Electric Spaser in the Extreme Quantum Limit

0.16 MB·English

by  Dabing Li

Electric Spaser in the Extreme Quantum Limit
Dabing Li
State Key Laboratory of Luminescence and Applications,
Changchun Institute of Optics, Fine Mechanics and Physics,
Chinese Academy of Sciences, Changchun 130033, China
Mark I. Stockman
Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA
3 (Dated: February 1, 2013)
1
0 We consider theoretically the spaser excited electrically via a nanowire with ballistic quantum
2 conductance. Weshowthatintheextremequantumregime, i.e.,forasingleconductance-quantum
nanowire, the spaser with the core made of common plasmonic metals, such as silver and gold, is
n
fundamentallypossible. Forballisticnanowireswithmultiple-quantaornon-quantizedconductance,
a
J the performance of the spaser is enhanced in comparison with the extreme quantum limit. The
electrically-pumpedspaser ispromisingas anoptical source, nanoamplifier,and digitallogic device
1
for optoelectronic information processing with speed ∼100 GHz to ∼100 THz.
3

PACSnumbers: 73.20.Mf42.50.Nn71.45.Gm73.23.Ad
]
l
l
a
h Active or gain nanoplasmonics was introduced [1] tum electron transport instead of classical, dissipative
- by spaser (surface plasmon amplification by stimulated one. Another important difference is that the contra-
s
e emission of radiation). The spaser is a nanoscale quan- dicting theoretical work [24, 25] on spasers (sometimes
m tum generator and ultrafast nanoamplifier of coherent also called plasmonic nanolasers)has ignoreddistinction
localized optical fields [1–5]. The spaser is a nanosystem between the threshold condition of spasing and the con-
.
t
a constituted by a plasmonic metal and a gain medium. dition of developed spasing with Nn 1 quanta (SPs)
∼
m The spaser is based on compensation of optical losses in per generating mode. While for macroscopic lasers with
- metalsbygainintheactivemedium(nanoshell)overlap- enormously large number of photons this distinction is
d ping with the surface plasmon (SP) eigenmodes of the not significant, for spasers, where N 1, it is very im-
n
n ∼
metal plasmonic nanosystem. There are many exper- portant as we show below in conjunction with Fig. 1.
o
imentally observed and investigated spasers where the One of the most important problems in nanoplas-
c
[ gain medium consisted of dye molecules [6–8], unstruc- monics, both fundamentally and from the application

tured semiconductor nanostructures and nanoparticles standpoint, is the development of electrically-pumped
2
v [9–18], or quantum-confined semiconductor heterostruc- 3d-confined nanospaser, which can serve as a versatile
6 tures: quantum dots (QDs) [19, 20], quantum wires nanoscopicsourceofopticalnear-fieldenergyorultrafast
6 (QWs), or quantum wells [21]. nanoscaleamplifier whichhas the same formformfactor
3 Classified by mode confinement, there are the spasers as the field-effect transistor(.30 nm linear dimensions)
0 with one-dimensional [9, 11, 13], two-dimensional [10], but is 103 times faster [3] with prospects of the large-
1. or three-dimensional (3d) confinement [6, 7, 18]. The scale in∼tegration in petahertz-bandwidth systems. Such
1
spasers can also be classified by the spasing-eigenmode a spaser has not yet been observed. This Letter demon-
2
type, which can be either localized surface plasmons strates that the electric spaser in the extreme quantum
1
: (SPs)[6,7,14,18],orsurfaceplasmonpolaritons(SPPs) regime,i.e.,pumpedundertheminimumrequiredvoltage
v
[22, 23] as in the rest of the cases. Among the observed via a quantum wire with a single quantized conduction
i
X spasers,mostarewithopticalpumping,includingallthe channel, is fundamentally possible.
r spasers with the strong 3d confinement [6, 7, 14, 18]. Ourgoalistoformulateaconditionofspaseroperation
a
Only few SPP spasers,whose confinement and lossesare in the extreme quantumlimit where it is only limited by
not strong, are with electric pumping [9, 11, 13]. lawsofquantummechanicsandexpressedthroughfunda-
There have been doubts expressed in the literature mentalconstants. Thecorrespondingmodeliscomprised
regarding viability of the nanospaser with the strong byasingle-modespaserwithelectricexcitationviaaQW
3d confinement [24], especially with electrical pumping (or a quantum point contact) with N open conduction
c
[25]. In a direct contrast, the possibility of the optically channels. The conductivity of such a QW is [28–31]
pumped strongly3d-confinedspaserhas beenboth theo-
e2
retically shown [1, 3, 26, 27] and experimentally demon- G=NcG0 , G0 = π~ , (1)
strated[6,7,14,18]. Herewetheoreticallyestablishthat
the electrically-pumped spaser is fundamentally possi- whereG0istheconductionquantum,andeiselementary
ble. A principal difference of our theory from the pre- charge; N = 1 in the extreme quantum limit. Such a
c
vious works [24, 25] is that we consider ballistic, quan- QW may be a ballistic semiconductor channel [30, 31], a
2
Nn metal point contact [28, 32, 33], carbon nanotube (Nc =
(a) 2) [34], or a graphene sheet [35, 36].
30 In the same spirit of the extreme quantum limit, we
Ag
assume perfect (hundred percent) quantum efficiency of
the electron to SP energy conversion. This requires that
20
the potential difference U along the QW accelerates an
electron to acquire just the energy of one plasmon, U =
10 ~ωn/e,whereωn isthe spasingfrequency. InviewofEq.
Au
(1), this results in the current along the QW,
Al
J =N eω /π . (2)
QW c n
1 2 3 4 5 6
ħωn (eV) Weassumedevelopedspasingwherethestimulatedemis-
sion into the spasing mode dominates over the sponta-
neousemission. Astheoryofthespaserhasdemonstrated
g (cm-1) (b)
5×104 th [3],duetotheverystrongfeedbackinthenanospaser,the
developedspasingoccursalreadyforN &1. Undersuch
Al n
4×104 conditions, the above-described excitation process leads
to accumulation of the mean number of SPs per spasing
Au
3×104 modeinthestationary(continuouswave,orCW)regime,
Ag
2×104 Nn =τnJQW/e , (3)
104 where τn = 1/(2γn) is the SP lifetime, γn = γn(ωn) is
the SP amplitude spontaneous decay rate [1, 5] at ω ,
n
1 2 3 4 5 6 ∂Reε (ω )
m n
ħω (eV) γn =Imεm(ωn) , (4)
n (cid:30) ∂ωn
J (µA) andεm isthepermittivityofthespaserplasmonicmetal.
QW (c)
200 Substituting Eq. (2) into Eq. (3) we obtain a funda-
mentalexpressionforthemeannumberoftheSPquanta
per the spasing mode for an spaser pumped electrically
150
Al via a single-quantum-of-conduction channel,
100 N =N Q(ω )/π , Q(ω )=ω /(2γ ) , (5)
n c n n n n
Au where Q(ω ) is the SP quality factor at the spasing fre-
50 n
quency in its standard definition [5]. Note that the ex-
Ag
pressionforN (5)dependsonlyonthespasingfrequency
n
and the permittivity of the plasmonic metal, which in
1 2 3 4 5 6
turn, is determined by the fundamental constants (e, ~,
ħω (eV)
n andelectronmassm)only. Itdoes notdepend explicitly
ongeometryofthespaserorpropertiesofthespasergain
medium and the QW. As we show below (see the follow-
FIG. 1: For three plasmonic metals, Ag, Au, and Al, depen-
ing paragraphs), the condition of the developed spasing,
dencies of working parameters of the CW-regime spaser on
spasing frequency ωn expressed as the corresponding SP en- Nn & 1, where Nn is given by Eq. (5), is fundamentally
ergy ~ω. (a) Mean number of SPs per spasing mode Nn in different from the spaser threshold condition.
theextremequantumlimit: computedfromEq.(5)forasin- In the extreme quantum regime of N = 1, the mean
c
gleconductionquantumchannel,Nc =1. (b)Thresholdgain number of SPs per spasing mode, N (5), is plotted in
n
gth required for spasing as computed from Eq. (6). (c) The Fig. 1 (a) for three plasmonic metals: silver (Ag), gold
nanowirecurrentrequiredfordevelopedspasingwithasingle
(Au), and aluminum (Al). The values of ε of Ag and
SP quantum per the spasing mode for the three plasmonic m
Au used are from Ref. 37 and of Al from Ref. 38.
metals as indicated (solid lines). The current of the single-
channelQWundertheminimumrequiredpotentialdifference Figure 1 (a) clearly demonstrates that the electrically
isshownbythedashedline. Thedevelopedspasingispossible pumped spaserin the extremequantumlimit fundamen-
for the spectral regions where the corresponding JQW curves tally can undergo the developed generation: the number
are below this line. oftheSPquantaperspasingmodeisN =15 30forAg
n
−
inthenearinfrared(ir)andvisible(vis)spectralregions,
3
in the near-ir and red-yellow vis spectrum, N = 6 7, the threshold of spasing becomes unstable. Above the
n
−
whileforAlN =2 3intheultravioletspectralregion. threshold,thisinstabilityisresolvedbyanon-equilibrium
n
−
Comparetheconditionofthedevelopedspasing,N phase transition with a spontaneousbreak down of sym-
n
≥
1, with the condition [3] for the spaser threshold, which metry: a coherent state of the spaser emerges with a fi-
is obtained for N = 0 and infinitely strong pumping nite mean SP population, N whose phase is established
n n
(fullpopulationinversion). Forself-containment,wegive completely randomly but is sustained afterwards. Note
below this threshold condition in the form of Ref. 5, thatthethresholdconditionsEqs.(6)-(7)imposerequire-
mentstothegainmediumanddocontainitsmicroscopic
g g , g = ωn Res(ωn) Imε (ω ) , (6) parameters d12, nc, and Γ12.
th th m n
≥ c√εd1−Res(ωn) Confusion of the conditions of the developed spasing
and the spasing threshold is not unusual in the litera-
whereg isthegainthataninfinitegainmediummadeup
ture–see,e.g.,Ref.25. Notethatthedifferencebetween
ofthespasergainmaterialshouldhave,cisspeedoflight,
the conditions of the developed spasing and the spas-
ε isthepermittivityofthedielectricmediumintowhich
d
the spaser is embedded, and s(ω) = [1 ε (ω)/ε ]−1 is ing threshold is numerically much more pronounced for
m d
− a nanoscopic spaser, which spases at N &1, than for a
Bergman’s spectral parameter [39]. The gain, g, of the n
macroscopic laser, which lases at N 1.
gain medium is given by the standard expression, n ≫
It is also important to investigate what currents are
4πωn√εd d12 2nc required for the spaser pumping, and whether real
g = | | , (7)
3c ~Γ12 nanowires can withstand them. Consider a current re-
quiredtosustainasingleSPperthespasingmode,which
whered12 isthetransitiondipolematrixelementsforthe can be found from Eq. (3) as J = 2eγ . This is il-
QW n
spasing transition in the gain medium, n is the density
c lustrated in Fig. 1 (c) for the three plasmonic metals
of this transitions (i.e., the density of the excitons or
under consideration. To compare, a good semiconduc-
recombiningelectron-holepairsinthegainmedium),and
tor nanowire, such as the channel in a high-performance
Γ12 is the spectral width of the spasing transition. metal-oxide-semiconductor field-effect transistor (MOS-
To illustrate quantitative differences between the de-
FET) [40], has a 20 30 nm-width and supports
velopedspasingandthespasingthreshold,weplotinFig. ∼ −
working (drive) current 20 30 µA. As comparison
1 (b) threshold gain g as a function of the spasing fre- ∼ −
th with Fig. 1 (c) shows, such a nanowire is sufficient to
quency expressed as energy ~ω of a SP quantum. Note
n pump both the Ag- or Au-based spasers in their respec-
thatarealisticgainthatonecanobtainfordyemolecules
tive plasmonic regions. However,the currentrequiredto
or direct band gap semiconductors is g 104 cm−1.
pump the Al-core nanospasers is large, J 200 µA.
∼ QW
Thus, it is obvious from Fig. 1 (b) that one can build ∼
Notethatevenifsuchcurrentsaresupplied,theAl-based
nanospasers with silver or gold as plasmonic metals, in
nanospasers may not generate because the existing gain
agreementwithexperimentaldemonstrations[6,18],but
mediamainnotprovidethegainrequiredforthespasing,
analuminum-basedspaserwouldrequireg 105cm−1,
th ∼ which is an independent condition – cf. Fig. 1 (b).
which is very high, thus making spasing on aluminum
Another quantity of interest for the extreme quan-
problematic. At the same time, the meanSP population
tumregimeisthe currentthatasingle-quantum-channel
numbers for Au and Al in the developed spasing regime
nanowire conducts at the minimum potential drop re-
illustratedinFig.1(a)areonthesameorder,N 1 5.
n
∼ − quired for the one-electron-to-one-SP conversion, U =
Hence, the condition of developed spasing, N & 1, is
n ~ω/e,which is givenby Eq. (2) for N =1. This is plot-
fundamentally different from the spasing threshold con- c
ted in Fig. 1 (c) by the dashed line. As one can see that
ditions, e.g., Eqs. (6)-(7). The reason is that the ex-
current is rather large: it is sufficient not only for Ag-
pression (3) for the mean number of quanta per a single
and Au- but also for Al-core spasers.
generating mode is only valid for the case of the devel-
opedspasing,i.e., wellabovethe spasingthresholdwhen Now we discuss the role of optical phonons, and
the stimulated emission dominates over the spontaneous whether they can dissipate so much energy that the bal-
one. Thisisevidentalreadyfromthefactthatthecoher- listic electron acceleration to the required energy of ~ωn
ent SP population, N , in Eq. (3) is proportional to the becomes impossible. In a semi-classical picture, excita-
n
pumping current. It depends only on spasing frequency tionofopticaloscillationswithfrequencyωop(emissionof
ω andSP qualityfactorQ(ω ),which, inturn,only de- anopticalphonon)occurswhenanelectronreachesopti-
n n
pends on metal permittivity εm. Equation (3) does not calphononenergy~ωop,whichrequiresaccelerationtime
dependwhatsoeveronthepropertiesofthegainmedium. t = (2m∗ω /~)1/2L/ω 10 fs, where m∗ is electron
a op n
In a sharp contrast, the threshold conditions (6)-(7) effective mass, the applied≈electric-field force is ~ω /L,
n
are those in the absence of the coherent SP population, L 100nmisthenanowire’slength,andweadaptvalues
i.e.,forN =0. Physically,theymeanthattheunphased for∼GaAs: m∗ =0.067m,and~ω =36meV. Obviously,
n op
system of the nanoplasmonic core and the gain shell at this acceleration time is too short, t t , comparing
a op
≪
4
toperiodofopticalphononst =2π/ω 100fs. Thus for Al as the spaser-coreplasmonic metal.
op op
∼
this classicalpicture is inapplicable: on the time scale of Above in this Letter, we have shown a fundamental
t 10fs the nanowirelattice appears“frozen”,andthe possibility of the electrically-pumped spaser, or rigor-
a
∼
optical phonons cannot manifest themselves. ously, that such a spaser is not fundamentally impos-
Instead, one has to invoke quantum mechanics where sible even for a single-channel QW. We have considered
an electron acquires energy by undergoing a quantum anidealizedspaserintheextremequantumlimitbutnot
transition with frequency ω rather than the gradual problemsassociatedwithitsexperimentalrealization,in-
n
classical acceleration. In Landauer’s electron transport cluding coupling of the QW to the gain medium needed
picture [29], such quantum transitions occur with a pe- toutilizethekineticenergyaccumulatedbytheelectrons
riodof2π/ω 1 3fsforω intheopticalrange. Thus upon their passage of the QW. Note that the ballistic
n n
∼ −
theemissionofopticalphononsinourcaseisanalogousto nanowireswith necessarycharacteristicsare wellstudied
that following an electronic transition in semiconductor and widely used, in particular, as channels of the high
crystals. Totakeallorasignificantpartoftheelectronic performance MOSFETs.
transition energy, ~ω 1 3 eV, it would require Because currents J 10 100 µA that we pre-
∼ n ∼ − QW ∼ −
emission of ω /ω 25 80 optical phonons, whose dict to be sufficient for the developed spasing are on
n op
∼ ∼ −
probability is obviously negligible. the same order as those in the common MOSFETs [40],
The electron-optical phonon coupling in direct which also have comparable nanoscopic sizes, we expect
bandgap semiconductors such as InGaAs is known to that there will be no severe problems with management
be particularly weak [41]: Fr¨olich coupling constant of the heat released, which is mostly due to the Ohmic
α = 0.05 0.06 1 in contrast to ionic solids such losses in the plasmonic metal. The heat dissipation in
− ≪
as, e.g., KCl where α=3.4. Consequently, the effects of the electrically- and optically pumped spasers should be
the electron-phononcoupling may only be significantfor onthesameorderofmagnitude. Notethattheoptically-
resonantintersubband-typetransitionswhosefrequencies pumped strongly 3d-confinednanospasersdo work with-
are in a 30 40 meV range [42] but not for optical- outadamageinthewiderangeofintensities[6,7,14,18].
−
frequency transitions: the emission of optical phonons With regard to potential experimental implementa-
accompanying a transition in the 1 3 eV range can- tions, one of the possibilities is provided by conven-
∼ −
not consume a significant part of the transition energy. tionaldiodegainmediausedinbothspasers(nanolasers)
Moreover, in our case there is not only poor temporal [9, 11, 13] and in microlasers [43]. In such a case, the
matching (ω ω ) and weak coupling (α 1) be- diode’s emitting region should be within the plasmonic
n op
≫ ≪
tween the electronic transitions and optical phonons but eigenmodespatialextension. Anotherpossibilityformid-
alsoaverypoorspatialoverlap: thecharacteristicoptical infraredspasersistouseaquantum-cascadegainmedium
phonon wavelength, which is on the order of the lattice [44] also located within the plasmonic eigenmode exten-
constant, a 0.4nm, is much less than the extension, sion. In both these cases, the diode or the quantum-
∼ ≈
L 100 nm, of the electron transition current. This cascade wells should be properly biased so that the en-
∼leads∼to another suppression factor of (a/L)2 1. ergy accumulated by an electron as a result of Lan-
∼ ≪
Based on the above arguments, we can safely conclude dauer’s transition is matched to the quantum energy of
thattheemissionofopticalphononsunderourconditions the electron-emitting region. Yet another possibility is
isunlikelytosignificantlydepleteenergy~ω acquiredby theproposeduseoftheSchottkycontactasthegainsys-
n
electrons in the quantum wire. tem [45] where the semiconductor electrically contacts
Astheconcludingdiscussion,wehavefoundconditions the plasmonic metal. In such a case, the kinetic energy
of the developedCW spasing for the electric pumping in of the hot electron is directly transformed into that of
the extreme quantum case, i.e., via a ballistic contact the plasmon [45] in a process reversed with respect to
QW with a single conduction quantum, N = 1, under generation of hot electrons by plasmon decay [46, 47].
c
theminimumrequiredpotentialdrop,U =~ω /e. Itisof BasedonthisLetter’sresults,weconcludethatachiev-
n
fundamental importance that the contact nanowire pos- ing an electrically-pumped nanospaser is possible funda-
sessesballisticconductancetoallowforaccumulatingthe mentally and plausible practically. Such a nanospaser
maximumelectronenergyandtopreventheatproduction will be especially important for future optoelectronic
in the nanowire and preserve its integrity under the re- processors. It will serve as an optical source for SPP-
quiredcurrentload. Notetheregimethatwehaveconsid- waveguide [22, 23] interconnects between transistors,
eredforasingleconductionquantum,N =1,isthemost which will eliminate delays and heat production related
c
stringent: QWs with N > 1 or non-quantum-confined tothecapacitivecharging/rechargingoftheelectricinter-
c
ballistic nanowires will always supply higher currents, connects and increase the processing rate to 100 GHz
∼
relaxing the spasing conditions and allowing for higher transistor-limited speed. In perspective, the nanospaser
SP numbers. As we have shown, the Ag- and Au-core canalsoreplacethetransistorasthelogicactiveelement,
electrically-pumped nanospasers have excellent spasing whichwillpotentiallybringaboutall-opticalinformation
characteristics for realistic QWs but there are problems processing to 10 100 THz spaser-limited speed.
∼ −
5
This work was supported by Grant No. DEFG02- Meyer, A. J. M¨akinen, K. Bussmann, L. Cheng, F. S.
01ER15213fromthe ChemicalSciences, Biosciences and Choa, and J. P. Long, Opt.Express 19, 8954 (2011).
Geosciences Division and by Grant No. DE-FG02- [22] D.K.GramotnevandS.I.Bozhevolnyi,Nat.Phot.4,83
(2010).
11ER46789fromthe MaterialsSciences andEngineering
[23] Z. Han and S. I. Bozhevolnyi, Rep. Prog. Phys. 76,
Division of the Office of the Basic Energy Sciences, Of-
016402 (2013).
fice of Science, U.S. Department of Energy, and by Chi- [24] J. B. Khurginand G.Sun,Appl.Phys.Lett.99, 211106
nese Academy of Sciences Visiting Professorship for Se- (2011).
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