Beyond electronics, beyond optics: single circuit parallel computing with phonons

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by  Sophia Sklan

Beyondelectronics,beyondoptics: singlecircuitparallelcomputingwithphonons
Sophia Sklan1 and Jeffrey C. Grossman2
1DepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts,02139,UnitedStates.
2DepartmentofMaterialsScienceandEngineering,MassachusettsInstituteofTechnology,Cambridge,Massachusetts,02139,UnitedStates.
Phononic computing – the use of (typically thermal) vibrations for information processing – is a nascent
technology;itscapabilitiesarestillbeingdiscovered. Weanalyzeanalternativeformofphononiccomputing
inspiredbyoptical,ratherthanelectronic,computing. UsingtheacousticFaradayeffect,wedesignaphonon
gyratorandtherebyameansofperformingcomputationthroughthemanipulationofpolarizationintransverse
phonon currents. Moreover, we establish that our gyrators act as generalized transistors and can construct
digitallogicgates.Exploitingthewavenatureofphononsandthesimilarityofourlogicgates,wedemonstrate
parallelcomputationwithinasinglecircuit,aneffectpresentlyuniquetophonons.Finally,agenericmethodof
3 designingtheseparallelcircuitsisintroducedandusedtoanalyzethefeasibilityofmagneto-acousticmaterials
1 inrealizingthesecircuits.
0
2
PACSnumbers:85.70.Ec,63.20.kk,72.55.+s,43.25.-x

n
a
Electronic Computing
J Computers are unquestionably useful, thanks in no small
3 part to their silicon-based electronic architecture. After all, gate
1 thesizeandexpenseofvacuumtubecomputersrenderedper- source e drain
sonalcomputingimpossible. Today,therearemanyproposed

] alternativestosiliconcomputers.Quantumcomputing,forex-
r
e ample,promisestoperformcalculationsmoreefficientlythan
h any classical computer could [1]. Moreover, biological and
t
o fuzzylogiccomputersoutperformclassicalcomputersinspe- transistor
. cificapplicationsbutareunlikelytosupplanttraditionalcom-
t
a puters[2,3]. Thecreationofphonon(vibrational)equivalents φ1
m Phonon Computing
of electronic computer elements has recently been explored.
φ
- Phononic(oftencalledthermal)computing’stypicalapplica- 2
d tion is to use heat from electronic computers, where waste ~B
n
heatischeap,plentiful,andwasted[4,5].
o
c The development of further applications for phonon com-
L
1 [ ppuroteprasghaatisobne.enTsthyims iiesdabgygtrhaevdatieffidcublytyrienlicaonncterolulpinognphanoanloon- p olariz atio n g yrator
v giesbetweentemperatureandvoltage,thustakinginspiration
7 fromelectronics[4–8]. Suchfocushasyieldedrelianceupon
0 nanostructuredmaterials[9,10]and1D,nonlinearchains[4– ~k
8 6,11]toconstructcircuitelements. Whilethishasresultedin
2 difficulties with constructing phonon computers, electronic-
FIG. 1: (COLOR ONLINE) Comparing electronic and phononic
.
1 inspired computation has also impacted how information is computing.Electroniccomputingusesastreamofelectronsthrough
0 encoded. Prior efforts encoded information in temperature asolid-statetransistortocontrolthemagnitudeofthevoltageatthe
3 differences.Butwhenphononsaloneareinformationcarriers, drain. Conversely, our implementation of phonon computing has
1
one could take inspiration from optical computing, encoding transversephononsundergoanacousticFaradayrotation,controlling
:
v information in the polarization of the phonon current. Log- thepolarizationofthetransmittedsignal. Notetheusetwoincident
i ical 1 (0) would be vertically (horizontally) polarized trans- phonon currents with different polarizations and frequencies, each
X
rotatedbyadifferentanglewhiletraversingthegyrator.
verse phonons. Logic operations in this computer therefore
r
a correspondtomodifyingthepolarizationofphononcurrents.
Kittel first realized the possibility of controlling phonon po-
larization. He discovered the acoustic Faraday effect (AFE, terialsbeginningwiththefundamentals-sources,wires,and
or acoustic Faraday rotation, it rotates the angle of polariza- operators. Attention will be paid to gyrators, where we find
tion)forultrasonicwavesinferromagnetssubjecttoauniform that,inadditiontoisolators(theopticalanalogofdiodes),gy-
magneticfield[12]. Interestinthisapplicationseemstohave ratorscancreatetransistors. Logicgatesareconstructedfrom
ended there. Instead, the focus on magneto-acoustic (MA, thesetransistors. Weanalyzehowindividualgatescouldper-
also called magneto-elastic) effects (the influence of a mag- formanewformofparallelcomputing(whichwetermrain-
neticfieldonphonons)hasbeentodiscovernewmechanisms bowparallelization,RP)(Figure1).
forMAeffectsandhowtheymightaidincharacterizingma- It is helpful to consider where a signal comes from and
terials[13–15]orcontrollingmagnons[16]. how it reaches computing elements. Waste heat would be
Here we consider the construction of logic from MA ma- the ideal source of cheap, convenient phonons, but it lacks
2
coherence, intensity, and polarization. At the other extreme, control A) B)
thephononlaser[17]hasgoodbeamcharacteristicsbutisin X X
theearlystagesofdevelopment. ForMA,acoustictransduc-
ers are a promising compromise. Piezoelectrics (e.g. AC- Y
cut quartz) convert electronic signals into polarized phonons controller 0 π Z 0 π Z
2 2
currents,withpolarizationdependingonthecrystallographic polarization
plane at the interface [18]. At higher frequencies (GHz to
THz, instead of MHz to GHz), femtosecond lasers can be C) D)
transduced into picosecond phonon pulses via thin metallic
X
films(e.g. polycrystallineAlonZn)[19,20]. Oncephonons X
gyrator
havebeenproduced, theycanbesteeredviawires. Anyma- Y Z
terialwithgoodlatticethermalconductivitywoulddo,asthis φ 0 π Z Y π
implies that phonons propagate quickly and retain polariza- 2
tionoverlongdistances(longphononmeanfreepath).
FIG. 2: Implementing transistor logic with phonon gyrators. Ele-
While the previous elements are similar to their electrical
mentsaredefinedinsidebar. XandYareinputs,Zisoutput,and0
counterparts, operators are not. In digital electronics, oper-
isaninputfixedtological0. (a)NOTgate(b)NANDgate(c)OR
ators (e.g. transistors) control voltage levels (logic states)
gate(d)coherentXORgate(usingOR).
by blocking or admitting electrical current; whereas opera-
tors in polarization-based computing always admit currents
(states are modified by switching polarizations). Given our
the gyrator activated by 0). This design is good for chang-
basis, a change of state is accomplished by having a gyrator
ing OR/NOR and AND/NAND, but an amplifying design is
rotate the polarization by π/2. Gyrators can be constructed
straightforward(Figure2(a)). Logical0isfedintothesource
from any material possessing the AFE (e.g. MA) (Figure
ofagyratorandthesignaltobeinvertedisappliedatthegate.
1). In the AFE, a longitudinal magnetic field (B~ ~k,~k is the Logical0atthegateactivatesthegyrator,rotatingthesource
k
phononwave-vector)inducescircularbirefringence(twocir- signalbyπ/2,yieldinglogical1atthedrain. However,logi-
cularlypolarizedmodestravelatdifferentspeeds). Everylin- cal1atthegatedoesnottriggerthecontrol,sothegyratoris
early polarized mode is composed of a superposition of cir- notactivatedandthedrainisatlogical0. Nowconsidertwo
cularly polarized modes, which reach the end of the gyrator signals at the gate (Figure 2(b)). When the phonons are in a
atdifferenttimes. Thisphasedifferencebetweenthecircular definitepolarization,twosignalswillsuperimpose(nopolar-
modesyieldsalinearlypolarizedcurrentwithanewpolariza- ization change for identical inputs and π/4 polarization for
tion. Themagneticfieldstrengthdeterminesthemagnitudeof orthogonal signals). Control activates w±hen this total signal
thegyration[21]. TherearemanyMAmaterialsknown[13], hasahorizontallypolarizedcomponentgreaterthanacritical
although yttrium iron garnet (YIG) is a good candidate here value. Thus,ifeitherinputislogical0,thencontrolwillacti-
astheAFEhasbeenobservedatroomtemperature[21]. vateandtheoutputwillbelogical1. Onlywhenbothinputs
Currently, when gyrators are used, they are paired with a areverticalwilltheoutputbelogical0. Therefore,thiscircuit
fixed magnet (often 0.01-10T) to give a predefined gyration. isNAND.Controlcouldalternativelyselectlogical1(Figure
This design has some uses in computing (e.g. when creat- 2(c)). Now,theoutputwillonlybelogical0whenbothinputs
ing isolators), but gyrators have other uses. We propose that areaswell,ergothiscircuitisOR.WitheitherNAND/ORand
a phonon current could control the strength of the magnetic NOT,wecanuseDeMorgan’slawstocreateanylogicgate.
field, allowing the gyrators to operate in a completely new But for coherent phonons, XOR is simple (Figure 2(d)). Es-
way. Onecontrolmechanismistotransduceasecondcurrent, sentially,thereisnowdestructiveinterferenceintheORgate
rectifying and amplifying the resultant signal, and using this for signals of identical polarization, so the current can only
signaltoalloworblockafixedcurrentfromreachinganelec- exceedthecriticalamplitudewhenthesignalsareorthogonal.
tromagnet(FigureS1). Withthissortofcontrolledmagnetic And since a π/4 polarized signal activates control, this OR
±
field,gyratorsbecomethreeterminaldevicesthatactlikegen- gateisnowanXOR.
eralized transistors. Signals propagate from one side of the While these designs are quite simple, there are other real-
gyratortotheother(sourcetodrain),butthegatingcurrenton izationsoflogicgates. ExploitingDeMorgan’slaws,wecan
the magnet controls the outgoing polarization. Importantly, constructanyoperationusingonlyπ/2gyrationsandvertical
treating the gate as the input and the drain as the output, the control elements (Figure S2). Using gates in fixed arrange-
signal is amplified (as in electronic transistors). This is an mentandselectively(de)activatingthem,wecanperformar-
advantageoverisolator-baseddesigns(e.g.someformsofop- bitrary operations (as in an FPGA). From this, single circuit
ticalcomputing[22]),aslossesaccumulateandhinderdetec- parallelcomputing(RP)follows. Considermultiplesignalsof
tion. differentfrequenciesallenteringthesamegenericcircuit. In-
With these transistors, we can construct basic logic gates. dependentlycontrollingwhichgyratorsandcontrolelements
First, consider a NOT gate. Given our basis, the easiest im- workatwhichfrequency,wecanindependentlyselectthelog-
plementationisaπ/2gyrator(0ingives1out,1ingives 0 ical operation for each frequency. Such control is feasible
−
out. To remove the negative sign, which is an overall phase, with multistage gyrators (Figure 3) – several gyrators in se-
theoutputcanbefedintothesourceandgateofaπ gyrator, ries where the sum of the gyrations at a given frequency is
3
B~i B~i+1 real space (Figure 4 insets). The higher frequency mode has
almost no loss (< 10 6%), while the lower has a loss of
−
19.8%. Finally, consider the number of stages possible for
agivenfrequencyandmagneticfieldstrengthrange. Wecal-
culate (see supplementary materials) the maximum number
of channels currently accessible is about 2700. Such a large
numberofchannelsisprincipallyduetothelargeQfactorof
φ φ
i i+1
FIG.3: (COLORONLINE)Multistagegyrator: aconcatenationof
several gyrators. The lengths and magnetic field strengths at each
stagearetunedtocontrolthegyrationsatmultiplefrequencies.
equaltothedesiredgyration
Φ(ω )= L φ(ω ,B ) (1)
(i) (j) (i) (j)
X(j)
B is the magnetic field strength of the jth stage, L its FIG.4:(COLORONLINE)RainbowParallelization:designinga2-
(j) (j)
stagegyratortoworkattwofrequencies.Totalgyrationasafunction
length,ω thefrequencyoftheithinput,Φ(ω )itgyration,
(i) (i) of frequency plotted in blue. Absorbtion (Im[Φ]) in dashed green.
andφthegyrationperunitlength. Thenumberofstagesre-
Circled frequencies have Φ = 0 and Φ = π/2. ω is rescaled by
quired for a multistage gyrator scales as O(N) for ungated η (the MA coupling rate). (inset) Real space representation of the
elementsandO(Nα)(α [1,2]anddependsuponthedesign waveformsforthetwoselectedfrequenciesselected.Lisrescaledby
∈
used)forgatedelements. Thesescalingscanbeimprovedby c(p)/η(c(p)isthetransversephononspeed).Bottom/topfigureisfor
t t
combining stages (allowing fringing fields to come to bear, lower/higherfrequencymodewithblue(orange)denotinghorizontal
slightlycomplicatingthedesignanddiscussedinthesupple- (vertical)polarization.Verticallinesdenoteboundaries(2ndand4th
mentary materials). Control of different frequencies can be intervalsaregyrators).
accomplishedbyfilteringsignalsbeforeamplification.
Assuming uniform magnetic field, the polarization shifts
obey Eq.1, one can easily find L ( B , Φ , ω ) by in-
(i) { } { } { } YIG, which makes it possible to distinguish very small fre-
vertingφ(ω ,B )( denotesets). Theexceptiontothisis
(i) (j) {} quencyseparations.
whentheφ(ω ,B )isseparable(φ = f(ω)g(B)). While
(i) (j)
We have demonstrated a new mode of operation of an
this is not generically the case with phonons, it is for optics
(where φ = V(ω)B) [23], the exceptions principally due to acousticgyratorandusedittocreatebasiclogicgates. Byex-
ploitingthewavenatureofphononsandthetunablenonlinear-
absorption(oftenincoldgases[24]).RPis(toourknowledge)
ityofMA,wedemonstratethatMAmaterialscanperformRP,
presentlyuniquetophonons(voltageisscalar, somanipulat-
an effect that is presently unique to this computing architec-
ing polarization is impossible in classical digital circuits), at
ture. Thissuggeststhattheremightbecircumstanceswherea
leastatroomtemperature. Toshowthefeasibilityofthistype
ofdesignforphonons,wecalculateL ( B , Φ , ω )for phononcomputercouldoutperformanelectroniccomputer.
(i)
{ } { } { }
a2-frequency,2-stagegyratorwithnogyrationinducedatthe This material is based upon work supported by the Na-
lowerfrequencyandaπ/2gyrationinducedatthehigherfre- tionalScienceFoundationGraduateResearchFellowshipun-
quency. ThegyrationΦ(ω; B , L )forthemultistagegy- derGrantNo. 1122374. Supportingdataandfiguresrelevant
{ } { }
rator is plotted in Figure 4. Im[Φ]<0 denotes absorbance. tothisworkarepresentedonlineundersupplementarymate-
Waveforms at the two frequencies are found analytically in rial.
[1] L. Grover, in Proceedings of the twenty-eighth annual ACM [2] L.M.Adelman,Science266,1021–1024(1994).
symposium on Theory of Computing (ACM, New York, NY, [3] A.Robinson,Nature463,736(2010).
USA,1996)p.212-219. [4] N.Li, J.Ren, L.Wang, G.Zhang, P.Ha¨nggi, andB.Li, Rev.
4
Mod.Phys.84,1045-1066(2012). [15] A. Sytcheva, U. Lo¨w, S. Yasin, J. Wosnitza, S. Zherlitsyn, P.
[5] L.WangandB.Li,Phys.Rev.Lett.99,177208(2007). Thalmeier,T.Goto,P.Wyder,andB.Lu¨thi,Phys.Rev.B81,
[6] B. Li, L. Wang, and G. Casati, Phys. Rev. Lett. 93, 184301 214415(2010).
(2004). [16] M.Bombecketal.,Phys.Rev.B85,195324(2012).
[7] W.C.Lo,L.Wang,andB.Li,J.Phys.Soc.Japan77,054402 [17] K.Vahala,M.Herrmann,S.Knnz,V.Batteiger,G.Saathoff,T.
(2008). W.Ha¨nsch,andT.Udem,Nat.Phys.5,682-686(2009).
[8] L.WangandB.Li,Phys.Rev.Lett.101,267203(2008). [18] B.Lu¨thi,PhysicalAcousticsintheSolidState(Springer,New
[9] C.Chang,D.Okawa,A.Majumdar,andA.Zettl,Science314, York,NY,2007).
1121(2006). [19] D.Hurley, O.B.Wright, O.Matsudaa, V.E.Gusevc, andO.V.
[10] R.-G.Xie,C-T.Bui,B.Varghese,M.-G.Xia,Q.-X.Zhang,C.- Kolosov,Ultrasonics38,470(2000).
H.Sow,B.Li,andJ.T.L.Thong,Adv.Funct.Mater.21,1602 [20] O.Matsuda,O.B.Wright,D.H.Hurley,V.E.Gusev,andK.
(2011). Shimizu,Phys.Rev.Lett.93,095501(2004).
[11] M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett. 88, [21] H.MatthewsandR.C.LeCraw,Phys.Rev.Lett.8,397(1962).
094302(2002). [22] J.BallatoandE.Snitzer,Appl.Opt.34,6848-6854(1995).
[12] C.Kittel,Phys.Rev.110,836(1958). [23] A.K. Zvezdin and V.A. Kotov, Modern Magnetooptics and
[13] J.D. Gavenda, V.V. Gudkov, Magnetoacoustic Polarization MagnetoopticalMaterials,(CRCPress,NewYork,NY,1997).
PhenomenainSolids(Springer,NewYork,NY,2000). [24] D. Budker, W.Gawlik, D.F.Kimball, S.M.Rochester, V.V.
[14] M.Boiteux,P.Doussineau,B.Ferry,J.Joffrin,andA.Levelut, Yashchuk,andA.Weis,Rev.Mod.Phys.74,1153(2002).
Phys.Rev.B4,3077(1971).
Beyondelectronics,beyondoptics: singlecircuitparallelcomputingwithphonons
SupplementaryMaterial
Sophia Sklan1 and Jeffrey C. Grossman2
1DepartmentofPhysics,MassachusettsInstituteofTechnology,Cambridge,Massachusetts,02139,UnitedStates.
2DepartmentofMaterialsScienceandEngineering,MassachusettsInstituteofTechnology,Cambridge,Massachusetts,02139,UnitedStates.
I. CLASSESOFMAGNETO-ACOUSTICMATERIALS interaction for magnon-phonon coupling schemes (the con-
clusionsthatwedrawherestillworkqualitativelyiftheseef-
fectsarepresent, butthequantitativedescriptionwouldneed
There are several criteria for a material to be able to sup-
to be modified). Since the acoustic Faraday effect is most
3 portthemagnetoacousticinteractionsnecessaryfortheacous-
1 extensively studied at room temperature in magnon-phonon
ticFaradayeffect[13]. First,itmustbeacrystallinematerial
0 couplingschemes(inparticularinYIG),weshallYIGforour
with at least C (3-fold rotation) symmetry. Second, there
2 3 magnetoacousticelement.
mustbesomeintermediatefieldtocouplethephononmodes

n to the magnetic field. This is necessary because the Lorentz
a
forceactingonaphononisgenerallynegligible. Theinterme-
J II. CONTROLMECHANISMS
diatefieldcanbephotons,electroniccharge,ormagnons. For
3
photon-phononcoupling,therelevantinteractionsariseinlow
1 There are two basic classes of magnetic control. Control
temperaturemetals,whereamagneticfieldallowsthepropa-

can either be direct or indirect. In direct control, the mag-
] gation of helicon (for uncompensated metals) and doppleron
r neticfieldismodifiedbythephononcurrentitself,andinin-
or Alfven wave (for both compensated and uncompensated
e directcontrolthephononcurrenttransducessomeintermedi-
h metals)modes.Thesemodeshavebeenstudiedinpuremetals
atesignalwhichcancontrolthemagneticfield. Directcontrol
t (copper,nickel,etc),whereitisfoundthatdopplerons(which
o can be performed by a ferromagnet (see Fig. S1(a)). When
havephaseandgroupvelocitiesantiparalleltoeachother)are
at. onlysupportedforverysmallsetsofwavevectors. a phonon current is incident upon the magnet, a portion can
be absorbed or converted into localized modes that heat up
m In electron-phonon coupling, the relevant interaction is
the magnet. This increase in temperature will result in in-
- a coupling between the octupole moment of the 3d (as in creaseddisorderofthemagntizationwithinthecrystal,form-
d
CeAl2), 4f (as in TGG), or higher order charge distributions
ingdomainsandweakeningthetotalmagnetization.Sincethe
n
and the crystalline electric field (CEF). These interactions
o spontaneous magnetization is temperature dependent, the re-
arise because the nuclei in the crystal are charged, and so
c sultantmagneticfieldwillbemodifiedafteranegligibledelay
[ distortions of the lattice will induce distortions of the elec- (sincechangesinfieldpropagateatthespeedoflight). Ifthe

1 tronic structure and vice versa. For materials of high rota- temperatureachievedisneversogreatastodemagnetizethe
tionalsymmetry,thelowerordermodesofthechargedistribu-
v entiremagnet,therewillbealwaysbeanetmagnetizationin
tion (monopole, dipole, quadrupole) are less significant. Be-
7 a fixed direction, which means that upon cooling the down
0 causethesematerialshavemultiplephononbands(i.e. acous- thedirectionofthetotalmagnetizationwillremainfixed(thus
8 tic and optical branches), [15, 25] found that they have two keepingourgyratorsactinglikegyratorsandnotpolarizers).
2 typesofmagnetoacousticinteractions. Inadditiontothetyp-
Whilethisapproachisclearlythesimplesttoengineer, there
1. ical coupling scheme (magnetic field to electronic charge to are some limitations to direct control. First, if the thermal
0 crystallinefieldtophonons)thereisalsoanindirectcoupling, conductivityandspecificheatofthemagnetarenot(approxi-
3 where magnetoelastic effects in one branch can induce them
mately)linearwithintherangeoftemperaturesachieved,then
1 inotherbranchesduetophonon-phononcouplings.Addition-
it is exceedingly difficult to control the strength of the mag-
:
v ally,thereisasecondclassofelectron-phononinteractionsca- netic field. What’s more, the heating of the magnet depends
i pableofsupportingmagnetoacousticeffects. Thisisdoppler- upon the total thermal current, rather than any particular po-
X
shiftedcyclotronresonance(DSCR),whichoccursundersim-
larizationorfrequency. Whilepolarizerscaneliminatetrans-
r ilar circumstances to helicon-phonon and doppleron-phonon
a versepolarizations(andevenfrequenyintervals,whenpairsof
couplings(i.e. lowtemperaturepuremetals).
orthogonalpolarizersareused),thereisnoeasywaytoelim-
Lastly,inmagnon-phononcoupling,thedependenceofthe inate the longitudinal modes. While we can normally safely
exchange interactions in the spin Hamiltonian upon the lo- ignorethesemodes(asineveryotherelement,theyarecom-
cation of the nuclei implies a dependence of the magnon pletely decoupled), here controlled operation would require
bands upon the vibrational distortion of the lattice. This is ustoeitheraccountfororeliminateanylongitudinaleffects.
essentially an ionic magnetostriction effect. It is found in Lastly,becausethephononsarebeingconvertedintomagnetic
both ferromagnets (for example YIG [12]), antiferromagnets disorder, they are being destroyed in this process and cannot
or flopped antiferromagnets (as in Cr2O3 [14]), or paramag- beusedinothercircuitelements.
nets doped with magnetic impurities (single molecule mag- Indirect control gets around many of these disadvantages.
nets,asinKMgF3+Ni[26]). Itisanopenquestionifphonon- Inthecontrolschemegivenhere, therearethreestages(Fig.
phonon effects can induce a second type of magnetoacousic S1(b),stagesshownininsets).First,thephononcurrentissent
2
through a piezoelectric material (or some other linear, polar- A) B)
itodzhfiaesttthiofoaernctbetsshaesenoissfcirttoyihvfseettarctalrr,naynissnvtsaddelurusccceoienrrmrg)e.osaTdphfleosenu.dcptTiuteohazetotihnetelrgeapcneotslrelviaccertriirszsiezacatumitoitoonsnduvecweshchwttohiciralhstl M(T) δVxcosp(ωiet)∝zoe(cid:15)lxezctric ∝yyzδVcos(ωt)(cid:15) VδVcos(ωt)oltageMultiplyingRe.cKN.ti.Nfie|δrV|
wdpihielonlntpowrnoidlcluubcreereadnnitreoacnstcdeidlhlaaptvaienragmllveaolgltntoaigttuehdegerpaloidnliaeeranirztl.aytTidohenipsvevenocdltetoanrgtoeufgptrohane- electromagnet KNδV HHmaxmaxV/ZI
the phonon amplitude, that is ∂ V (cid:15) (where (cid:15) is the
i iz iz
∝
satlroaningzc)o.mPplaocnienngtwinirethseatitthhedfiareccetsioonfpoifezaomeloedcteripcrwopilalgraetsiunlgt pCiez)oelectric VoltageM D|)H|
tsiponmhroaaonlvflo,etntrhhyemescopmrdhyaeosltnl(aiolA.ensC.ucr∆uvfaroVrcletean)∝gt.elBenˆtaehivc(cid:15)aeaitzsu,dstewehpehtehenperdieeesznnˆoseoerilsgleeytlchyteerxiucntprooearscnmsteeaadnlgtiivisaveleslcoyn- KNδV Hout ultiplyingRectifier
without loss. At the second stage, this voltage is converted Vin electromagnet (cid:15)
intoafinite,constantvoltage(removingallnon-transienttime FIG S1: Magnetic control designs. (a) Direct control by
dependenceisnecessarytokeepthemagneticfieldfromfluc- phonons. (b)Linearindirectcontrol. (c)Digitalindirectcon-
tuating). Thisconversionisaccomplishedbyusingaconcate- trol. (d)Magneticfieldasafunctionofamplitude. Thedotted
nationofvoltagemultiplyingrectifiers. Thenumberofrecti- linedenotesdirectcontrol. Thedashedlinedenoteslinearin-
fiersusedwillincreasetheoutputvoltageatthisstage,soarbi- directcontrol. Thesolidlinedenotesdigitalindirectcontrol.
traryvoltagescanbeachievedinprinciple.Thisvoltagecould
beusedtodrivethecurrentinanelectromagnet(stagethree),
which would give a magnetic field that is approximately lin-
ear in the amplitude of a polarized phonon current (linearity
ismaintainedsolongasthetotalmagneticfieldremainswell III. SIMILARITYOFLOGICGATES
below the maximum possible). Strict linearity, however, can
beadrawbackinbuildingdigitalcircuits. Forexample,inthe
To determine the minimum number of gates required to
construction of an OR gate it is preferable to have signals of
construct all the two-terminal logic functions, we specialize
strength(cid:15)and2(cid:15)giveidenticalfields(correspondingtoinputs
to only using π/2 gyrations and veritcal control. By explicit
ofX=1,Y=0andX=1Y=1,respectively). Toachivethissort
construction(seeFig. S2), weseethatwerequirethreecon-
ofdigitalcontrol,thevoltageproducedattheendofstagetwo
trolelements(oneforeachinput,onefortheirsuperposition),
isusedtocontrolthegateofanelectronictransistor(seeFig.
and seven gyrators (three controlled, four fixed). Given that
S1(c)). Sincethecurrentproducedistheonethatflowsfrom
controlelementsarealwayspairedtogyrators,thisgivesato-
sourcetodrain,thetransistoroutputis(approximately)adig-
talofsevenelementstoperformallsixteenlogicoperations.
ital response to the gate voltage. The amplitude dependence
ofthesethreetypesofcontrolregimesissummarizedinFig. X X X X
Y Y Y Y
S1(d). Since large magnetic fields are typically required to 0 0 0 XY 0 X¯Y 0 X
achievestrongmagnetoacousticeffects(typicallyintherange X X X X
Y Y Y Y
of.5to10T[14,21],sometimesfieldsofupto20Thavebeen
0 XY¯ 0 Y 0 X Y 0 X+Y
used[15]), theelectromagnetwillhavetobequitepowerful. X X X ⊕ X
Bitterelectromagnets, forexample, havebeenconstructedto Y Y Y Y
0 X¯Y¯ 0 X Y 0 Y¯ 0 X+Y¯
produce continuous fields of at least 35 T (superconducting X X ⊕ X X
magnets can exceed this, but not at room temperature) [27], Y Y Y Y
0 X¯ 0 X¯+Y 0 X¯+Y¯ 0 1
farhigherthananythefieldsrequiredbyoursystem.
FIG S2: Alternative transistor logic using only vertical
control and 90 degree gyrations. Implementations of all 16
two-terminallogicfunctions.
IV. INCORPORATIONOFINHOMOGENEITYINTO
MULTI-STAGEGYRATORDESIGN
TheFaradayrotationinducedbyauniformmagneticfieldis
straightforwardtocalculate.Whenthemagnetoacousticinter-
3
actionisduetomagnon-phononresonance,ittakestheform V. DESIGNOFAMULTISTAGEGYRATOR
k k
+
Φ = dz − − Whiletheuseofinhomogeneousmagneticfieldsmakesfor
2
Z amoreelegantformofsinglecircuitparallelization(inprinci-
ω2 ple, only a single magnetoacoustic material is required, with
V dz
≈ ω2 Ω2 themagneticfieldprogrammablebyasetofcontrolelements),
Z −
= Lφ itdoesmakedesignmoredifficult. Foruniformfields,there-
lation
where Φ is the total rotation, V is a constant (the analog
of the Verdet constant), ω the phonon frequency, Ω the ef- Φ(ω ) = L φ(ω ,B )
i (j) (i) (j)
fective magnon frequency, z the runs over the thickness of (j)
the gyrator along the direction of propagation (L), and φ is P
is clearly satisfied, meaning that we can invert the φ matrix
the gyration per unit length [13]. In a strong, uniform mag-
netic field (B~ Bzˆ), ~Ω γ B , where γ is the gyro- to determine the necessary length. To demonstrate how this
≡ ≈ | | design works, we consider the problem of designing a two
magnetic ratio (g µ for spins). When the magnetic field is
s B frequency circuit which flips one mode and passes the other.
notuniform(whichcanoccurformultistagegyrators),calcu-
TocalculatetheexactFaradayrotationweusek2 = k2(1+
lating the Faraday modes is more complicated. We assume 0
aofgvraadryuianlginthheommoaggennoenitym(otdheast itsh,emonsleylviensc,luidginnogrinthgetehfefedcets- sωosu±nηωd−),ηω)swthheermeacg(tpn)okn0fr=eqωuen(cc(typ,)ainsdth~eηt=ran4sGve4±r4se2Ssp3e/eMd oisf
| |
viation of the coupled mode Hamiltonian from the Faraday magnetoelasticcouplingenergy(G isthecouplingconstant,
44
regime). Consider the inhomogeneous magnetic field B = S isspin,andM istheeffectivemass). TheFaradayrotation
1B (z)(~x+~y)+B(z)zˆ, where B (z) = dB(z) B(z) equation given previously is based upon the assumption that
−aenss2darthy0etotersmatipsrfoyportioBnal=to0th.eTdheirsivga0itvivees oafvthedceztofirel(cid:28)pdoitsennteica-l kap+p−roxki−m=atio(kn+2is−gekn−2er)a/l2lyk0us(ei.feu.lk,i+td+oeks−no≈tg2ikv0e)a.nWehasilyewthaiys
A= 1B(z)(yxˆ∇x·yˆ),therebymaintainingtheorbitalZee- tocalculatetheabsorptionoftheincidentwave.
−2 −
man term proportional to B(z)L . The spin Zeeman term, We work in units such that the equations simplify, i.e.
z
which is more important in this context, is proportional to Ω ωs η γB η + iΓ B + iΓ, and rescale
≡ − ≈ − ≡
B S. Theelectronicdegreesoffreedomwillarrangethem- all the variables (ω ω/η , B B/η, Γ Γ/η,
soeuls·vleysaptrofipxeerdlyzs,oit.he.at Bth(azt)Sωz(mc)an be ωev(amlu)(a0t)e(dz)u,n(aωm(bmi)g(u0)- kk2→=ωc(t2p()1k/+η, L1→ →η)LF/ocr(tcpl)a,riφty,→we→φk)eegpivthinegmthage→nroenlaltiifoen-
is the magnon frequency for thneqnth→branncqh/polarizationnqand ti±me(τ = 1/ΓB)+lioΓn±gω,settingΓ=0.0023. WeselectB =1.32
s 1
wavevector q, calculated to 0th order in transverse compo- andB =0.17, thusmakingitunlikelythatourvalueshappen
2
nent) which can be integrated over in determining the Fara- tosimplifytheproblem(wealsowantthemagneticfieldval-
dayrotation. Thetermsperpendiculartozˆ,ontheotherhand, uesreasonablywellseparated, assignificantlossesappearin
will perturb the magnon Hamiltonian, shifting the frequen- the range Re[ω] = [B,B +1]). For Φ = π/2, it is best to
2
cies. We treat these additional terms as a pertubation upon selectafrequencyfarfromresonance,sowesetω =3.05. To
1
ωn(mq)(z),keepingtermslessthanO(α3),aswedidinourpre- make Φ1 = 0, it is necessary for it to be close to resonance
vious derivation. The perturbation Hamiltonian, in Holstein- withamagnonmode(thisisimpliedbytheshapeoftheFara-
Primakoffcoordinatesis: day rotation curve). Thus, ω22 = 1.32 −Γ2. Solving for the
lengths then gives L = 0.403 and L =2.51. A plot of the
1 2
ΣqB0(z)√2S[x−( q)α(q)+x+( q)α†(q)] totalgyrationasafunctionoffrequencyquicklyconfirmsthat
− −
thisdeviceactsaspredicted. Totranslatethisplotintothereal
where
space wavefunction, we assume impedance matching at the
∂ ∂ boundaries(i.e. ignoringlossesduetoreflection). Thisallows
x (q) = i[ i ]
± ustoisolatethelossesduetothefinitelifetime,whichwefind
− ∂q ± ∂q
x y tobe3.81 10 7%atω and19.8%atω .
− 1 2
3 ×
= −i 8πF[rY1∓1(rˆ)]
r
VI. THEMAXIMUMNUMBEROFCHANNELS
( is the Fourier transform operator, r is distance, and Ym
F l CURRENTLYACCESSIBLE
are the spherical harmonics), which (to lowest order) can be
considered a constant so far as spin degrees of freedom are
Thenumberofpossiblechannelsforamulti-stagegyrator
concerned. The first order correction vanishes trivially, and
is difficult to determine. In principle, an infinite number of
thesecondorderis
stagescouldsupportaninfinitenumberofchannels,butsince
~22Sω[n(Bmq0)((0z))(]z2) |hx−−qi|2−|hx+−qi|2 , tihnecrceoamsepsl.exMitoyroeofvtheer,dtehveirceeairnecrleimasietastaiosntshetonuthmebferreqoufestnacgieess
(cid:2) (cid:3) andfieldstrengthsthatarefeasible.Assuch,onlyasmallsub-
whichisgenericallynonzero. setofthechannelsphysicallyaccessible.Sincethestrengthof
4
themagneticfieldcanbecontrolledexperimentallysolongas frequencies accessible. Experimentally, the majority of MA
theelectromagnetsusedarebelowtheirsaturationmagnetiza- experimentsconcentrateintherangeω/2π 10MHztoGHz
∈
tion,wearelimitedmorebytheresolutionofphononmodes. [13].Formagneticfieldsof 10T,however,thecyclotronfre-
≤
We estimate the spectral resolution of the phonon modes us- quency(i.e.themagnonfrequency)is152MHz.Wetherefore
ing the Q-factor. To make sure that the frequencies are dis- lacktheabilitytoeasilytunegyrationsabovethisfrequency.
tinguisable, we assume ω + 1∆ω = ω 1∆ω Q for YIG is typically quite large in this frequency, values
N 2 N N+1 − 2 N+1
where ω is the Nth frequency used and ∆ω is the band- between 104 and 107 are regularly found for frequencies be-
N
width. Essentially, this relation implies that frequencies are tween10MHzand1GHz(Q 1/ω)[28].However,whenthe
∝
separatedbytheiraveragebandwidth. Plugginginthedefini- intensity of the phonon current is too great (1mW is the rule
tionofQ =ω /∆ω ,wefindthat ofthumbinthe500MHz-1GHzrange),themagnoncoupling
N N N
isnolongerlinearandtheQfactordegrades(thephononslose
ω /ω =ΠN 2Qn+1 (2Qmin+1)N. energy to magnons, which quickly decay). This degradation
N 0 n=02Qn 1 ≤ 2Qmin 1 is typically of order unity, but we shall take a conservative
− −
estimate of Q as 103. This gives N =2700. This could be
Therefore
improved even further by countering the reduction of the Q
ω 2Q +1 ω factor, as well as extending the range of accessible frequen-
N min N
N ln( )/ln( ) Qminln cies. Conversely, given that this many channels would make
≥ ω 2Q 1 ≈ ω
0 min 0
− fabrication dramatically more complicated, the a portion of
(the error in this approximation is <1% for Q>10). Setting thechannelscouldbesacrificed(e.g. reducingthemaximum
ω = ω ,ω = ω givesthenumberofdistinguishable strengthmagneticfieldtoamorereadilyachievablevalue).
N max 0 min
[1] L. Grover, in Proceedings of the twenty-eighth annual ACM [15] A. Sytcheva, U. Lo¨w, S. Yasin, J. Wosnitza, S. Zherlitsyn, P.
symposium on Theory of Computing (ACM, New York, NY, Thalmeier,T.Goto,P.Wyder,andB.Lu¨thi,Phys.Rev.B81,
USA,1996)p.212-219. 214415(2010).
[2] L.M.Adelman,Science266,1021–1024(1994). [16] M.Bombecket.al.,Phys.Rev.B85,195324(2012).
[3] A.Robinson,Nature463,736(2010). [17] K.Vahala,M.Herrmann,S.Knnz,V.Batteiger,G.Saathoff,T.
[4] N.Li, J.Ren, L.Wang, G.Zhang, P.Ha¨nggi, andB.Li, Rev. W.Ha¨nsch,andT.Udem,Nat.Phys.5,682-686(2009).
Mod.Phys.84,1045-1066(2012). [18] B.Lu¨thi,PhysicalAcousticsintheSolidState(Springer,New
[5] L.WangandB.Li,Phys.Rev.Lett.99,177208(2007). York,NY,2007).
[6] B. Li, L. Wang, and G. Casati, Phys. Rev. Lett. 93, 184301 [19] D.Hurley, O.B.Wright, O.Matsudaa, V.E.Gusevc, andO.V.
(2004). Kolosov,Ultrasonics38,470(2000).
[7] W.C.Lo,L.Wang,andB.Li,J.Phys.Soc.Japan77,054402 [20] O.Matsuda,O.B.Wright,D.H.Hurley,V.E.Gusev,andK.
(2008). Shimizu,Phys.Rev.Lett.93,095501(2004).
[8] L.WangandB.Li,Phys.Rev.Lett.101,267203(2008). [21] H.MatthewsandR.C.LeCraw,Phys.Rev.Lett.8,397(1962).
[9] C.Chang,D.Okawa,A.Majumdar,andA.Zettl,Science314, [22] J.BallatoandE.Snitzer,Appl.Opt.34,6848-6854(1995).
1121(2006). [23] A.K. Zvezdin and V.A. Kotov, Modern Magnetooptics and
[10] R.-G.Xie,C-T.Bui,B.Varghese,M.-G.Xia,Q.-X.Zhang,C.- MagnetoopticalMaterials,(CRCPress,NewYork,NY,1997).
H.Sow,B.Li,andJ.T.L.Thong,Adv.Funct.Mater.21,1602 [24] D. Budker, W.Gawlik, D.F.Kimball, S.M.Rochester, V.V.
(2011). Yashchuk,andA.Weis,Rev.Mod.Phys.74,1153(2002).
[11] M. Terraneo, M. Peyrard, and G. Casati, Phys. Rev. Lett. 88, [25] P.Thalmeier,Phys.Rev.B80,214421(2009).
094302(2002). [26] J.W.Tucker,J.Phys.C:SolidStatePhys.13,1767(1980).
[12] C.Kittel,Phys.Rev.110,836(1958). [27] S.HannahandE.Palm,J.LowTemp.Phys.159,366(2010).
[13] J.D. Gavenda, V.V. Gudkov, Magnetoacoustic Polarization [28] R.C.LeCrawandR.L.Comstock,inPhysicalAcoustics:Prin-
PhenomenainSolids(Springer,NewYork,NY,2000). ciples and Methods, edited by W.P. Mason (Academic Press,
[14] M.Boiteux,P.Doussineau,B.Ferry,J.Joffrin,andA.Levelut, NewYork,1964),vol.3B,p.127.
Phys.Rev.B4,3077(1971).