Observation of vacancy-induced suppression of electronic cooling in defected graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Observation of vacancy-induced suppression of electronic coolingin defected graphene
Qi Han, ∗ Yi Chen, ∗ Gerui Liu, Dapeng Yu, and Xiaosong Wu † State Key Laboratory for Artificial Microstructure and Mesoscopic Physics,Peking University, Beijing 100871, ChinaCollaborative Innovation Center of Quantum Matter, Beijing 100871, China
Abstract
Previous studies of electron-phonon interaction in impure graphene have found that static dis-order can give rise to an enhancement of electronic cooling. We investigate the effect of dynamicdisorder and observe over an order of magnitude suppression of electronic cooling compared withclean graphene. The effect is stronger in graphene with more vacancies, confirming its vacancy-induced nature. The dependence of the coupling constant on the phonon temperature implies itslink to the dynamics of disorder. Our study highlights the effect of disorder on electron-phononinteraction in graphene. In addition, the suppression of electronic cooling holds great promise forimproving the performance of graphene-based bolometer and photo-detector devices. NTRODUCTION
In recent years, there has been considerable interest in utilizing graphene as photo-detectors[1–8]. Most of these detectors are based on a hot electron effect, i.e. the electronictemperature being substantially higher than the lattice temperature. Two properties ofgraphene strongly enhance the effect. First, low carrier density gives rise to a very smallelectron specific heat. Second, weak electron-phonon (e-p) interaction reduces the heattransfer from the electron gas to the lattice. Thus, it is of practical interest to understandthe e-p interaction in graphene. Both theoretical and experimental efforts have been devotedto this topic. Earlier work was mainly focused on clean graphene and considered the Diracspectrum of electrons[9–15]. As the important role of impurities in electronic transport hasbeen revealed, its effects on the e-p interaction began to draw attention[16–18]. For instance,due to the chiral nature of electrons, long range and short range potentials scatter electronsdifferently in graphene[19–21]. Recently, a strong enhancement of electronic cooling via e-pinteraction in presence of short range disorder has been predicted[18]. This is achieved viaa so-called supercollision process. When the carrier density is low, the Bloch-Gr¨uneisentemperature T BG can be quite small. Since T BG sets the maximum wave vector of phononsthat can exchange energy with electrons, when T BG < T , only a portion of phonons cancontribute to the energy relaxation. Interestingly, in presence of short range potentials, thetheory has found that a disorder-assisted scattering process can occur, in which all availablephonons are able to participate. As a result, the energy relaxation is strongly enhanced.Shortly, two experiments confirmed the supercollision[22, 23], although long range potentialscattering usually dominates in such samples[24, 25]. In the case of long range potentials,Chen and Clerk have also predicted an increase of electronic cooling at low temperaturefor weak screening[17]. Note that besides the different potential profiles, e.g. long range orshort range, disorder can be static or dynamic. Despite these studies, in which only staticdisorder was considered, the dynamics of disorder has not been addressed.Here, we present an experimental investigation of the effect of vacancy on electroniccooling in both monolayer and bilayer defected graphene. In contrast to typical scatteringpotentials previously treated in theories or encountered in experiments, which are static,vacancies in our defected graphene are dragged by phonons, hence highly dynamic. Bystudying the nonlinear electric transport of defected graphene, a strong suppression of e-2 energy relaxation, instead of an enhancement in the case of static potentials, has beenobserved. The more disordered the graphene film, the stronger the suppression is. Our workprovides new experimental insight on the effect of scattering potential on e-p interaction.Moreover, the suppression suggests that the performance of graphene hot electron photo-detectors can be further improved by introducing vacancies. EXPERIMENT
In this work, we have investigated four exfoliated graphene samples on Si/SiO substrates.Thickness of all the monolayer (SM1 and SM2) and bilayer (SB1, SB2) samples were es-timated by optical contrast and confirmed by Raman spectroscopy[26]. Graphene flakeswere patterned into ribbons, using e-beam lithography. 5 nm Ti/80 nm Au were e-beamdeposited, followed by lift-off to form electrodes. Typical sample geometry can be seen inthe inset of Fig. 1a. In order to introduce vacancies, samples were then loaded into a Femtoplasma system and subject to Argon plasma treatment for various periods (from 1 to 5s)[27]. Four-probe electrical measurements were carried out in a cryostat using a standardlock-in technique. Room temperature π -filters were used to avoid heating of electrons byradio frequency noise. Information for four samples are summarized in Table I. RESULTS AND DISCUSSION
Previously, we have already demonstrated a hot electron bolometer based on disorderedgraphene[28]. It has been shown that the divergence of the resistance at low temperature canbe utilized as a sensitive thermometer for electrons. By applying Joule heating, the energytransfer rate between the electron gas and the phonon gas can be obtained. The samemethod has been employed in this work. As showing in Fig. 1a, the resistance of defectedgraphene exhibits a sharp increase as the temperature decreases. The divergence becomesstronger as one approaches the CNP. The R − T behavior can be well fitted to variablerange hopping transport, described as R ∝ exp[( T /T ) / ][29]. Here, the characteristictemperature T = 12 / [ πk B ν ( E F ) ξ ], with k B the Boltzmann constant, ν ( E F ) the density ofstates at the Fermi level E F , and ξ the localization length. By fitting to this formula, thelocalization length ξ is determined. It is employed as a measure of the degree of disorder. ξ i.e. via electron diffusion into electrodes or e-p interaction into thelattice. In our strongly disordered graphene, the former is significantly suppressed due toa very low carrier diffusivity. It has been found that e-p interaction dominates the energydissipation in such devices[28]. Then, the electronic temperature can be directly inferredfrom the resistance. Furthermore, it is estimated that the thermal conductance between thegraphene lattice and the substrate is much higher than that due to e-p interaction. Thus, thephonon temperature T ph is approximately equal to the substrate temperature T [5, 22, 30].Under these conditions, the energy balance at the steady state of Joule heating can bewritten as P = A ( T δ e − T δ ph ) (1)where P is the Joule Heating power, A is the coupling constant and T e is the electronictemperature. δ ranges from 2 to 6, depending on the detail of the e-p scattering process[12].Upon Joule heating, the electronic temperature is raised, leading to decrease of the resis-tance, depicted in Fig. 1b. Based on the resistance as a function of temperature, we obtainthe P − T e relation at different carrier densities, plotted in the insets of Fig. 2. P is alsoplotted against T − T . The linear behavior agrees well with Eq. (1) with δ = 3 for bothmonolayer and bilayer graphene at all carrier densities. It has been theoretically shown thatboth clean monolayer and bilayer graphene can be described by Eq. (1) with δ = 4 at lowtemperature [9, 12]. In presence of disorder, e-p interaction is enhanced and δ is reduced to3[17, 18]. δ obtained in our result is consistent with these theories, indicating the effect ofdefects. T dependence has also been reported in some other experiments. In the following,we will compare our results in detail with previous theoretical and experimental results.The e-p interaction is usually considered in two distinct regimes, high temperature and lowtemperature. In normal metals, Debye temperature θ D demarcates two regimes. Below θ D ,the phase space of available phonons increases with temperature, while it becomes constantabove it(all modes are excited). In graphene, because of its low carrier density, the Bloch-Gr¨uneisen temperature T BG becomes the relevant characteristic temperature. It is definedas 2 k B T BG = 2 hck F . Here k B is the Boltzmann constant, h the Plank constant, c thesound velocity of graphene and k F the Fermi wave vector. T BG stems from the momentum4onservation in e-p scattering. Because of it, when T ph > T BG , only a portion of phonons canparticipate in the process[31]. Considering the band structure of graphene, we have T BG =2( c/v F ) E F /k B in monolayer graphene and T BG = 2( c/v F ) √ γ E F /k B in bilayer graphene[12].Here v F ≈ m/s is the Fermi velocity, c ≈ × m/s and γ ≈ . n ≈ × cm dueto charge puddles[32, 33], it can be readily estimated that even at the CNP, T BG >
34 K.It is much higher than T e = 1 . T dependence. It has been theoretically shown that in the caseof weak screening, static charge impurities leads to enhanced e-p cooling power over cleangraphene and δ = 3[17]. For comparison, we plot our data, the theoretical cooling power ofclean graphene in Fig. 3. The theoretical prediction of the cooling power per unit area inclean monolayer graphene is [12] P clean = π D E F k B ρ ¯ h v c ( T − T ) (2)where ρ ≈ . × − kg/m is the mass density of graphene and D is the deformation po-tential chosen as a common value 18 eV [23, 34, 35] (this choice will be discussed later). Thetheoretical cooling power P clean as a function of the carrier density and electron temperatureis depicted as a transparent surface (with T ph =1.5 K) in Fig. 3a. It can be clearly seenthat the cooling power of our disordered samples SM1 and SM2 (green and blue lines) arewell below the surface at all carrier densities. For comparison, we also plot the data fromtwo other experiments in which T -dependence were observed at low temperatures[35, 36].These results (with similar T ph ) are either on or above the surface. The suppression of thecooling in Fig. 3a is considerable. For instance, at n = 4 × cm and T e = 20 K, the the-ory predicts P clean =4.7 nW/ µ m . In Ref. 36 the cooling power was found to be 27 nW/ µ m .In sharp contrast, our experiment gives a cooling power of 0.33 nW/ µ m for SM1, over anorder of magnitude lower than that in clean graphene. For the more disordered sample,SM2, it is even smaller.Similar suppression occurs in bilayer graphene samples, too. The cooling power per unit5rea in clean bilayer graphene is given by [12] P clean = π D γ k B ρ ¯ h v c r γ E F ( T − T ) (3)Fig. 3a shows the plot of Eq. (3), the cooling power of the bilayer samples SB1, SB2 and thedata from Ref. 36. Although not as pronounced as monolayer graphene, our data still belowthe theoretical surface. The weaker suppression may result from the fact that the bottomlayer of bilayer graphene has experienced less damage by our low energy plasma than thetop one[27]. Therefore, this less disordered layer provides a channel of substantial cooling.The e-p coupling strength depends on the deformation potential D , which characterizesthe band shift upon lattice deformation[37–39]. For the theoretical cooling power surface inFig. 3, we use D = 18 eV. Note that D for graphene ranges from 10 to 70 eV in variousexperiments, but 18 eV is the most common value for graphene[35]. If the suppression isdue to an over-estimated D , to account for the small cooling power, one would require D to be only about 5 eV, one-half of the lowest value reported. Therefore, we believe that thesuppression cannot be explained by a small D .By linear fits of P versus T − T , the coupling constant A can be obtained. In Fig. 4, A is plotted as a function of carrier density n . A for all samples decreases when approachingthe CNP. This is because fewer carriers at Fermi level could contribute to total cooling powerof the sample.We now take a look at the dependence of the coupling constant on the degree of disorder.As listed in Table I, the samples have been subject to various periods of plasma treatment.Consequently, the degree of disorder is different, indicated by the localization length ξ . Forinstance, ξ for SM1 and SM2 is 156 nm and 21 nm, respectively. As plotted in Fig. 4a,the coupling constant A of the less disordered SM1 is only about one-third of the value forthe more disordered SM2. The dependence of A on ξ is consistent with the suppression ofthe e-p scattering by disorder. For the two bilayer samples, SB1 and SB2, the localizationlengths are close. The n dependence of A for both samples aligns reasonably well and isconsistent with the monolayer samples, see Fig. 4b.The Joule heating experiment has also been carried out at different phonon temperatures T ph . In Fig. 4c, the coupling constant A is plotted as a function of T ph . Usually, A isindependent of T ph , which is actually seen at low temperature for SB1. However, as thetemperature goes above 7 K, A is enhanced. Later, we will show that the unexpected6 -dependence is likely related to the dynamic nature of vacancies.At first glance, the suppression of electronic cooling by vacancies seems surprising, inthat previous theories have predicted that disorder would enhance the cooling[17, 18]. Mostof earlier experimental results have confirmed the enhancement[22, 23, 35, 36]. However,there is a key difference between those earlier studies and ours. In the former, disorder istheoretically considered to be static. This is indeed true in other experimental work, inwhich the dominant disorder is due to charge impurities[24, 25]. However, in our samples,the dominant disorder is vacancies, which are completely dragged by phonons. The effect ofdisorder on the e-p interaction has been studied in disordered metals and found to depend onthe character of disorder[40–43]. In the case of static disorder, diffusive motion of electronsincreases the effective interacting time between an electron and a phonon, leading to anenhancement of interaction. However, dynamic disorder modifies the quantum interferenceof scattering processes[41]. As a result, the interaction is suppressed, in accordance withthe famous Pippard’s inefficient condition[44]. It is reasonable to believe that the observedsuppression results from dynamic disorder, vacancies. Furthermore, since the dynamics ofdisorder apparently depends on T ph , the dependence of the coupling constant A on thephonon temperature T ph is then conceivable. As described in Schmid’s theory[40, 41], thee-p scattering is suppressed due to strong disorder. The resultant energy relaxation rate τ − is of the order of ( q T l ) τ − where τ − ∝ T is the relaxation rate in pure material, q T is thewave vector of a thermal phonon and l is the mean free path. As q T ∝ T ph , the relaxationrate increases with T ph , in agreement with our result.It is also worthy to note that charge impurities are long range potentials that preservethe sublattice symmetry. This is in contrast to vacancies, which are short range potentialsand break the sublattice symmetry. The theory for supercollision models disorder as shortrange potential[18], while in Ref. 17, disorder potential is long-ranged. This character ofdisorder strongly affects scattering of chiral electrons in graphene. Our samples representa graphene system that is quite different from what was commonly seen, in that dynamicand short-ranged potentials dominate. Therefore, the quantitative understanding of ourexperimental results, including the power index δ , relies on future theory that takes boththe dynamics and the symmetry of disorder into account.7 ONCLUSION
In conclusion, we have observed significant suppression of electronic cooling in defectedgraphene. The cooling power of both monolayer and bilayer graphene samples show T dependence, consistent with disorder-modified electron-phonon coupling in graphene [17, 18].However, the magnitude of the cooling power is over an order of magnitude smaller than thatof clean graphene predicted by theory[9, 12] and also less than other experiments [35, 36].The more disordered a graphene film is, the lower cooling power is observed, confirming theeffect of disorder. The suppression of electronic cooling is attributed to the dynamic natureof vacancies, which has not been studied in graphene. This effect can be utilized to furtherimprove the performance of graphene-based bolometer and photo-detector devices.This work was supported by National Key Basic Research Program of China (No.2012CB933404, 2013CBA01603) and NSFC (project No. 11074007, 11222436, 11234001). ∗ These two authors contributed equally. † [email protected][1] X. Xu, N. M. Gabor, J. S. Alden, A. M. van der Zande, and P. L. McEuen, Nano Lett. ,562 (2009).[2] N. M. Gabor, J. C. W. Song, Q. Ma, N. L. Nair, T. Taychatanapat, K. Watanabe, T. Taniguchi,L. S. Levitov, and P. Jarillo-Herrero, Science , 648 (2011).[3] N. G. Kalugin, L. Jing, W. Bao, L. Wickey, C. D. Barga, M. Ovezmyradov, E. A. Shaner, andC. N. Lau, Appl. Phys. Lett. , 013504 (2011).[4] D. Sun, G. Aivazian, A. M. Jones, J. S. Ross, W. Yao, D. Cobden, and X. Xu, Nat Nano ,114 (2012).[5] J. Yan, M.-H. Kim, J. A. Elle, A. B. Sushkov, G. S. Jenkins, M. H. M., M. S. Fuhrer, andH. D. Drew, Nat Nano , 472 (2012).[6] H. Vora, P. Kumaravadivel, B. Nielsen, and X. Du, Appl. Phys. Lett. , 153507 (2012).[7] K. Yan, D. Wu, H. Peng, L. Jin, Q. Fu, X. Bao, and Z. Liu, Nat Commun , 1280 (2012).[8] X. Cai, A. B. Sushkov, R. J. Suess, G. S. Jenkins, J. Yan, T. E. Murphy, H. D. Drew, andM. S. Fuhrer (2013), arXiv: 1305.3297.
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PSfrag replacements R ( k Ω ) T (K)0 10 20 3001020304050 PSfrag replacements R ( k Ω ) I (nA) −
500 0 50010203040 xT e T ph ElectrodesLatticeHot electrons (a) (b) (c)
FIG. 1. Resistance of defected graphene. (a) Temperature dependence of resistance in sample SM1at different gate voltages, showing divergence at low temperature. Inset: Optical micrograph ofa typical device configuration. (b) Resistance of SM1 as a function of Joule heating current atdifferent gate voltages at T = 1 . V g = 0V, 6V, 9V, T e − T ph ( × K ) P ( n W / µ m ) (a) SM1 T ph =1.5K P (nW/ µ m ) T e ( K ) V g = 0V, 8V, 16V, 24V, 32V, 40V, 44V, 48V T e − T ph ( × K ) P ( n W / µ m ) (b) SB1 T ph =1.5K P (nW/ µ m ) T e ( K ) IG. 2. Cooling power of monolayer and bilayer defected graphene. (a)(b) Cooling power P against T − T shows a linear dependence for both monolayer and bilayer samples. Inset: P versus T e .FIG. 3. Suppression of electronic cooling in defected graphene. (a) Cooling power of clean mono-layer graphene is depicted as a transparent surface, with a logarithmic z-axis scale and as a functionof T e and n . P for SM1 and SM2 are over an order of magnitude smaller than clean graphene at allcarrier densities, while the data from others’ work is either on or above the surface. n is chosenas 4 × cm to account for charge puddles near CNP. T ph is 1.5 K in SM1 and the theoreticalsurface, 7K in SM2, 0.8 K in the data from Ref. 35 and 1.8 K in the data from Ref. 36. (b) Similarsuppression is observed in bilayer defected graphene samples. n is chosen as 4 × cm . T ph is1.5K in SB1, SB2 and the theoretical surface, and 1.8 K in the data from Ref. 36. −5 SM2 ξ =21nmSM1 ξ =156nm n ( × cm −2 ) A ( n W / µ m K ) −6 −4 −2 0 2123456 x 10 −4 SB1 ξ =50nmSB2 ξ =54nm n ( × cm −2 ) A ( n W / µ m K ) −4 T ph (K) A ( n W / µ m K ) SB1SB2 SM2 468x 10 −5 (a) (b) (c) FIG. 4. Coupling constant A . (a) Extracted coupling constant A as a function of carrier density n in monolayer samples. The more defective sample, SM1, exhibits a smaller coupling constant.(b) Dependence of A on carrier density n in bilayer graphene samples. The curves for two sampleswith similar degree of disorder align reasonably well. (c) Dependence of A on phonon temperature T ph . The data of SM2 are plotted with respect to the right y-axis. TABLES
TABLE I. Sample information of four investigated devices. Different Ar gas flow rates and plasmatreatment times have been applied to produce different amount of vacancies. V CNP is the chargeneutrality point (CNP) of samples and ξ is the localization length near the CNP.Devices Length( µ m) Width( µ m) Ar flow rate(sccm) Plasma treatment period(s) V CNP (V) ξ (nm)SM1 2 3 3 1 14.5 156SM2 6.7 2.7 4 3 30 21SB1 3 2.7 4 3.5 70 50SB2 6 2.7 4 5 57 54(nm)SM1 2 3 3 1 14.5 156SM2 6.7 2.7 4 3 30 21SB1 3 2.7 4 3.5 70 50SB2 6 2.7 4 5 57 54