Mobility enhancement and temperature dependence in top-gated single-layer MoS2
aa r X i v : . [ c ond - m a t . m t r l - s c i ] S e p Mobility enhancement and temperature dependence in top-gated single-layer MoS Zhun-Yong Ong and Massimo V. Fischetti ∗ Department of Materials Science and Engineering, University of Texas at Dallas,800 W Campbell Rd RL10, Richardson, TX 75080
The deposition of a high- κ oxide overlayer is known to significantly enhance the room-temperatureelectron mobility in single-layer MoS (SLM) but not in single-layer graphene (SLG). We give aquantitative account of how this mobility enhancement is due to the non-degeneracy of the two-dimensional electron gas system in SLM at accessible temperatures. Using our charged impurityscattering model [Ong and Fischetti, Phys. Rev. B , 121409 (2012)] and temperature-dependentpolarizability, we calculate the charged impurity-limited mobility ( µ imp ) in SLM with and withouta high- κ (HfO ) top gate oxide at different electron densities and temperatures. We find thatthe mobility enhancement is larger at low electron densities and high temperatures because offinite-temperature screening, thus explaining the enhancement of the mobility observed at roomtemperature. µ imp is shown to decrease significantly with increasing temperature, suggesting thatthe strong temperature dependence of measured mobilities should not be interpreted as being solelydue to inelastic scattering with phonons. We also reproduce the recently seen experimental trendin which the temperature scaling exponent ( γ ) of µ imp ∝ T − γ is smaller in top-gated SLM than inbare SLM. Finally, we show that a ∼ percent mobility enhancement can be achieved by reducingthe HfO thickness from 20 to 2 nm. I. INTRODUCTION
In recent years, two-dimensional metal dichalcogenideshave attracted much attention as viable alternatives tographene [1] for post-CMOS nanoelectronic applications[2]. In particular, single-layer MoS (SLM) has been thefocus of much research [3–8]. Like single-layer graphene(SLG), SLM is an atomically thin two-dimensional crys-tal. Given its atomic thickness and close proximity tothe substrate, the electron density in SLM can be tunedvia a vertical electric field. However, this means SLMis highly susceptible to the local electrical field gener-ated by charged impurities near or at the substrate sur-face. Therefore, the electron mobility is expected to bestrongly affected by charged impurity (CI) scattering [9]and/or remote phonon scattering [10–14].Radisavljevic and co-workers [3] recently measured theelectron mobility ( µ e ) in SiO -supported SLM to be be-tween 0.1 and 10 cm V − s − . However, when a thin layerof HfO ( κ = 22 ) was deposited on the SLM to form a topgate, they reported a 20-fold mobility increase of ∼ cm V − s − at room temperature. More recent and accu-rate mobility measurements [8] based on the Hall effectyield a maximum mobility of ∼ cm V − s − in top-gated SLM and ∼ cm V − s − in bare uncovered SLMat 260 K, an almost 4-fold improvement. This mobilityenhancement was attributed to screening from the HfO which reduces CI scattering, believed to be the dominantscattering process. Amani and co-workers also found asimilar 3-fold enhancement in Al O -covered SLM grownwith chemical vapor deposition [15].This mobility enhancement from dielectric screeningis puzzling given that the same effect has not been seenin top-gated SLG. When Fallahazad and co-workers de-posited HfO on SiO -supported SLG, they did not ob- serve any mobility enhancement although they did findthat a thinner gate oxide increases the mobility in SLG[16]. This has been explained as consequence of greaterscreening of the charged impurities by the metal gate [17].In every instance that we know of [16, 18–21], the depo-sition of an oxide layer on high-mobility, non-epitaxial SLG has lead to a mobility decrease, probably as a resultof more CI and defect scattering. Thus, it is surprisingto observe a several-fold improvement for SLM. This sug-gests that CI scattering is qualitatively different in top-gated SLM [3]. The variance between the data from Refs.[3] and [16] is striking, and may be due to the differentelectronic band structures, the nature of the interactionbetween the substrate and the SLG/SLM, or the type ofcharge screening. In both cases, the substrate material isSiO while the gate oxide is HfO (30 nm thick in Ref. [3]and 11 nm in Ref. [16]), and the mobility measurementmethods (two-probe) are similar. This and the similarstack structure rule out the possibility of the differencebeing due to the top gate capacitance [22, 23].Another salient feature of electron transport in SLM isthat the deposition of the top gate oxide alters the tem-perature dependence of the electron mobility. At roomtemperature (300 K), the phonon-limited electron mobil-ity is predicted to scale as µ e ∝ T − γ with γ = 1 . and µ e ≈ cm V − s − in bare SLM and γ = 1 . and µ e ≈ cm V − s − in top-gated SLM where the ho-mopolar optical phonon mode is assumed to be quenched[5]. Measurements by Radisavljevic and Kis of the high-temperature ( T = 80 to 280 K) Hall mobility in bareSLM yield γ ≈ . , in good agreement with Ref. [5], al-though the absolute value of the mobility is about one or-der of magnitude smaller with µ e < cm V − s − at 260K [8]. Their measurements on top-gated SLM also yield γ = 0 . to 0.73 with µ e = 57 to 63.7 cm V − s − at 260K in samples exhibiting the metal-insulator transition.Their bare SLM results are also in good agreement withthe more recent data from Baugher and co-workers whosemeasurements on bare SLM give µ e < cm V − s − and γ = 1 . at 300 K [24]. Although experimentally deter-mined values of γ from Refs. [8, 24] ( γ = 1 . and 1.7 re-spectively) agree with the theoretically predicted value of γ = 1 . in bare SLM, the experimental values ( µ e < cm V − s − ) are one order of magnitude smaller than thetheoretical value ( µ e ≈ cm V − s − ) and suggest thatintrinsic phonon scattering is not the dominant factor inthe temperature dependence of µ e .In this article, we study temperature-dependent,charged impurity-limited electron transport in bare andtop-gated single-layer MoS by adapting the model de-veloped in Ref. [17] and including not only the effectof the dielectric environment but also the temperaturedependence of the charge polarizability. HfO -coveredSLM on a SiO substrate is used as a model system herealthough the theory can be easily generalized to othergate dielectrics and single-layer transition metal dichalco-genides (TMDs). Our use of the temperature-dependentcharge polarizability is motivated by the electron trans-port data from Ghatak and co-workers [25], which havebeen interpreted to imply that charged impurities areweakly screened at room temperature. For simplicity,electron-phonon interaction is mostly ignored here to iso-late the effects of screening by the charge polarizabilityas well as the dielectric environment although scatteringwith the intrinsic phonons is included when it comes tothe mobility scaling with temperature. The differencebetween the charge impurity-limited electron mobility( µ imp ) in bare and top-gated SLM at different tempera-tures ( T ) and electron densities ( n ) is used to explain thescreening effect of the gate oxide on room-temperatureelectron transport. We also show that the lower mobil-ity at higher temperatures can be due to temperature-dependent screening. Lastly, we predict the scaling of µ imp with the gate oxide thickness ( t ox ) at room temper-ature. II. METHODOLOGYA. Charged impurity scattering potential
A schematic of the setup is shown in Fig. 1. The modelconsists of a SLM sheet sandwiched between two oxidelayers with the interface at z = 0 on the x - y plane. Thesubstrate oxide (SiO ) is semi-infinite ( z < ) while thegate oxide has a thickness of t ox (i.e. ≤ z < t ox ).We approximate SLM as an ideal zero-thickness two-dimensional electron gas (2DEG). To determine µ imp , wecompute the scattering rate Γ imp for the single CI scat-tering potential φ scr q (0) . The expression for the φ scr q (0) is Figure 1: (Color online) Basic model used in our calculation.The SLM is an infinitely thin layer at the interface ( z = 0 )between a semi-infinite substrate and a top oxide layer ofthickness t ox . The dielectric is capped with metal, which weassume to be a perfect conductor. The charged impurity atthe interface has image charges under and above it in thesubstrate and top gate, respectively. [17]: φ scr q ( z = 0) = e G q (0 , ǫ ( q, T ) (1)where q , e and G q (0 , are the wave vector, the ab-solute electron charge quantum, and the Fourier trans-form (with respect to x and y ) of the Green’s func-tion solution of the Poisson equation, respectively; ǫ ( q, T ) is the generalized static dielectric function,given by ǫ ( q, T ) = 1 − e G q (0 , q, T, E F ) where Π( q, T, E F ) is the temperature-dependent static chargepolarizability. The expression for G q (0 , is G q (0 ,
0) = { [ ǫ tox coth( qt ox ) + ǫ box ] q } − where ǫ tox and ǫ box are thestatic permittivity of the top and bottom oxides, re-spectively. The electrostatic boundary conditions are in-cluded in G q (0 , . B. Fermi temperature and temperature-dependentscreening
While graphene remains degenerate even at low den-sity around room temperature, in TMDs the tempera-ture dependence of the dielectric response can play a sig-nificant role. We take it into account by first examin-ing the long-wavelength, finite-temperature approxima-tion for Π( q, T, E F ) [26], i.e. lim q → Π( q, T, E F ) = − gm eff π ~ (cid:20) − exp (cid:18) − π ~ n m eff k B T (cid:19)(cid:21) , (2)where g and m eff are the valley-spin degeneracy ( g = 4 )and the effective electron mass, respectively; E F is thechemical potential and is related to n via the equation E F = k B T ln { exp[ π ~ n/ (2 m eff k B T )] − } ; k B and ~ arethe Boltzmann and Planck constant, respectively. Fora given electron density n , the 2DEG can be considereddegenerate when T ≪ T F where T F = π ~ n/ (2 m eff k B ) isthe characteristic Fermi temperature . At n = 10 cm − , T F = 29 K. Therefore, we need to use finite-temperaturescreening for the range of electron densities and temper-atures in our calculations later. At finite q , we can usethe more general expression [27–29]: Π( q, T, E F ) = ˆ ∞ dµ Π( q, , µ )4 k B T cosh ( E F − µ k B T ) , (3)where Π( q, , µ ) = Π(0 , , µ ) { − Θ( q − k F )[1 − (2 k F /q ) ] / } with k F = √ m eff µ/ ~ and Π(0 , , µ ) = − gm eff / (2 π ~ ) . Figure 2 shows the q -dependence of Π( q, T, E F ) at T = 0 , 50, 100 and 300 K for (a) n = 10 cm − and (b) n = 10 cm − . For the same given T , thechange in the polarizability relative to the 0 K case isgreater at n = 10 cm − ( T F = 29 K) than at n = 10 cm − ( T F = 290 K). We also observe that Π( q, T, E F ) is significantly smaller at 300 K than at 0 K. In general, Π( q, T, E F ) in Eq. (2), which appears in the denomina-tor in Eq. (1) and corresponds to charge screening, van-ishes as n → or T → ∞ , i.e. charge screening weak-ens with decreasing electron density or increasing tem-perature. Hence, the CI scattering strength increases as n → or T → ∞ . To illustrate this, we plot the cor-responding scattering potential φ scr q in top-gated SLM at T = 0 , 50, 100 and 300 K, normalized to φ scr q =0 at T = 0 K, in Fig. 2 for (c) n = 10 cm − and (d) n = 10 cm − .For n = 10 cm − , the scattering potential remains rel-atively unchanged as T increases, unlike the scatteringpotential for n = 10 cm − which increases by up toan order of magnitude as T increases from 0 K to 300K, because the Fermi temperature at n = 10 cm − is T F = 290 K.Following Ref. [5] we approximate the electron dis-persion in SLM with a parabolic expression E ( k ) = ~ k / (2 m eff ) with effective mass m eff = 0 . m (where m is the free electron mass) and minimum at the sym-metry point K . The use of a single valley should not con-stitute a big error since at low fields no interband tran-sitions are expected to take place [5]. The expression forthe CI scattering rate is [17]: Γ imp ( E k ) = n imp π ~ ˆ d k ′ | φ scr | k − k ′ | ( d ) | × (1 − cos θ kk ′ ) δ ( E k − E k ′ ) , (4)where θ kk ′ is the scattering angle between the k and k ′ states, and n imp is the CI concentration which is a fittingparameter. The expression for the CI-limited electronmobility is: µ imp = eπ ~ k B T ˆ ∞ f ( E )[1 − f ( E )]Γ imp ( E ) − E dE , (5)where f ( E ) is the equilibrium Fermi-Dirac distributionfunction. By using Eq. (5), we assume that electron q / k F Π ( q , T , E F ) / Π ( , , E F ) (a) n = 10 cm −2 q / k F (b) n = 10 cm −2 q / k F φ q sc r ( T ) / φ q = sc r ( ) (c)
300 K100 K50 K0 K 0 1 2 3 4 q / k F (d) Figure 2: (Color online) Plot of the normalized polarizabil-ity Π( q, T, E F ) / Π(0 , , E F ) for (a) n = 10 cm − and (b) n = 10 cm − at T = 0 K (solid), 50 K (dash-dot), 100 K(dotted) and 300 K (dashed). We also plot the correspondingnormalized scattering potential φ scr q ( T ) /φ scr q =0 ( T = 0) for (c) n = 10 cm − and (d) n = 10 cm − in top-gated SLM. transport is described by semiclassical band transport,as in Refs. [5, 30] and opposed to hopping transport assuggested in Ref. [25], and that the dominant scatter-ing mechanism is CI scattering, which is mostly at theFermi surface. The main momentum relaxation processcorresponds to the momentum change of q ∼ k F andthe related Fourier component of the scattering poten-tial φ scr k F , which is inversely proportional to the dielec-tric function ǫ (2 k F , T ) and strongly affected by thetemperature broadening of Π(2 k F , T, E F ) [28, 31]. III. RESULTS AND DISCUSSIONA. Electron density dependence of mobility at lowand high temperature
The variables µ imp and µ TGimp denote the CI-limited mo-bility in bare SiO -supported ( ǫ tox = ǫ and t ox = ∞ )and 30-nm-HfO -top-gated, SiO -supported ( ǫ tox = 22 ǫ and t ox = 30 nm) SLM. We assume n imp = 4 × cm − for ease of comparison with the results in Ref. [3]. At n = 2 × cm − and T = 10 K, this yields µ TGimp ∼ cm V − s − , comparable to that measured by Radisavl-jevic and Kis [8] at low temperatures. We first calculateand plot in Fig. 3 µ imp and µ TGimp at T = 10 K, from n = 10 to × cm − in steps of ∆ n = 10 cm − . n (10 cm −2 ) µ i m p ( c m V − s − )
10 K300 KBareHfO Figure 3: (Color online) Plot of µ imp (‘Bare’) and µ TGimp (‘HfO ’) at T = 10 K (hollow symbols) and T = 300 K (solidsymbols) for n = 10 to × cm − for n imp = 4 . × cm − . At 300 K, the mobility scales almost linearly with theelectron density. The corresponding Fermi temperature range is T F = 29 to 580 K. We find that both µ imp and µ TGimp increase mono-tonically with n , in good agreement with the Hall mobil-ity data given in Ref. [24], with the density dependencestronger for µ TGimp . At low densities ( n < × cm − ),the density dependence is markedly greater.Our results indicate that µ TGimp is higher than µ imp , withthe relative difference increasing with n ; at n = 10 cm − , we have µ TGimp /µ imp = 1 . while at n = 2 × cm − , we have µ TGimp /µ imp = 1 . . This suggests that themobility enhancement from overlaying SLM with a high- κ material is modest at low temperatures. This is becauseat low temperatures ( T ≪ T F ), screening is dominatedby the charge polarizability. To see how, we rewrite thescattering potential of a single CI in Eq. (1) as: φ scr q (0) = e G q (0 , − e G q (0 , q, T, E F ) . In the long-wavelength limit, the second term inthe denominator, which corresponds to the screeningcharge, dominates, giving us lim q → φ scr q (0) = − Π( q =0 , T, E F ) − . Thus, the scattering potential is indepen-dent of the dielectric environment in the long-wavelengthlimit and depends only on the polarizability. At low T and q < k F , the polarizability is nearly independent of q , i.e. lim T → Π( q < k F , T, E F ) = − m eff / ( π ~ ) . Thisexplains why µ TGimp /µ imp is close to unity. The decreaseof µ imp and µ TGimp with smaller n is due to the fact that atsmall n , we have lim n → Π( q, T, E F ) ∝ n which impliesthat the scattering potential strength scales as ∼ n − .On the other hand, experimental measurements revealthat covering SLM with a high- κ dielectric leads to signif-icant room-temperature mobility enhancement [8]. Thissuggests that screening by the dielectric plays a greaterrole in the temperature regime T & T F . Hence, the n (10 cm −2 ) µ i m p T G / µ i m p0
300 K10 K
Figure 4: (Color online) Plot of the mobility enhancement µ TGimp /µ imp at 10 K (circle) and 300 K (triangle). The mo-bility enhancement increases at higher temperatures or lowerelectron densities. screening effect of the charge polarizability in SLM isless significant. To show this, we repeat our calcula-tion of µ imp and µ TGimp but now at room temperature(300 K). The room-temperature results are also shownin Fig. 3. In contrast to the low-temperature results inFig. 3, µ imp and µ TGimp are strongly density-dependent andscale almost linearly with n , in good agreement withthe room-temperature data for bare SLM by Ghatakand co-workers [25]. The linear density-dependence isa signature of weak or absent screening by the polar-ization charge in SLM. Thus, the role of screening bythe surrounding dielectric media becomes more impor-tant. At low n , µ TGimp is significantly larger than µ imp . At n = 10 cm − , µ imp ≈ . cm V − s − while µ TGimp ≈ . cm V − s − , nearly an order-of-magnitude increase. Thisagrees very well with the measured several-fold mobilityenhancement reported in Refs. [8, 15]. We plot the mo-bility enhancement µ TGimp /µ imp in Fig. 4 at 10 and 300 K.The mobility enhancement is much greater at 300 K thanat 10 K because of the temperature-induced weakeningof the charge polarizability. At 300 K, the mobility en-hancement decreases and converges to that at 10 K as n increases because charge screening becomes stronger athigher densities. B. Temperature dependence of electron mobility
The temperature dependence of the electron mobilityin experiments is often used to determine the nature ofelectron transport in semiconductors. When the mobilitydecreases with increasing T , it is commonly interpretedto be a signature of phonon-limited electron transport inthe metallic phase [5, 8, 24, 32]; in the insulating phase,the rise in mobility with increasing T is usually character-ized as originating from hopping transport [25]. Kaasb-jerg and co-workers predict the intrinsic phonon-limitedmobility to vary as µ e ∝ T − γ ( γ = 1 . ) in top-gatedSLM. Measurements of γ by Radisavljevic and Kis haveit varying between 0.3 and 0.73 [8], which is suggestiveof phonon-limited transport. For ease of comparison, wesummarize the representative theoretical and experimen-tal mobility results from Refs. [5, 8, 24, 30] in Table I, to-gether with our results. However, Li and co-workers [30]and Kaasbjerg and co-workers [5] predict the K valley-dominated, intrinsic phonon-limited mobility values tobe around several hundred cm V − s − at room temper-ature, which are at least an order-of-magnitude largerthan measurements [8, 24]. Thus, the temperature de-pendence of the measured mobility is probably due toextrinsic factors such as charged impurities and remotephonons.The disparity between our calculated low- and room-temperature µ imp implies that CI scattering is stronglytemperature-dependent and plays an important role inthe overall mobility temperature dependence. Hence, itis important to quantify the temperature dependence inour model for direct comparison with experiments, in or-der to understand the causes of this temperature depen-dence. In particular, we are interested in the temper-ature scaling of the high-temperature electron mobility( µ e ∝ T − γ ), which has been investigated theoreticallyand experimentally in Refs. [5, 8, 24, 30], and the differ-ence in this temperature scaling between bare and top-gated SLM. Radisavljevic and Kis recently reported asubstantial decrease in γ , from γ = 1 . in bare SLM to γ = 0 . − . in top-gated SLM [8], much greater thanthat expected from the quenching of homopolar opticalphonons [5]. By studying the difference in the temper-ature dependence of the mobility in bare and top-gatedSLM with our model, we hope to shed light on this phe-nomenon.Since the temperature variation of the electron mobil-ity may depend on scattering with phonons, we computethe CI/phonon-limited electron mobility µ e = µ imp+phon ,taking into account charged impurity as well as intrinsicphonon scattering, in addition to the computation of theCI-limited mobility µ imp . The intrinsic electron-phononinteractions include the longitudinal acoustic (LA), thetransverse acoustic (TA), the intervalley longitudinal op-tical (LO) and the intravalley homopolar optical (HP)phonons, with the scattering rate formulas and parame-ters taken directly from Ref. [5]. In our calculation of theCI/phonon-limited electron mobility ( µ imp+phon ) in bareSLM, we include electron scattering with the LA, TA,LO and HP phonons while in top-gated SLM, we assumethat the HP phonons are quenched (as in Ref. [5]) andwe do not include them in our calculation of the mobility( µ TGimp+phon ).Figure 5 shows (a) µ imp+phon and (b) µ TGimp+phon for n = 10 to × cm − in steps of ∆ n = 10 cm − , which we take to be representative of the low-density regime, and n = 10 to × cm − in steps of ∆ n = 2 × cm − , which we take to be representative of the high-density ‘metallic’ regime, from T = 10 to 300K. Figures 5(a) and (b) show that the relative variation ofthe mobility with T increases as n becomes smaller. Thedecrease in µ imp+phon is very large as we go from 10 to 300K. For example, at n = 10 cm − , µ imp+phon decreasesby > percent. The sensitivity to changes in tempera-ture is significantly greater for µ imp+phon than µ TGimp+phon .The corresponding results for the CI-limited mobilities( µ imp and µ TGimp ) are not shown here since they exhibit asimilar trend with respect to temperature change.From T = 200 to K and n = 10 to × cm − , µ imp+phon and µ TGimp+phon exhibit a power-law de-pendence on T , i.e. µ imp+phon ∝ T − γ , similar to thatreported in Refs. [8, 24]. We plot γ as a function of n for µ imp+phon , µ TGimp+phon , µ imp and µ TGimp in Fig. 5(c). Theexponent γ decreases with n and is also much larger for µ imp+phon ( γ = µ TGimp+phon ( γ = γ was found for top-gated SLM.The γ values for µ imp+phon are comparable to the T − behavior expected for a dilute, high-temperature 2DEG[31] but lower than the γ = 1 . and 1.4 from Refs. [24]and [8], respectively. The values for µ TGimp+phon are how-ever within the range measured for top-gated SLM sam-ples ( γ = γ values for theCI-limited mobilities µ imp and µ TGimp are slightly smaller(0.70 to 0.98 and 0.30 to 0.36 respectively in the case of µ imp and µ TGimp ) since the temperature dependence onlycomes from the finite-temperature charge polarizability.Nevertheless, we observe a similar decrease in γ whencomparing µ TGimp to µ imp . This implies that the change in γ is due to the modification of CI scattering in top-gatedSLM.In Ref. [8], γ increases with n (from γ = 0 . at n = 0 . × cm − to γ = 0 . at n = 1 . × cm − ) in contrast to our results for µ TGimp and µ TGimp+phon where γ decreases as n increases. This suggests thatother more strongly temperature-dependent scatteringprocesses may be involved. In Fig. 3(a) and (b), µ TGimp increases with n , i.e. CI scattering becomes less impor-tant at higher densities. Hence, the relative contributionof the other scattering processes may become more sig-nificant.
C. Gate Oxide Thickness Dependence
Having shown that screening by the top gate enhancesthe mobility at room temperature and low n i.e. when T ≫ T F , we explore the possibility of using a thinnergate oxide to screen the charged impurities. We compute µ TGimp for n = 10 to × cm − and t ox = 2 to nmat 300 K. Figure 6 shows the calculated µ TGimp values nor-malized to the µ TGimp for a semi-infinite top oxide layer. Asexpected, µ TGimp increases as t ox decreases because a thin- Bare Top-gatedReference T (K) Method µ e γ µ e γ Kaasbjerg et al. (phonon-limited) [5] 300 Theory 410 1.69 480 1.52Li et al. (phonon-limited) [30] 300 Theory 320 – – –Baugher et al. [24] 300 Expt. <20 1.7 – –Radisavljevic and Kis [8] 260 Expt. 17.2 1.4 56.9 to 63 0.3 to 0.73Ong and Fischetti (CI-limited) 300 Theory 17.4 0.98 56.5 0.36Ong and Fischetti (CI/phonon-limited) 16.2 1.0 48.9 0.46Table I: Comparison of representative electron mobility µ e (in units of cm V − s − ) and power-law exponent γ (where µ e ∝ T − γ )values for bare and HfO top-gated SLM from Refs. [5, 8, 24, 30]. The results from Li et al. [30] and Kaasbjerg et al. [5] assume K valley-dominated, intrinsic phonon-limited electron transport. The CI-limited results by Ong and Fischetti are computedwith an impurity concentration of n imp = 4 × cm − at the electron density of n = 10 cm − while the CI/phonon-limitedresults are computed using the same n imp and phonon parameters from Ref. [5]. Our CI-limited mobility results show that asignificant temperature dependence can arise even in the absence of phonon scattering. ner oxide places the image charges in the metal closer tothe SLM and screens the charged impurities more effec-tively. At n = 10 cm − , a 37 percent enhancement in µ TGimp can be achieved by reducing t ox from 20 to 2 nm.This implies that reducing t ox can significantly mitigatethe effects of charged impurities especially when T ≫ T F . IV. FURTHER DISCUSSION AND SUMMARY
The underlying physics of our findings stems from thetransition of the 2DEG in SLM from degeneracy to non-degeneracy at higher accessible temperatures. At hightemperatures ( T & T F ), charge screening within the2DEG becomes weaker with increasing temperature, andthe charged impurity-limited mobility becomes more de-pendent on screening by the dielectric environment of theSLM. The non-degeneracy-to-degeneracy transition alsoexplains why the mobility enhancement is not seen intop-gated SLG, the question posed at the beginning thepaper. The linear band structure of SLG ensures thatit remains degenerate even at room temperature. Forexample, the Fermi temperature in SLG exceeds 1300K at n = 10 cm − whereas the corresponding Fermitemperature in SLM is 29 K. Thus, charge screeningwithin SLG is effectively temperature-independent anddominates the screening of charged impurities at acces-sible temperatures. On the other hand, charge screeningwithin SLM weakens with temperature and allows screen-ing by the dielectric environment to play a bigger role athigh temperatures.We also point out that mobility enhancement has beenobserved in top-gated epitaxial SLG [33, 34]. However,it is known that the band structure of epitaxial grapheneis unlike that of ideal exfoliated SLG as a result of theformation of a substrate-induced band gap [35, 36]. As-suming that electron transport in epitaxial SLG is limitedby CI scattering, the low mobility in epitaxial SLG (rela-tive to exfoliated SLG) suggests that its intrinsic chargescreening is weakened, possibly from the aforementionedband structure modification. With regard to our results, we have calculated thecharged impurity-limited mobility ( µ imp ) in SLM withelectron density and temperature-dependent screening.Our results agree with the several-fold improvement inroom-temperature mobility reported in Refs. [8] and[15] when a high- κ overlayer is introduced, and theyare consistent with the weak charge screening foundin Ref. [25]. We have found that µ imp decreaseswith increasing temperature primarily as a result oftemperature-dependent polarizability, suggesting thatthis temperature-dependent phenomenon is not neces-sarily a signature of phonon scattering. Our model alsoqualitatively reproduces the change in the temperaturescaling of µ e when HfO is deposited on SLM [8]. How-ever, we are unable to reproduce accurately the magni-tude and temperature-scaling exponent γ of the mobilityin our model, even with the inclusion of intrinsic phonons.This suggests that other scattering mechanisms, possiblyremote phonons [10, 12, 13, 37], must be accounted for ina more realistic model of electron transport in bare andtop-gated SLM. Lastly, we have shown that a thinnertop oxide can lead to a significant improvement in µ imp at low electron densities for temperatures greater thanthe Fermi temperature. Our results highlight a possi-ble strategy to optimize the device geometry for superiorelectron transport properties in single-layer MoS andother transition metal dichalcogenides.We gratefully acknowledge the support provided byTexas Instruments, the Semiconductor Research Corpo-ration (SRC), the South-West Academy of Nanotechnol-ogy (SWAN) under Task 4.3 Theme 2400.011, and Sam-sung Electronics Ltd. ∗ Electronic address: max.fi[email protected][1] K. Novoselov, A. Geim, S. Morozov, D. Jiang, Y. Zhang,S. Dubonos, I. Grigorieva, and A. Firsov, Science ,666 (2004). T (K) µ i m p + phon ( c m V − s − ) (a) T (K) (b)
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