Top oxide thickness dependence of remote phonon and charged impurity scattering in top-gated graphene Zhun-Yong Ong and Massimo V. Fischetti Citation: Appl. Phys. Lett. 102, 183506 (2013); doi: 10.1063/1.4804432 View online: http://dx.doi.org/10.1063/1.4804432 View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v102/i18 Published by the American Institute of Physics.
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APPLIED PHYSICS LETTERS 102, 183506 (2013)
Top oxide thickness dependence of remote phonon and charged impurity scattering in top-gated graphene Zhun-Yong Onga) and Massimo V. Fischettib) Department of Materials Science and Engineering, The University of Texas at Dallas, 800W Campbell Rd RL10, Richardson, Texas 75080, USA
(Received 30 December 2012; accepted 24 April 2013; published online 7 May 2013) We have calculated the substrate-limited electron mobility in top-gated, SiO2-supported single-layer graphene from remote-phonon and charged impurity scattering rates. The mobility dependence on gate insulator thickness is explained in terms of the dielectric screening of remote phonons and charged impurities by the high-j/metal gate. We also find that the effects of high-j/metal gate screening are reduced at high carrier densities. Of the top gate dielectrics considered (h-BN, HfO2, and Al2O3), h-BN results in a better overall performance and most mobility improvement with a C 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4804432] thinner top gate. V Electron mobility in suspended single-layer graphene (SLG) has been demonstrated to be as high as 200 000 cm2 V1 s1 at room temperature,1 and this high value is central to many potential nanoelectronic applications, especially high-frequency analog devices.2,3 In suspended SLG, the mobility is limited by scattering with intrinsic acoustic and optical phonons.4,5 However, in most SLG-based heterostructures, the SLG is supported on an insulating dielectric substrate, usually amorphous SiO2, which introduces new scattering sources such as surface roughness, charged impurities and remote phonons, degrading mobility. Thus, the mobility in SiO2-supported SLG is limited by defect, charged impurity5–8 and remote phonon scattering,9–11 and is usually between 103 and 2 104 cm2 V1 s1 (Ref. 6) at room temperature. Defects in SLG, such as ripples, grain boundaries, and local tensile strain12 are not affected by its dielectric environment, unlike charged impurities, plasmons, and substrate dipole modes which are electrodynamic phenomena. In supported SLG, the substrate surface optical (SO) phonons,13,14 also known as surface polar phonons (SPPs), in the dielectric couple to the graphene plasmons, resulting in dynamic screening of the remote phonons.15,16 A large gate capacitance is advantageous in highfrequency application, necessitating the integration of a top gate.17 However, the top oxide in top-gated SLG can degrade the mobility through increased impurity and, especially with high-j dielectrics like HfO2, remote phonon scattering.15,16 On the other hand, the top oxide modifies the dielectric environment and screens charged impurities at or near the SLG-oxide interface. Jang and co-workers found that by depositing ice on SLG at 77 K, the mobility was found to increase by up to 30%.18 Similar mobility improvements were observed by Ponomarenko and co-workers who covered SLG with high-j liquids.19 More realistic top-gated SLG devices have been fabricated without significant mobility degradation.9,20,21 Curiously, Fallahazad and co-workers found that increasing the top oxide thickness lowers the mobility in HfO2-covered SLG,20 and attribute this to increased defects and charged impurities. a)
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b)
0003-6951/2013/102(18)/183506/4/$30.00
To understand the impact of the top gate on mobility, we compute the remote phonon (RP) and charged impurity (CI) scattering rates for three different dielectrics (HfO2, h-BN and Al2O3) at different carrier densities n and dielectric thickness tox. HfO2 and Al2O3 are the more commonly used top gate dielectrics20–22 while h-BN offers great promise for integration with graphene.23 The contributions of the intrinsic phonons are neglected, as in Ref. 24. The scattering rates are used to compute the substrate-limited carrier mobility lsub . This sets an upper bound to the mobility in SLG since its intrinsic mobility (2 105 cm2 V1 s1 ) is much greater than lsub . To distinguish the effects of RP and CI scattering, we also calculate lrem (remote phonons only) and limp (charged impurities only). From our results, we try to understand mobility scaling with tox at different carrier densities and temperatures. The RP scattering rates are computed using our recently developed theory of interfacial plasmon-phonons (IPPs),15,16 modified to take into account the polar phonons in the top dielectric, with the IPP parameters taken from Refs. 15 and 16. This theory25 incorporates the dynamic screening phenomenon and the coupling of the dielectric polar modes to graphene plasmons. The CI scattering rates are calculated using another recently developed model of CI scattering in top-gated SLG.26 Both models account for the modification of the dielectric environment caused by a different tox. Charge-neutral defects are ignored in our calculations because they do not scale with tox. We assume the impurity concentration to be nimp ¼ 5 1011 cm2 as in Ref. 24. A schematic diagram of our top-gated SLG structure is shown in Fig. 1. The SiO2 substrate is assumed to occupy the semiinfinite region below the SLG. The SLG-dielectric gap size is finite and equals d ¼ 0.35 nm. The top dielectric is capped with metal which is assumed to be an ideal conductor. To compute the semi-classical substrate-limited electron gs gv e Ð 1 f ðE EF Þ mobility, we use the formula lsub ¼ 4pn 2 h kB T 0 ½1 f ðE EF ÞCtr ðEÞ1 EdE, where gs, gv, e, kB, T, EF, and f are the spin degeneracy, the valley degeneracy, the electron charge, the Boltzmann constant, the temperature, the chemical potential, and Fermi-Dirac distribution function, respectively. The momentum relaxation rate Ctr is assumed to be the sum of the CI scattering (Cimp ) and RP scattering rates
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Appl. Phys. Lett. 102, 183506 (2013)
FIG. 1. Schematic of top-gated SLG. A vacuum gap of d separates the SLG from the top and bottom oxides. The substrate is assumed to be semi-infinite while the top oxide has a thickness of tox. The top oxide is capped with metal which is assumed to be an ideal conductor. The impurities are assumed to be in the graphene plane.
(Crem ), i.e., Ctr ¼ Cimp þ Crem . The expression for Cimp is given in Ref. 26 while that for Crem , which is complicated, is described in detail in the supplementary material.27 Although details of the theory of interfacial plasmonphonons in top-gated SLG are not presented here and deferred to a future publication,25 some discussion of its basic elements is necessary. In the uncoupled system, we have two SPP branches from the substrate (SiO2), two from the top dielectric (HfO2, h-BN or Al2O3) and a single one from the graphene plasmons. In the stack, these excitations are coupled, forming IPP modes.27 In Fig. 2, we plot the IPP dispersion (x vs. Q, where Q is the wave vector), in which there are five branches, for a 20 nm HfO2 top-gated SLG at n ¼ 1012 cm2. To illustrate the effect of tox on the electron-remotephonon interaction, we calculate and plot the electron-IPP ðIPP1Þ ), corresponding to the lowest coupling coefficient (MQ IPP branch (IPP1) in Fig. 2, in Fig. 3 for (a) tox ¼ 2 and (b) 20 nm at n ¼ 1011, 1012, and 1013 cm2. The insets show the corresponding bare electron-SPP coupling coefficient ðSPP1Þ ðIPP1Þ ðSPP1Þ ). Note that limn!0 MQ ¼ MQ . For tox ¼ 2 nm (MQ ðIPP1Þ MQ
ðSPP1Þ MQ
11
2
¼ at n ¼ 10 cm , indicating [Fig. 3(a)], the absence of dynamic screening. This is a consequence of
ðIPP1Þ
FIG. 3. Plot of the electron-IPP coupling coefficient (MQ ) for the lowest IPP branch for (a) tox ¼ 2 and (b) 20 nm at n ¼ 1011 cm2 (dotted line), 1012 cm2 (dotted-dashed line) and 1013 cm2 (dashed line). The insets ðSPP1Þ show the corresponding electron-SPP coupling coefficient (MQ ), correðIPP1Þ
sponding to MQ
at zero carrier density.
the softening of the plasmons from screening by the metal gate. At n ¼ 1012 cm2, the plasmon frequencies are higher. Plasmonic coupling and hence, dynamic screening are stronðIPP1Þ ðSPP1Þ < MQ . At n ¼ 1013 cm2, ger. Thus, for small Q, MQ ðIPP1Þ
ðSPP1Þ
< MQ for the entire range of Q values in Fig. MQ 3(a) because the much higher carrier density leads to even more dynamic screening. For tox ¼ 20 nm [Fig. 3(b)], screening by the metal gate is weaker because of the larger distance ðSPP1Þ for tox ¼ 20 between the SLG and the metal. Hence, MQ ðSPP1Þ
nm is larger than MQ
for tox ¼ 2 nm. On the other hand, ðIPP1Þ
ðSPP1Þ
< MQ is also the range of Q values for which MQ larger because metal gate screening of the plasmonic coupling is reduced, leading to more dynamic screening. For ðIPP1Þ example, at n ¼ 1012 cm2 and Q < 2:6 108 m1 ; MQ ðIPP1Þ
FIG. 2. The dispersion of the five IPP branches (from bottom to top, IPP1 to IPP5) in solid lines, with IPP1 being the lowest branch, for a 20-nm HfO2 top-gated SLG at n ¼ 1012 cm2. The asymptotes corresponding to the uncoupled SPP branches and the graphene plasmon are drawn in dashed and dotted lines, respectively. The effects of Landau damping are ignored here.
for tox ¼ 20 nm is smaller than MQ for tox ¼ 2 nm. As we will see later, the increase of dynamic screening with a thicker top oxide at higher n leads to reduced RP scattering. We vary the oxide thickness from 2 to 20 nm in order to understand the effect of tox on lsub . In such ultrathin dielectrics, the metal is very close to the SLG, screening the carrier interaction with remote phonons and charged impurities because of the close proximity of the image charges in the metal. Given that the scattering strength of the remote phonons is affected by dynamic screening which emerges from the plasmonic coupling to the bare SPPs,15,16 we also vary the carrier density. Figures 4–6 show the (a) lsub , and (b) lrem and limp at n ¼ 1011, 1012, and 1013 cm2, respectively. At low carrier
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FIG. 4. Calculated (a) lsub , and (b) lrem (open symbols) and limp (solid symbols) vs. tox at n ¼ 1011 cm2 for HfO2 (squares), h-BN (triangles) and Al2O3 (circles). The mobility decreases with increasing tox.
densities (n ¼ 1011 cm2, Fig. 4), lsub scaling is strongly dominated by RP scattering. When n is small, plasmonic coupling to the bare SPPs is weak. Both RP and CI scattering decrease with lower tox because of greater screening of the bare SPPs and charged impurities by the image charges in the metal, resulting in smaller lrem and limp . Therefore, lsub increases significantly with decreasing tox. h-BN has the lowest limp because it screens the charged impurities most weakly, but it has the highest lsub as a result of its very weak RP scattering. This trend of decreasing mobility in larger tox has been reported in several experiments.20,28 At a technologically relevant carrier density (n ¼ 1012 2 cm , Fig. 5), dynamic screening of the remote phonons by the plasmons becomes important. Unlike Fig. 4(b), lrem decreases in HfO2 and Al2O3 with lower tox because the metal gate weakens plasmonic coupling to the bare SPPs, reducing dynamic screening. However, limp still decreases with increasing tox. Overall, lsub increases for HfO2 and Al2O3 with increasing tox, because RP scattering dominates CI scattering, while it decreases for h-BN given the weak RP scattering in the latter. At high carrier densities (n ¼ 1013 cm2, Fig. 6), lsub scaling is dominated by CI scattering. In Fig. 6(b), lrem is much higher than limp for all three dielectrics because the high carrier concentration results in strong plasmonic coupling and dynamic screening, and consequently, very low RP scattering rates. Although lrem decreases with larger tox like
FIG. 5. Calculated (a) lsub , and (b) lrem (open symbols) and limp (solid symbols) vs. tox at n ¼ 1012 cm2 for HfO2 (squares), h-BN (triangles) and Al2O3 (circles). The mobilities for HfO2 and Al2O3 increase with increasing tox at small tox.
Appl. Phys. Lett. 102, 183506 (2013)
FIG. 6. Calculated (a) lsub , and (b) lrem (open symbols) and limp (solid symbols) vs. tox at n ¼ 1013 cm2 for HfO2 (squares), h-BN (triangles) and Al2O3 (circles). The inset in (b) shows a magnified plot of limp . There is no significant dependence on tox.
in Fig. 4(b), its high values mean that it has no effect on the scaling of lsub . Instead, we observe that lsub limp , i.e., the carrier mobility is dominated by CI scattering. Also, limp for HfO2, h-BN and Al2O3 vary weakly with tox and are all around 8 103 to 104 cm2 V1 s1, because screening at high n is dominated by the polarization charge which depends only on n, and the image charge distribution has much less effect on CI screening. Again, we find that h-BN gives the highest lsub because of its weak RP scattering. From our results, we find that the tox-scaling of lsub depends on the top oxide and carrier density. In HfO2 and Al2O3 where RP scattering dominates, lsub decreases with smaller tox at low n but increases at higher n because plasmonic coupling is weaker in thinner oxides. In h-BN where RP scattering is weak and CI scattering dominates, lsub decreases with smaller tox for all n because a thinner dielectric screens the charged impurities more effectively. At very high n, the tox-scaling of lsub is insignificant for all three dielectrics. To understand how the scattering mechanisms vary with carrier density and temperature, we compute lsub at different n for a 100 nm thick HfO2 top gate from 50 to 400 K. We choose a 100 nm thick top dielectric to avoid, in our analysis, the complication of the image charge effect. The results are shown in Fig. 7. At 50 K, lsub is very close to limp (dashed line) indicating that CI scattering dominates at low temperatures. As n decreases, the temperature variation of lsub becomes larger. At high n, the plasmonic coupling to the
FIG. 7. Calculated lsub vs. carrier density for a 100 nm thick HfO2 top gate at different temperatures. The temperature-induced change in mobility becomes smaller as carrier density increases.
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SPPs is strong. Thus, the temperature variation is smaller because the RP scattering is heavily screened and has significantly weaker effect on mobility. We predict that the temperature dependence of carrier mobility in top-gated SLG is reduced as the carrier density increases. In summary, we have computed the dependence of the substrate-limited mobility of top-gated SLG on the top oxide thickness, and find that it varies with the carrier density. This variation depends on the strength of the plasmon coupling to the SPPs in the top and bottom oxides as well as the screening of this plasmonic coupling by the top gate. At low carrier density, the increase in lsub with decreasing tox is substantial for all dielectrics. Our results also indicate that the temperature dependence of the mobility diminishes with increasing carrier density. Of the top gate dielectrics considered, h-BN results in most mobility improvement with a thinner top gate. We acknowledge financial support from Texas Instruments, the Semiconductor Research Corporation, the Microelectronics Advanced Research Corporation, the Focus Center Research Project for Materials, Structure and Devices (MSD), and Samsung Electronics Ltd. Valuable technical discussions were provided by Matthew J. Hollander and Suman Datta (PSU), Emanuel Tutuc (UT Austin), Eric Pop (UIUC), and David K. Ferry (ASU). 1
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