Supplementary material for " Coherent and Tunable Terahertz Radiation from Graphene Surface Plasmon Polarirons Excited by Cyclotron Electron Beam "

Transcription

1 Suppleenty teil fo " Coheent nd Tunble Tehet Rdition fo Gphene Sufce Plson Polions Excited by Cycloton Electon Be " To Zho,, Sen Gong,, Min Hu,, Renbin Zhong,,Diwei Liu,,Xioxing Chen,, Ping hng,, Xinn Wng, 3, Cho Zhng, 4, Peiheng Wu, 3, Shenggng Liu *,. Tehet Resech Cente, School of Physicl Electonics, Univesity of Electonic Science nd Technology of Chin, Chengdu, Sichun, 654, Chin,. Coopetive nnovtion Cente of Tehet Science, Chengdu, Sichun, 654, Chin 3. School of Electonic Science nd Engineeing, Nnjing Univesity, Nnjing, Jingsu,, Chin 4. School of Physics nd nstitute fo Supeconducting nd Electonic Mteils, Univesity of Wollongong, New South Wles 5, Austli

2 Appendix : The electognetic fields poduced by the cycloton electon be Fig.. The schetic of cycloton electon be, the electon be oves t velocity u v e v e with cycloton tjectoy. The schetic of n cycloton electon be (CEB) is shown in Fig.., the electon be oves t velocity v ve ve with cycloton tjectoy. The vecto nd scl potentils A nd e used to deive the electognetic fields poduced by the CEB. The vecto nd scl potentils obey the following d'alebet equtions, given below in MKS unit, A t A t J (.) (.) With Loent guge: A t Then, the electognetic fields E nd H cn be obtined: A A E H t (.3) (.4) The position nd velocity vectos of the CEB cn be descibed in Ctesin co odinte syste. (.5) t e x t e y t e t e x y (.6) v t e v t e v t

3 Then, the chge nd cuent density cn be expessed. Whee q is the chge quntity, e q t q x x t y y t t (.7) J v t e v t e v t q x x t y y t t (.8) v is the coponent of velocity, v is the cycloton velocity of the be ound xis. n the cylindicl coodinte syst e, (.9) x t cos ct (.) y t sin ct v t (.) t Whee is the dius of the tjectoy pojection of cycloton electon be in X-Y plne, v c. Siilly, the obseved position of fields cn be expessed in the cylindicl coodinte syste. x cos (.) y sin (.3) (.4) Now the chge nd cuent density cn be expessed. cos c c cos t, t q e t q sin sin t vt cos c cos t J, t v t ev ev q sin sin ct vt (.5) (.6) The field cn be obtined by using the Fouie tnsfo, ssue wve fcto is in the fo jk e j t., 3 jk jt A t A k e e dk (.7) 3 jk e t J, t J k e dk (.8) 3

4 Subitting Eq. (.7) nd (.8) into Eq. (.) nd (.), the vecto A cn be obtined by Fouie tnsfotion in the wve vecto spce. A, t 3 jk e t qe v eve dk (.9) k k k Bsed on Loent guge, we cn get the scl potentil. A j (.) Bsed on the Eq. (.4), we obtin the coponents of the electic field. Appendix : The theoy of cicul cylindicl onolye gphene stuctue Fig.. Schetic of the cicul cylindicl onolye gphene stuctue with dielectic loding ( ), the dius of the dielectic ediu is, the dius of the tjectoy pojection of cycloton electon be in X-Y plne is, it oves t velocity v ve ve bove the gphene lye with cycloton tjectoy.. Dispesion eqution As shown in Fig.., the schee cn be divided into two egions. Without chge souces, solving Eq. (.) nd (.) togethe with the boundy conditions, the fields in egion nd cn be obtined. The fcto n the egion ( ): e jkjj t is oitted. 4

5 E A k J ( k ) H A k J ( k ) c c c c E ja k kc J ) A J ) E A k J ) jakc J ) H A J ) ja kkc J ) H A j kcj ) Ak J ) (.) whee: k c k k, J ) [ J ) J )], J ) J ) n the egion ( ): whee: E A k K ( k ), H A k K ( k ) 3 c c 4 c c E jkkc A3 K ) A4 K) E k A3 K) jkca4 K ) H jkkca4 K ) A3 K) k H A4 K) j kca3 K ) kc k k, K ) [ K ) K )], K ) K) (.) Assuing tht onolye gphene is toiclly thin, it cn be egded s conductive sufce with conductivity. The boundy conditions e shown s below: E E, E E ( H H ) E, ( H H ) E g g g (.3) Afte substituting Eq. (.) nd (.) into the bove boundy conditions, the dispesion eqution of cicul cylindicl gphene stuctue with dielectic loding is obtined, 5

6 k J ( k ) k j K k k J k k J k K k Q c c [ ( c ) c ( c )] g c ( c ) ( c ) kck ) [ k J ( k ) k K ( k ) Q ] g c c c (.4) k [ K ( k c ) Q J ( k c )] [ k K ( k ) Q k J ( k ) j k J ( k )] whee: Q c c c c g c c k ( )( ) c jk J kc kc J ), kc Q kck kc kck kc ( ) ( ). Powe density.. Excittion of fundentl SPPs ode by linely oving electon be Becuse the fundentl SPPs ode (=) is tnsvese gnetic (TM) ode, it cn be excited by the TM evnescent fields poduced by linely oving electon be (the pticul cse of CEB without otting velocity). The electognetic fields poduced by the linely oving electon cn be expessed s below []. E i H Ae i ci J ( k ) H k () jk ci () Ji) H kci j A e k ci jk q v whee A ( / ), c ci J ( k ) H k J k H k () ci () ( ci ) ci Fo the fundentl ode, the TM fields in the stuctue cn be expessed s below. n the egion ( ):, (.5) E A k J ( k ) c c E ja k k J ( k ) c c H A j k J ( k ) c c (.6) whee: k c k k, J ) J ) n the egion ( ): E A k K ( k ) 3 c c E jk k A K ( k ) c 3 c H j k A K( k ) c 3 c (.7) 6

7 whee: kc k k, K ) K) The boundy conditions cn be witten s E i i ( E E ),( H H H ) E (.8) g By solving the bove boundy conditions, the fields coefficients cn be deteined... Excittion of hybid odes by CEB The electognetic fields in the stuctue nd poduced by CEB e obtined in the pevious pgphs. The boundy conditions cn be witten s, i i E ( E E ), E ( E E ), ( H H H ) E, ( H H H ) E i i g g (.9) Subitting the electognetic fields into the bove boundy conditions, the field coefficients cn be obtined. The excited SPPs e tnsfoed into dition in the dielectic ediu when the Cheenkov dition condition is stisfied. The dition powe density cn be clculted by the following eqution, Re[ * ] P E H dd (.) Appendix : The theoy of cicul cylindicl double-lye gphene stuctue Fig. 3. Schetic of cicul cylindicl double-lye gphene stuctue with dielectic loding, the dius of the dielectic ediu is, the dielectic fil is in the egion b, nd the dius of the tjectoy pojection of electon be is. 7

8 3. Dispesion eqution As shown in Fig. 3., the schee cn be divided into thee egions. Without chge souces, solving Eq. (.) nd (.) togethe with the boundy conditions, the fields in egion, nd cn be obtined. The fcto n the egion ( ): e jkjj t c c c c is oitted. E A k J ( k ) H A k J ( k ) E ja k kc J ) A J ) E A k J ) jakc J ) H A J ) ja kkc J ) H A j kcj ) Ak J ) (.) whee: k c k k, J ) [ J ) J )], J ) J ) n the egion ( b ): E A k ( k ) A k K ( k ), H A k ( k ) A k K ( k ) 3 c c 4 c c 5 c c 6 c c E jkkc[ A3 ) A4 K )] [ A5 ) A6 K)] E k [ A3 ) A4 K )] jkc [ A5 ) A6 K )] (.) H jkkc[ A5 ) A6 K )] [ A3 ) A4 K)] k H [ A5 ) A6 K)] j kc[ A3 ) A4 K )] whee: kc k k, ) [ ) )], ) ), K ) [ K ) K )], K ) K) n the egion ( ): b 8

9 E A k K ( k ), H A k K ( k ) 7 c3 c3 8 c3 c3 E jkkc3a7 K 3) A8 K3) E k A7 K3) jkc3a8 K 3) H jkkc3a8 K 3) A7 K3) k H A8 K3) j kc3a7 K 3) whee: kc3 k k, K 3) [ K 3) K 3)], K 3) K3) The boundy conditions cn be witten s, (.3) E E, E E, ( H H ) E, ( H H ) E g g E E, E E, b b b b ( H H ) E, ( H H ) E g g b b b b (.4) (.5) Subitting the electognetic fields into the bove boundy conditions, the dispesion eqution cn be obtined. 3. Powe density 3.. Excittion of fundentl ode by linely oving electon be Fo the fundentl ode (=), the electognetic fields in the stuctue cn be expessed s, n the egion ( ): E A k J ( k ) c c E ja k k J ( k ) c c H A j k J ( k ) c c (.6) whee: k c k k, J ) J ) n the egion ( b ): 9

10 whee: E A k ( k ) A k K ( k ) 3 c c 4 c c E jk k [ A ( k ) A K ( k )] c 3 c 4 c H j k [ A ( k ) A K( k )] c 3 c 4 c (.7) kc k k, ) ), K ( k ) ( ) c K k c n the egion ( ): b E A k K ( k ) 7 c3 c3 E jk k A K ( k ) c3 7 c3 H j k A K( k ) c3 7 c3 (.8) with k k k c3 The boundy conditions cn be witten s, E E, ( H H) E g E ( E E), ( H H H) E (.9) i i b g b b b By subitting the electognetic fields into the bove boundy conditions, the fields coefficients cn be obtined... Excittion of hybid odes by CEB The hybid odes cn be excited by the CEB, the boundy conditions cn be witten s, E E, E E ( H H ) E, ( H H ) E g g (.) i i E E E, E ( E E ) b b b b ( H H H ) E, ( H H H ) E (.) i i g g b b b b Subitting the electognetic fields into the bove boundy conditions, the fields coefficients cn be obtined. Appendix V: nfluence of elxtion tie of gphene on dition pefonce The pefonce of dition fo electon be excited SPPs is inly dependent on the qulity of gphene, especilly its elxtion tie. n this section, we

11 discuss the influence of on the dition pefonce. n the nuscipt, the vlue of. ps is used bsed on ecent high-qulity gphene []. While, it is known tht the ost CVD gphene is with low-qulity, nd its elxtion tie would futhe decese when plced on dielectic substte due to the extinsic sctteing [3], which leds to the sll elxtion tie of the ode of. ps [4]. Howeve, ecent dvnces in CVD fbiction of high-qulity gphene ke long elxtion tie vilble [5]. Fig. 4. shows the esults of nolied ttenution constnts of gphene SPPs nd Fouie spect of dition intensity s function of fequency fo elxtion tie. ps nd.ps, espectively. And othe petes e the se s those used in Fig. in the nuscipt. The ttenution constnt of elxtion tie.ps is uch lge thn tht of.ps, especilly in the long-wvelength egie whee the dition intensity is vey wek. Fig. 4. () Nolied ttenution constnts of gphene SPPs nd (b) Fouie spect of dition intensity s function of fequency fo elxtion tie.ps nd.ps. Howeve, the sitution is uch bette when the elxtion tie inceses to. ps, s shown in Fig. 4.. The tio of ttenution constnts fo elxtion tie. ps nd. ps is less thn, so the dition intensity is stonge fo.ps

12 Rdition intensity thn tht fo thitieth of tht fo.ps. But the dition intensity t the dition pek is only. ps. Fig shows Fouie spect of dition intensity fo diffeent elxtion ties. Highe elxtion tie leds to stonge nd shpe dition pek, which hs been nlyed by [6]. This is cused by stonge esonnt stength nd lowe esonnt dping. While the dition intensity fo sevel ties highe thn those fo obtined in wide elxtion tie nge.. ps is.4 ps. So high pefonce dition cn be Fig. 4. () Nolied ttenution constnts of gphene SPPs nd (b) Fouie spect of dition intensity s function of fequency fo elxtion tie.ps nd.ps ps. ps.8 ps.6 ps.4 ps Fequency (TH) Fig Fouie spect of dition intensity s function of fequency fo diffeent elxtion ties. Refeence:. Liu, S. et. el. Phys. Rev. Lett. 9, 539 ().. Den, C. et l. Boon nitide substtes fo high-qulity gphene electonics.

13 Ntue Nnotechnol. 5, 7-76 (). 3. Chen, J., Jng, C., Xio, S., shigi, M. & Fuhe, M. Nt. Nnotechnol. ntinsic nd extinsic pefonce liits of gphene devices on SiO. 3, 6 (8). 4. Tssin, P., Koschny, T. & Soukoulis, C. Gphene fo Tehet Applictions. Science. 34, 6 (3). 5. Ho,Y. et l. The ole of sufce oxygen in the gowth of lge single-cystl gphene on coppe. Science. 34, 7 (3). 6. Zhn, T. et l. Tunble tehet dition fo gphene induced by oving electons. Phys. Rev. B. 89, (4). 3