESULTS AND DISCUSSION
Figure 1 shows the measured normalized magnetoresis-tivities, ∆ ρ ( B ) /ρ (0) = [ ρ ( B ) − ρ (0)] /ρ (0), as a func-tion of magnetic field B at several temperatures for tworepresentative ITO films, as indicated. Note that all theother films revealed similar behavior. The MRs are neg-ative at all temperatures and their magnitudes decreasewith increasing T . In our films, we evaluate the product k F l ≈ (with the function f defined in Ref. 27),∆ ρ ( B ) ρ (0) = e π ~ r eB ~ (cid:20) f (cid:18) BB ϕ (cid:19) − f (cid:18) BB (cid:19)(cid:21) , (2)where B ϕ = B i + B and B = B i + B / B so / B ϕ + 4 B ∗ so /
3. The characteristic field B j is related tothe characteristic scattering time τ j through the relation B j = ~ / eDτ j , where j = i, so and 0 refer to the inelastic,spin-orbit, and T -independent scattering times, e is theelementary charge, and D is the diffusion constant.
35 K25 K15 K7.5 K
No.1 B (T) (a)
35 K25 K15 K7.5 K4 K / ( ) ( - m - ) No.3 B (T) (b) FIG. 1. (Color online) Normalized magnetoresistivity∆ ρ ( B ) /ρ (0) as a function of magnetic field at several tem-peratures for two ITO thick films, as indicated. The magneticfield was applied perpendicular to the film plane. The solidcurves are least-squares fits to Eq. (2). Our MR data are least-squares fitted to Eq. (2). Wefound that, in all films, B ∗ so is ∼ B ϕ (4 K). That is, effectively, B ∗ so may be setto zero and B ϕ becomes the sole adjustable parameter.Indeed, the negative MRs at all measurement tempera-tures already suggest that the spin-orbit scattering mustbe negligibly weak. The solid curves in Fig. 1 representthe theoretical predications of Eq. (2). Clearly, the MRdata can be well described by the WL theory. We ob-tain the electron dephasing length L ϕ = p Dτ ϕ at 4 Kvarying from ∼
110 to ∼
190 nm in our films. Thus, ourthick films are 3D with regard to the WL effect.Figure 2 plots our extracted electron dephasing rate1 /τ ϕ as a function of T / for four representative films, TABLE I. Parameters for four representative ITO thick films. O p and T s are the percentage of oxygen and substrate temper-ature, respectively, during deposition, ρ is resistivity, n is carrier concentration, and D is diffusion constant. 1 /τ ϕ and A ee aredefined in Eq. (3). A thee is the theoretical e - e scattering strength predicted by the second term of Eq. (1).Film O p T s thickness ρ (10 K) n (10 K) D (10 K) k F l τ ϕ A ee A th ee (%) (K) ( µ m) (mΩ cm) (10 cm − ) (cm / s) (10 s − ) (K − / s − ) (K − / s − )1 0.8 670 1.39 1.93 2.3 1.3 3.3 1.6 2.3 × × × × × × × × / ( s - ) T (K ) FIG. 2. (Color online) Electron dephasing rate 1 /τ ϕ as afunction of temperature for four ITO thick films, as indicated.The solid curves are least-squares fits to Eq. (3). as indicated. Note that the product k F l ≈ Inspection of Fig. 2 indicatesthat 1 /τ ϕ varies linearly with T / in the wide measure-ment temperature range 4–35 K. We compare our 1 /τ ϕ data to the following expression1 τ ϕ = 1 τ ϕ + A ee T / , (3)where the first term on the right-hand side stands for a T -independent (or weakly T dependent) contribution, and the second term stands for the 3D small-energy-transfer e - e scattering rate. Our least-squares fitted val-ues of 1 /τ ϕ and A ee are listed in Table I. Also listed inTable I are the corresponding theoretical values A thee =( √ / ~ √ E F )( k B /k F l ) / . Inspection of Table I indi-cates that our experimental values of A ee are within afactor of ∼ A thee . This degree of agreement is sat- isfactory. Similar degree of agreement was found for allthe other films studied in this work. We mention that,in this study, we have intentionally focused on T ≥ /τ ϕ on extracting A ee . / ( s - ) k F-5/2 l -3/2 (¯) FIG. 3. (Color online) Electron dephasing rate 1 /τ ϕ as afunction of k − / F l − / at 5 and 15 K, as indicated. The solidlines are linear fits to the experimental data. Figure 3 shows a plot of the variation of our extracted1 /τ ϕ with k − / F l − / at two representative T values, asindicated. This figure clearly reveals a linear variation,as it should be according to the second term of Eq. (1).Our least-squares fits give slopes of ≃ . × m − at 5 K and ≃ . × m − at 15 K. The theoreticalslope can be expressed as (1 . √ m ∗ / ~ )( k B T ) / in thefree-electron model, where the effective mass m ∗ = 0 . m ( m being the free electron mass). We calculate the the-oretical slopes to be ≃ . × and ≃ . × m − at 5 and 15 K, respectively. These values agree with theexperimental values to within a factor of ∼
5. Thus, inaddition to the temperature dependence, our 1 /τ ϕ dataconcerning the carrier concentration ( k F ) and disorder ( l )dependence fully support the Schmid-Altshuler-Aronovtheory of 3D small-energy-transfer e - e scattering time indisordered conductors. Finally, we estimate the e -ph relaxation rate in ourITO thick films. It has recently been established thatthe electron scattering by transverse vibrations of defectsdominate the e -ph relaxation in the quasiballistic limitof q T l >
1, where q T is the wavenumber of a thermalphonon. The relaxation rate is predicted to be τ e - t, ph = 3 π k B β t ( p F u t )( p F l ) T = A e - t, ph T , (4)where β t = (2 E F / N ( E F ) / (2 ρ m u t ) is the electron–transverse phonon coupling constant, p F is the Fermimomentum, u t is the transverse sound velocity, ρ m isthe mass density, and N ( E F ) is the electronic density ofstates at the Fermi level. In the ITO material, u t ≈ / s, and the typical values of l in our films are ≈ q T l ≈ k B T l/ ~ u t ≈ . T ,where T is in K. Thus, above about 10 K, our films lie inthe quasiballistic limit. Substituting ρ m ≈ / m (Ref. 33) and the relevant electronic parameters intoEq. (4), we obtain the coupling strength A e - t, ph ∼ × K − s − . Thus, 1 /τ e -ph is still about one order of magni-tude smaller than 1 /τ ee even at T = 35 K. The smallnessof 1 /τ e -ph in ITO makes feasible our experimental ob- servation of the 3D small-energy-transfer e - e scatteringtime in the wide T range 4–35 K. IV. CONCLUSION
We have investigated the electron dephasing mecha-nisms in ITO thick films in a wide temperature range 4-35K. Our electron dephasing times were extracted from theweak-localization magnetoresistance measurements. Weobtained 1 /τ ϕ ∝ T / and 1 /τ ϕ ∝ k − / F l − / , which canbe well ascribed to the small-energy-transfer electron-electron scattering process in three-dimensional disor-dered systems. This observation was achieved becauseour ITO films possessed relatively low carrier concen-trations which resulted in a greatly suppressed electron-phonon relaxation rate. ACKNOWLEDGMENTS
This work was supported by the NSF of China throughGrant No. 11174216, Research Fund for the Doc-toral Program of Higher Education through Grant No.20120032110065 (Z.Q.L.), and by the Taiwan NationalScience Council through Grant No. NSC 101-2120-M-009-005, and the MOE ATU Program (J.J.L.). ∗ Electronic address: [email protected] † Electronic address: [email protected] B. L. Altshuler, A. G. Aronov, and D. E. Khmelnitsky, J.Phys. C , 7367 (1982). P. A. Lee and V. Ramakrishnan, Rev. Mod. Phys. , 287(1985). J. J. Lin and J. P. Bird, J. Phys. Condens. Matter ,R501 (2002). J. Rammer and A. Schmid, Phys. Rev. B , 1352 (1986). Y. L. Zhong and J. J. Lin, Phys. Rev. Lett. , 588 (1998). H. Fukuyama and E. Abrahams, Phys. Rev. B , 5976(1983). F. Pierre, A. B. Gougam, A. Anthore, H. Pothier, D. Es-teve, and N. O. Birge, Phys. Rev. B , 085413 (2003). A. Schmid, Z. Phys. , 251 (1974). B. L. Altshuler and A. G. Aronov, JETP Lett. , 482(1979). B. L. Altshuler and A. G. Aronov, in
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