Two-Dimensional Bipolar Junction Transistors

Transcription

1 wo-dimesioal iolar Juctio rasistors ehaz Gharekhalou, Sia Khorasai, ad Reza Sarvari School of Electrical Egieerig, Sharif Uiversity of echology, P. O. ox , ehra, Ira fax: Abstract Recet develomet i fabricatio techology of laar two-dimesioal (D) materials has brought u ossibilities of umerous ovel alicatios. Our recet aalysis has revealed that by defiitio of - juctios through aroriate attered doig of D semicoductors, ideal exoetial I-V characteristics may be exected. However, the theory of D juctios turs out to be very much differet to that of the stadard bulk juctios. ased o this theory of D diodes, here we costruct for the first time, a model to describe the D iolar Juctio rasistors (D-Js). We derive the smallsigal equivalet model, ad estimate the erformace of a D-J device based o Grahoe as the examle material. A curret gai of about 38 ad maximum threshold frequecy of 77GHz, together with a ower-delay roduct of oly 4fJ er m lateral width is exected at a oeratig voltage of 5V. Also, we derive ecessary formulae ad a ew aroximate solutio for cotiuity equatio i the D cofiguratio, which have bee verified agaist umerical solutios.. Itroductio Sice the discovery of Grahee i 004 [], lots of attetio has bee draw to the so-called twodimesioal (D) materials. he commo articular asect of D materials is that they are all oly oe moatomic layer thick. I geeral, basic roerties of D materials ca be routiely egieered by doig, fuctioalisatio, ad chemical modificatio [-6], such as hydrogeatio or oxidizatio. For istace, the fully-hydrogeated Grahee, or Grahae, should be obtaiable as a stable D hydrocarbo [7,8]. Sice the, a great deal of research has bee focused o the roerties of this uique material, ivestigatig its electroic, otical [9], aotube allotroes [0], ad eve suercoductive roerties []. It has bee show that although erfectly-ordered Grahae may be difficult to obtai [], however, as it has bee show [3], the ifluece of substrate material o the quality of D crystal domais caot be eglected. Recetly, a similar material with idetical chemical cofiguratio has bee created by hydrogeatio of Germaee, amed as Germaae [4]. hese moolayer hydrogeated dielectrics are very much similar ad are exected to be described by idetical theories. For most alicatios such as ours, siglesided hydrogeatio of Grahee is however much better suited, commoly referred to as Grahoe [3]. It has bee show that stacks of D materials could also lead to urecedeted alicatios i everyday s sciece ad techology. With the advet of isulatig D oro Nitride, further roerties of such layered D sadwichs have bee roosed [3,5]. As we have show usig tight bidig calculatios Grahae has otetial advatages over Grahee with regard to havig a cotrollable badga which makes it highly desirable for itegrated electroic alicatios [,6,7]. May articles are cocered with simulatios of material roerties oly, whereas studyig the oeratio of the roosed class of D structures would eed a sigificatly differet theory. he formulatio discussed i this article is based o the theory of D - rectifyig juctio diodes develoed i a earlier study of the authors [8]. I the revious research, we ivestigated a - juctio with abrut doig rofile made of a Grahoe sheet [8]. Evidetly, a air of similar D juctios may rovide a fairly good basis for oeratio of a D iolar Juctio rasistors (D-Js). We actually had made a basic study of D-Js [9], however, our iitial effort led to a device with a curret gai less tha uity. Here, we revisit the desig of a D-J with abrut doig rofile, ad Grahoe is used as the host D material. A electrostatic aalysis of the trasistor is doe together with Shockley s law. he small-sigal equivalet circuit of the D-J i its active regio is obtaied, ad we show that extremely large outut resistaces together with vaishigly small owerdelay roducts are ossible at oce, thus brigig the ultralow ower circuits oe ste closer to the reality. For the first time, we derive the ecessary formulae for descritio of D-Js, study the D cotiuity equatios, ad suggest useful aroximate aalytical solutios. Accuracy of aroximate aalytical solutios is established by comarig to the exact umerical solutios. We aticiate that D-Js based o Grahae, Germaae, ad similar D materials, would also share similar theories.

2 Figure. New charge model to illustrate surface charge.. Physical Parameters Some of the required hysical arameters are already reorted i literature, ad the remaiig data are derived from bad structure ad/or available relatios. hese iclude the effect of doat tye ad cocetratio of carrier mobility, effective dielectric costat, effective mass ad the D desity of states. here exist exerimets which have robed the ifluece of chemical doats o the carrier mobility i Grahee [0]. Hall measuremets revealed that NO ad H O adsortio act as accetors whereas NH 3 ad CO are doors. It is reorted that whe Grahee is treated with atomic hydroge, it starts exhibitig a isulatig behavior because of icreasig bad ga. his is while the electro mobility of Grahae [8] has bee also measured i exerimet, ad it is foud that by aealig, the mobility of Grahae is recovered to the Grahee mobility 3,500cm /Vs. Also, recet exerimets have reorted the ifluece of chemical doats o the carrier mobility i Grahee [0,]. It is observed that eve for surface chemical doat cocetratios i excess of 0 cm - there is o observable chage i the carrier mobility. We have foud the effective mass ad D desity of states from Grahoe bad structure [8]. For this urose, we used the tight-bidig method to obtai bad structure, leadig to the effective masses m * C.03, m * V 0.63 [8]. ased o these data, the total laar desity of states was obtaied to be D N m k [8]. C V C V he relative dielectric costat of Grahee is also reorted to be about.5 []. here is however o available measuremet o the dielectric costat of Grahae. I this study, we use the dielectric image method ad take o a value of.7 for the relative dielectric costat of Grahae layer o SiO substrate. It is ot difficult to show usig the electrostatic image method that the effective ermittivity see by charges lyig o a iterface betwee two dielectrics is simly Figure. Abrut - juctio i the thermal equilibrium: (a) Sace charge distributio; (b) Electric field distributio; (c) Potetial distributio. ( ). Hece, if the substrate is SiO with eff the relative ermittivity 4.4 ad the coverig layer o the Grahae is vacuum or air, the the effective ermittivity of the layer will be give by eff (4.4 ) he D P-N Juctio I our revious work we desiged a abrut - juctio with arbitrary dimesios. We assumed that the juctio has ifiity legth ad solved the cotiuity equatio with the cotiuity coditios at the two edges. Now we require desigig a abrut - juctio with submicro dimesios. his requiremet will be eeded to obtai high gai ad badwidth. For this urose, we have to resolve the cotiuity equatio for a limited legth juctio. Also we will drive the ecessary theoretical formulae ad comare them with umerical calculatios. 3.. uilt-i Potetial ad Deletio Layer I a bulk 3D - juctio we deal with sace charge withi the deletio regio. ut for a D juctio, there is a surface charge. So we have itroduced a ew charge model to illustrate this D system [8]. We divide this charge sheet ito ifiitely may thi lie charges with differetial lie charge desities A ad D i ad deletio regios, resectively. Figure shows this idea schematically. he electric field due to each lie charge is E( r) r where r is measured with

3 juctio it is roortioal tow. Now sice the electric field is o-zero iside the ad eutral regios, the otetial dros across these regios are foud as ) l( ) l( ) ( X x )l( X x ) W l( W ) V ( A D x 0 x 0 X X A 0 0 D ( X x )l( X x ), X x x D V (A D x 0 x 0 X X ) l( ) l( ) ( X x )l( X x ) W l( W ) A 0 0 A ( X x )l( X x ), x x X D (4) (5) o simlify the aalysis, the deletio aroximatio is here used. Hece, uder the thermal equilibrium, the total egative charge er uit width i the -side must be equal to the total egative charge er uit width i the -side N x N x (6) A 0 D 0 Figure 3. (a) Electric field distributio; (b) Electric otetial distributio for the desiged - juctio. resect to the symmetry axis of the lie charge. he total electric field had bee reorted for the four regios over the surface of the juctio i [8], which may be ow combied ito oe uified equatio as x x E( x) ( x, z x x 0 E ˆ l l () A D x x x 0 where, qn, A D A D. Figure shows this abrut - juctio uder the thermal equilibrium. It may be oticed that sice E ( x, z zˆ z 0, the derivative E( x) x does ot reroduce the iitial charge distributio, ulike the regular ste juctios. his is E( x) x E x, z 0. Now, from simly because the Poisso equatio, the otetial distributio is obtaied. he otetial across the deletio regio, gives the built-i otetial V ( ) l( ) l( ) bi A D x W x x W x () where W x 0 x 0. For a symmetric abrut juctio, x x W. Hece (3) is reduced to, A D 0 0 V W l bi (3) It ca be see that the built i otetial is roortioal to W while for a covetioal bulk - For a abrut juctio the built-i otetial is equal to V V l N N, V k q bi A D i (7) From (), (6) ad (7), the deletio width is calculated to be V ln N A D i W q N l N N N l N N D A D A D A 3.. Curret-Voltage Characteristics (8) he ijected miority-carriers distributio i the - side is govered by the cotiuity coditios at the two edges, subject to the Shockley s boudary coditios ( x) d d ( ( ) ) ( ) 0 0 E x V dx dx x ( x x ) ex( V V ) 0 0 ( x X ) 0 ad similarly i the -side ( x) d d ( ( ) ) ( ) 0 0 E x V dx dx x ( x x ) ex( V V ) 0 0 ( x X ) 0 (9) ( As metioed before, i the eutral regio the electric field is o-zero ad E(x) reresets the local electric field of each oit i this regio. 3

4 ex( N ) L N N N 0 V N( x V V N( x W ) 0 E (4) We ow defie two arameters M N ad f E V. Usig these arameters we obtai the oliear firstorder system of equatios M N ex( N ) M M fm L (5) Figure 4. Miority carrier distributios uder forward bias: (a) Hole cocetratio i the -regio; (b) Electro cocetratio i the -regio. o solve these equatios umerically, besides usig these equatios directly, we ca also use the logarithmic form discussed i the ext subsectio he Cotiuity Equatio i Logarithmic Form A useful ad stable method for solvig this equatio is to trasform the equatio ito Logarithmic form. We start from () ad first defie ( x) exn x. he we have ( x) ( x) N x () ( x) ( x) ( x) ( x) N x Now, by dividig both sides by ( x ), we get E x x L ( ) ( ) 0 0 ( x) V ( x) ( x) 0 () (3) where L V. Hece, we may rewrite ( as the secod-order oliear equatio 4 which ca be solved by the well-kow Ruge-Kutta method. he solutios i both exoetial ad logarithmic forms agree, but the logarithmic form is far more stable Aroximate Solutio to the Cotiuity Equatio Here, we suggest a aalytical aroximatio to (9,. First, we suose that the electric field at each oit located i the eutral regio is a costat arameter. he we solve the equatio just if as the electric field would have zero derivatives. Oce the exressio is foud, the ositio-deedet electric field is lugged i. Obviously, this method is sufficietly accurate for slowly varyig electric fields. hus, this aroximatio gives ( x) 0 0 ex( V V ) ex x L ( x) ex X L ( x) ex x L ( x) ex X L ( x) ex x0 L ( x) ex X L ( x) ex x 0 L ( x) ex X L ( x) where i (6) Li ( x) V E( x) E ( x) 4V Similarly, we obtai the ijected holes distributio i the -side theoretically as ( x) 0 0 ex( V V ) ex x L ( x) ex X L ( x) ex x L ( x) ex X L ( x) ex x 0 L ( x) ex X L( x) ex x 0 L ( x) ex X L ( x) where i i( ) ( ) ( ) 4 L x V E x E x V. (7) I ractice, these aroximatios have bee verified agaist umerical solutios of the exoetial ad logarithmic oliear differetial equatios, yieldig surrisigly good agreemets, as discussed later below. Now, we roceed to desig of the ste D - juctio. he overall device legth is take to be about 370m ad the surface doat cocetratios are N D = 0 cm - ad N A =5 0 cm - as deicted i figure. All calculatios are here doe er uit-width.

5 able. Numerical values of hole ad electro curret desities. J 0 (A/cm) J 0 (A/cm) of the device. Usig (8), the deletio width is foud to be 68.5m. Figure 3 illustrates the corresodig electric field ad otetial distributios. ased o this electric field distributio, we solve (9, both umerically ad usig the aroximate formulae (6,7). Figure 4 shows how the exact umerical solutio (dashed curves) ad aroximate theoretical formulae (solid curves) behave very closely. he results rove the accetable accuracy of the formulas. Now, we cotiue to derive the curret desities. At, the hole curret desity is x x 0 d J q ( E V ) J ex( V V ) 0 dx x x 0 q V ex x 0 L( x ex X L( x 0 J 0 q 0E ( x A( x L( x ex x 0 L ( x ex X L( x L( x x 0 X x X A( x ex 0 ex ex ex L( x L( x L( x L( x (8) Similarly, we obtai the electro curret i the -side at x x 0 as d J q ( E V ) J ex( V V ) 0 dx x x 0 q V ex x 0 L ( x ex X L ( x 0 J 0 q0e( x Ax L x 0 ( ex x 0 L ( x ex X L ( x L ( x x X x X Ax 0 ex 0 0 ex ex ex L ( x L ( x L ( x L ( x (9) able shows the umerical values of the hole ad electro curret desities exected for this - juctio. he total curret desity is give by the sum of the (8) ad (9) as V V J J J b a ex( ) ( J J ) ex( ) 0 0 V V ( Figure 5. Electrostatic model for -- trasistor to aroximate laar charge distributio usig lie charges. 4. he -- iolar rasistor I this sectio, we desig ad aalyze a -- D-J with abrut charge doig rofile ad fiite width. Figure 5 shows the electrostatic model to aroximate laar charge distributio usig lie charges with ifiitesimal width ad charge desity. ased o this charge model the uified equatio for the total electric field over the surface of the device is give by x x x( x ( X x )) E( x) E l l x W ( x x )( x X ) x ( X WC ) C l x ( X x ) E where N, E,, C. () Figure 6 illustrates the electric field ad otetial distributios over the surface of our desiged D-J with a total width of about 7.7μm. Here, the doat cocetratios are N E = 0 cm -, N = 0 cm - ad N C = 0 0 cm -. he legths of termials are suosed to be 4m, 464m ad 7.μm for emitter, base ad collector resectively. ased o (8), the deletio widths are foud to be x =4.5m, W E =66.75m, x =.58m, W C =.33m as show i Figure 6. Now, we may roceed to obtai the basic arameters itroduced for a laar D biolar trasistor. 4.. Curret Gai Figure 7 resets a schematic of a -tye D-J, coected i a exteral bias circuit. We here assume a commo base cofiguratio ad ormal (active) mode of oeratio to fid the corresodig curret gai. Calculatio of curret gai would obviously eed a thorough aalysis of carrier trasort across the width of structure. 5

6 Figure 7. A -- trasistor uder the ormal forward oeratig coditios. Similarly, i the seudo-eutral emitter ad collector regios, the equatios for miority-carriers are E d d dx V dx ( 0E C ) ex 0 E ( C ) 0 E( C ) x x V V E( C ) 0 E( C ) E( C ) ( x x ) E( C ) ed E ( C ) 0 E( C ) (5) Figure 6. (a) Electric field distributio; (b) Electric otetial distributio for the metioed -- trasistor. All the termial currets are artitioed ito hole ad electro comoets, ad the from the cotiuity equatio we obtai the steady-state currets. he trasort i the eutral base regio of the device is o-trivial ad cotais both diffusio ad drift terms. Hece, the distributio of ijected mioritycarriers (electros) i the base regio is govered by the cotiuity coditios at the two edges, subject to the Shockley s boudary coditios d d 0 dx dx 0 0 E V 0 ( x ex V V E ( x X ) ex V V C () Figure 8 shows the electro distributio i the eutral base regio based o umerical solutio of this equatio. he electro surface curret desities at the emitter edge J E ad the collector edge J C are give by with x0 E W E, x0 C X WC, xede X E ad xedc XC as illustrated i figure 7 Numerical solutios of these equatios are show i figure 8. he hole surface curret desities at the emitter edge J E ad the collector edge J C are give by J q ( E V d ) dx E( C ) E( C ) E ( C ) E ( C ) x x0e ( x0c ) a ex( V V ) E ( C ) E( C ) (6) Numerical arameter values for the costat arameters i the above sets of equatios (3,4,5) are elisted i able. Now, the termial surface curret desities ca be summed over resective electro ad hole currets as J E J E J, E JC JC J C, ad J J E J. Hece, C the commo-base ad commo-emitter curret gais are foud to be J J J J J J J J C E C C E E E C 38 (7) (8) JE q ( E V d ) dx x0 a E ex( VE V ) b E ex( VC V ) J q ( E V d ) dx C x X a C ex( VE V ) b C ex( VC V ) (3) (4) Figure 9 shows J C versus V CE for the commo-emitter cofiguratio uder ormal ad iverted bias cofiguratio. he saturatio regio is here defied as V CE < V CE,sat =0.V ad the curret gai i iverted mode is foud to be β r =0.9. I cotrast, stadard bulk Js start to eter the saturatio at V CE < 0.V, ad therefore D-Js may oerate at smaller voltages comarig to their bulk couterarts. 6

7 Miority Carrier Distributio (cm - ) Emitter Neutral Regio (m) ase Neutral Regio (m) Collector Neutral Regio (μm) Figure 8. Miority carries distributios i eutral regios of the emitter, base ad collector. able. Numerical values of costats for hole ad electro curret desities. J E(A/cm) J C(A/cm) J E(A/cm) J C(A/cm) a E be a C b C a E a C Small-Sigal Model he equivalet circuit for the behavior of this laar J is idetical to that of a bulk J. Small-sigal model arameters are here defied as trascoductace gm ic ve, outut resistace r o, ad iut resistace r i v g E m. ased o the flatess of the I C curves, the Early voltage V A is obtaied to be ifiity, ad hece the outut resistace exressed as r i v V I o C CE A C was obtaied as virtually ulimited. his is articularly imortat for obtaiig very large voltage gais at low oeratig voltages ad ower cosumtios usig roerly desiged -- D-Js as active loads, ad also for desigig earideal curret-sources. he diffusio caacitace reresets the ijected electro cocetratio versus distace i the eutral base give as ( x, V) C q dx V (9) Variatio of C as a fuctio of alied voltage is show i Figure 0. he geeral tred is that it attais a maximum value of roughly C,max 3( F / cm) at a bias of about VE 0.8( V ), ad starts to fall afterwards. his behavior is quite tyical ad also see i bulk Js as well. Also, the ucomesated doats i the trasitio regios of a - juctio cause dioles, causig the juctio caacitace Figure 9. Outut characteristics i commo emitter cofiguratio. C j JC (A/cm) ( A/ cm) J 0.40 ( A / cm) VCE (V) ( A/ cm) -0-6 J J J ( A/ cm) ( A / cm) dq dv (3 Figure also deicts the juctio caacitace of the reverse-biased -C juctio versusv C. Now we assume that a AC sigal is fed ito the device by a curret source i b coected to the base. he cutoff frequecy f is the frequecy at which the AC curret gai ac ic ib falls to uity. Figure shows the variatio of f as a fuctio of collector surface curret desity J C. he maximum value of this frequecy is about 77GHz at a fairly surface curret desity of 0.38 ma/m. he tyical cosumed ower is very small. A comariso to a existig desig [3] of bulk Js may be made by takig a idetical base width of μm. he at the maximum oeratig frequecy which J J J J J J 3.63( A/ cm) ( A/ cm) 0.0 ( A / cm) ( A / cm) ( A / cm) 7

8 able 3. Numerical values of Ebers-Moll arametres. J ED (A/cm) J CD (A/cm) N I corresods to a trasit time of f.s, ad takig a collector bias voltage of 5V, the oeratig ower (er m width) would be aroud.9mw. Hece, the estimated ower-delay roduct [4] will be aroud 4fJ, far less tha the tyical record values which are of the order of tes of femto-joules. It has to be metioed that this ower-delay roduct has bee derived whe the D-J is such biased to allow the maximum oeratio seed. As it has bee show i Figure, much smaller ower-delay roducts ca be exected if lower bias curret ad therefore the threshold frequecy is chose. Actually, the ower-delay roduct blows u at large bias currets, ad for istace if oe limits the cutoff frequecy to 70GHz, the resultig ower-delay roduct will be strictly limited to the record value of 0.0fJ/m. t Figure 0. Diffusio ad juctio caacitace versus voltage. Figure. Cutoff frequecy versus curret desity Large-Sigal Model he DC emitter ad collector currets i active mode are well modeled by a aroximatio to the Ebers Moll model. We have thus the relatioshis J E J ED ex( VE V ) IJ CD ex( VC V ) (3) JC NJ ED ex( VE V ) J CD ex( VC V ) (3) Numerical arameter values for these set of equatios are elisted i able Aroximate Formulae for a -- D-J Here we will derive arameters of -tye D-Js based o the theory derived for - juctio i sectio III. A dual aalysis will be evidetly alicable to - tye D-Js by chagig the carrier tyes ad voltage ad curret olarities. For calculatio of trasistor arameters, the ijected miority-carriers distributios are ecessary, ad here we reset the aroximatio solutios followig the equatios calculated i the above sectios. Startig with the miority-carriers i the base, we have to solve (3). he aroximate solutio to this equatio is Figure. Power-delay versus cutoff frequecy; the iset shows a magificatio of the kee. ( x) 0 ex x L ( x) ex x L ( x) 0 ex( VC V ) X L x X L x ex ( ) ex ( ) ex x L ( x) ex X L ( x) ex x L ( x) ex X L ( x) ex X L ( x) ex X L ( x) 0 ex( VE V ) where L x V E x E x V (33) i i( ) ( ) ( ) ( ) 4 he electro surface curret desity of the emitter is similarly give by. 8

9 JE ae ex VE V be ex VC V q 0V ae q 0E( ex X L ( ex X L ( ex X L ( ex X L ( L ( L ( q 0V be ex X L ( ex X L L ( L ( ( (34) he electro surface curret desity of the collector will be JC a C ex VE V b C ex VC V ex X L ( X) ex X L ( X) ac q V 0 ex X L ( X) ex X L ( X) L ( X) L ( X) q 0V bc q 0E( X) ex X L ( X) ex X L ( X) ex X L ( X) ex X L ( X) L ( X) L ( X) (35) he hole distributios i the emitter ad collector are obtaiable usig (5). he solutios are like those obtaied for a - juctio, give by 0 E ( C ) E ( C )( x) 0 E ( C ) ex( VE ( C ) V ) A( x) ex x LE ( C ) ( x) ex xede ( C ) LE ( C )( x) ex x LE ( C )( x) ex xede ( C ) LE ( C ) ( x) x 0 E ( C ) x ede ( C ) A( x) ex x E C x ex 0 ( ) ede ( C ) LE ( C )( x) ex ex L E ( C )( x) LE ( C )( x) LE ( C )( x) with (36) i L ( E C i x ) V E ( ( ) x ) ( ) E ( x ) 4 V Also, we have x0 E WE, x0 C X WC, xede XE ad xedc XC, as illustrated i Figure 7. he hole surface curret desities i the emitter ad collector are obtaied as JE ( C ) a E( C ) ex( VE ( C ) V ) q V a q E ( x ) 0 E( C ) E( C ) 0 E( C ) 0 E( C ) A( x0 E( C )) ( x ) ex x L ( x ) ex x L ( x ) 0 E( C ) 0 E( C ) E( C ) 0 E( C ) ede ( C ) E 0 E( C ) LE ( C )( x 0 E( C )) ) ex ( ) ex x 0 E( C ) LE ( C )( x0e( C ) xede ( C ) LE ( C ) x0 E( C ) LE ( C )( x0 E( C )) (37). 5. Coclusios We derived ecessary formulae i studyig D ste - juctios ad suggested a aroximate aalytical solutio for cotiuity equatios, which exhibited good agreemet with exact umerical solutios. he, we roceeded to develo the theory of D Js, ad used the oe-sided hydrogeated Grahee, or Grahoe, as the examle D material. Distributios of electric field ad otetial were calculated, ad the small sigal hybrid- model of the trasistor was obtaied. he results showed that this D-J would oerate somewhat similar to bulk Js i DC ad AC circuits, however, with very differet erformace arameters. he maximum frequecy ad curret gai were foud to be resectively as 77GHz ad 38, at a maximum ower-delay roduct of 4fJ/m. his is while much smaller ower-delay roducts would be ossible at the exese of less threshold frequecy. his relimiary aalysis demostrates the usefuless of such D-Js, i several asects. As we have show, a D-J would allow areciable curret gai, large oeratig frequecies, yet very small ower-delay roducts which may be ultimately attributed to the associated ultrathi D materials. Furthermore, whe used as active loads i a roerly desiged circuit, D-Js may be able to outerform their CMOS couterarts with regard to the large voltage gais at exceedigly small ower cosumtio. Refereces [] Geim A K, Novoselov, K S 007 Nature Mat [] Jug N, Kim N, Jockusch S, urro N J, Kim P, rus L 009 Nao Lett [3] Gierz I, Riedl C, Starke U, Ast C R, Ker K, 008 Nao Lett [4] Das A, Pisaa S, Chakraborty, Piscaec S, Saha S K, Waghmare U V, Novoselov K S, Krishamurthy H R, Geim A K, Ferrari A C, Sood A K 008 Nat. Naotech. 3 0 [5] Wag X, Li X, Zhag L, Yoo Y, Weber P K, Wag H, Guo J, Dai H 009 Sciece [6] Cervates-Sodi F, Csayi G, Piscaec S, Ferrari A C 008 Phys. Rev [7] Sofo J O, Chaudhari A S, arber G D 007 Phys. Rev [8] Elias D C, Nair R R, Mohiuddi M G, Morozov S V, lake P, Halsall M P, Ferrari A C, oukhvalov D W, Katselso M I, Geim A K, Novoselov K S 009 Sciece [9] Schäfer R A, Eglert J M, Wehrfritz P, auer W, Hauke F, Seyller, Hirsch A 03 Chemie [0] We X-D, Yag, Hoffma R, Ashcroft N W, Marti R L, Rudi S P, Zhu J-X 0 ACS Nao 6 74 [] Savii G, Ferrari A C, Giustio F 00 Phys Rev Lett [] Flores M Z S, Autreto P A S, Legoas S, Galvao D S 009 Naotechology [3] Kharche N, Nayak S K 0 Nao Lett. 574 [4] iaco E, utler S, Jiag S, Restreo O D, Widl W, Goldberger J E 03 ACS Nao [5] Khorasai S, Sodagar M 03 Scietia Iraica (submitted) arxiv

10 [6] Gharekhalou, Alavi M, Khorasai S 008 Semicod. Sci. echol [7] Gharekhalou, Khorasai S 0 i Grahee: Proerties, Sythesis ad Alicatio (edited by Z. Xu, New York: Nova Sciece) Cha [8] Gharekhalou, Khorasai S 009 IEEE ras. Electr. Dev [9] Gharekhalou,, ousaki S, ad Khorasai S 00 J. Phys.: Cof. Ser [0] Schedi F, Geim A K, Morozov S V, Hil E W, lake P, Katselso M I, Novoselov K S 007 Nature Mat [] Che J H, Jag C, Fuhrer M S, Williams E D, Ishigami M 008 Nature Phys [] Wehlig O, Sasioglu E, Friedrich C, Lichtestei A I, Katselso M I, lügel S 0 Phys. Rev. Lett [3] Cressler J D, Warock J, Harame D L, urghartz J N, Jekis K A, Chuag C- 993 IEEE Electro Dev. Lett [4] Reisch M 003 High-Frequecy iolar rasistors: Physics, Modellig, Alicatios (erli: Sriger) 0