Photon Upconversion with Hot Carriers in Plasmonic Systems

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by  Gururaj V. Naik

Photon Upconversion with Hot Carriers in Plasmonic Systems
Gururaj V. Naik, and Jennifer A. Dionne∗
Materials Science and Engineering, Stanford University,
496 Lomita Mall Stanford, California 94305 USA
We propose a novelschemeof photon upconversion based on harnessing theenergy of plasmonic
hot carriers. Low-energy photons excite hot electrons and hot holes in a plasmonic nanoparticle,
whicharetheninjectedintoanadjacentsemiconductorquantumwellwheretheyradiativelyrecom-
bine to emit a photon of higher energy. We theoretically study the proposed upconversion scheme
usingFermi-liquidtheoryanddeterminetheupconversionquantumefficiencytobeashighas25%in
5 5nmsilvernanocubes. Thisupconversionschemeislinearinitsoperation,doesnotrequirecoherent
1 illumination, offers spectral tunability,and is more efficient than conventional upconverters.
0
2
n Plasmons-thecollectiveoscillationsoffreeelectronsin for electrons and (EF − ~ω) < E < EF for holes are
a a metal or highly-doped semiconductor - enable tailored produced in the metal. If ~ω is greater than the largest
J light-matterinteractions[1]. Whenplasmonicnanostruc- Schottky barrier (Φ or Φ ), some hot electrons and hot
h e
7 tures absorb incident photons, energetic carriers known holes excited in the metal will have sufficient energy to
1
as hot carriers are created. Because hot carriers are ex- crossthecorrespondingSchottkybarriers. Someofthese

tremely short-lived [2] (with lifetimes on the order of hot carriers will be injected into SC-1 in a process sim-
]
s a few femtoseconds), extracting their energy into forms ilar to thermionic emission in a Schottky diode. The
c
otherthanheatischallenging. Nevertheless,manyrecent band offsets in the heterostructure trap the hot carri-
i
pt studies have shown that it is possible to extract hot car- ers in SC-2, extending the lifetime of otherwise rapidly
o rierstogenerateelectricityorcatalyzechemicalreactions decaying hot carriers and increasing the probability of
. [3,4]. Here,weproposeatechniquetoextracttheenergy radiative recombination in SC-2. This radiative recom-
s
c of plasmonic hot carriers in an optical form, enabling bination leads to photon emission of energy ~Ω ≈ E .
g2
i
s photon upconversion. This novel scheme of upconver- Clearly, it is possible to have 2ω ≥ Ω > ω, allowing for
y sion relies upon a metal/semiconductor heterostructure photon upconversion. Note that two incident photons
h
to trap plasmonic hot carriersand allows them to radia- arenecessarytocreateoneupconvertedphotoninaccor-
p
tively recombine and emit a higher-energy photon than dance with energy conservation. Also note that charge
[
that absorbed. conservationdictatesthatthesteady-stateinjectionrates

1
Photon upconversion is useful in many applications for both carriers are identical. Lastly, note that upcon-
v
such as photovoltaics, deep-tissue bioimaging, photody- verted photons are emitted as long as excitation illumi-
9
5 namic therapy, data storage, and security and surveil- nation persists as both electrons and holes are injected
1 lance applications [5–8]. In most of these applications, fromthemetaltothesemiconductor,precludinganycon-
4 either lanthanide-based solid-state upconverters or or- tinuous charge build-up.
0
ganic bimolecular upconverters are used. While organic Our upconversion scheme differs from previously re-
.
1 bimolecular upconverters can be as efficient as 16% [9], ported strategies for harnessing the energy of plasmonic
0 lanthanide upconverters are only about 2-5% efficient hotcarriersinthatitextractsboththeelectronandhole
5
[5, 10]. Moreover, the absorption and emission wave- photocurrents from the same interface. Accordingly, the
1
length ranges for these upconverters are fixed by the kinetic energy of hot carriers in a metal is converted to
:
v atomic or molecular energy levels and are challenging to potential energy in the semiconductor heterostructure.
i
X tune. Comparedtoexistingupconversiontechniques,the
proposedschemeusing hotcarriersinplasmonicsystems
r
a cluasnTtrhbaeetemhdootirnecaeFrffiirgice.iren1mt.eadnCidaotnoesffdiedruesprscaponemcvteerratsaliolt/nusnesmacbhicielomitnyed.uisctoilr- a)ħω MetalΦe SC-1 SC-2 SC-1EC b) SMemeitcaolndudctor
heterostructure with the electronic energy diagram as EF E E ħΩ x
shown in Fig. 1a. The metal forms a Schottky contact Φh g1 g2 z
with a wide-bandgapsemiconductor (SC-1). A narrower ħω EV yz-polaħriωzation
bandgap semiconductor (SC-2) is sandwiched between
SC-1 layers to form a quantum well. When photons of
FIG. 1. The proposed upconversion scheme: a) Energy band
energy ~ω illuminate the metal/SC-1 interface, hot car-
diagram of a metal/semiconductor (SC)-1/SC-2 heterostruc-
riers with energy E such that (E + ~ω) > E > E
F F ture. The carrier flow paths are indicated by the brown ar-
rows. b) The investigated geometry, consisting of a metallic
nanocubeadjacent toa semiconducting half-space. The inci-
dent light is z-polarized (normal to themetal/semiconductor
∗ [email protected] interface).
2
Since potential energy storage is accomplished by trap-
ping charge carriers, it is not necessary to simultane-
δρ δρ
ously inject an electron and a hole, eliminating the need nn nn
for temporally coherent illumination and rendering the ηinj = kn,Pz≥kb = kn,Pz≥kb (3)
δρ ∆N(E +E )
scheme linear. Further, unlike conventional upconvert- nn F b
En≥(PEF+Eb)
ers, this hot carrier scheme enables tuning the absorp-
tion and emission wavelengths across optical frequencies Notethatthismodelholdsbothforelectronsandholes,
by choosing the appropriate materials combinations. thoughthedistributionofholeswillbeforenergiesbelow
To determine the efficiency of hot carrier upconver- E . Therefore, determining the number of hot electrons
F
sion, we use the theoretical framework previously devel- alsogivesthenumberofhotholes. IftheSchottkybarri-
oped by Govorov et al. [11] and Manjavacas et al. [12]. ersforbothelectronandholesarethesame,theinjection
We first determine the carrier distribution upon illumi- rates of both of the carriers would also be equal. Oth-
nation of a metal nanoparticle, then the fraction of ex- erwise, calculations should be performed for the carrier
cited carriers that are injected into the semiconductor, with the higher Schottky barrier,since charge neutrality
and finally the internal quantum efficiency of upconver- at steady-state requires that both carriers be injected at
sion. Forsimplicity,weconsiderametalnanocubeplaced the same rate, and the injection rate will be limited by
adjacent to a semiconductor half-space as shown in Fig. the higher barrier.
1b. We assume that the nanocube is small compared The internal quantum efficiency of upconversion η
UC
to the wavelength of light (quasistatic approximation) isthenumberofupconvertedphotonsemitteddividedby
[13], and that the metal/semiconductor interface is per- the number of photons absorbed and may be calculated
fectly flat. The Schottky barrier heights for electrons as:
and holes are taken to be equal and given by E . Upon
b
illuminating this system with z-polarized light, the elec-
δρ
tric field strength inside the metal nanoparticle (E ) is nn
approximately constant. This electric field redistribzutes η = 1η kn,Pz≥kb =η .η ∆N(EF +Eb)
UC well well inj
2 δρ ∆N(E )
the metal free-electrons, resulting in a non-equilibrium nn F
En≥P(EF)
population distribution that can be computed using the
(4)
density matrix formulation [11, 12]. The change in pop-
This equation assumes that all carriers with sufficient
ulation δρ of state n arising from the interaction with
nn energy and the correct momentum (as computed in Eq.
the incident photons is treated as a perturbation and is
3) are injected into the semiconductor, and all carriers
given by [11]:
injectedintothesemiconductoraresubsequentlytrapped
inthe quantumwell,emitting photons with anefficiency
η . NotethatEq. 4hasaprefactorof1/2,accounting
1 well
δρnn =4e2Xn′ (fn0′ −fn0)|φnn′|2(cid:26)(~ω−En+En′)2+Γ2 faollrttwhoe-ipnhjeocttoinonuopfcobnovthereslieocntrpornosceasnsdesh,oinletsh.iAssscwhietmhewtihthe
1 theoretical maximum η is 50%.
UC
+ (1)
(~ω+En−En′)2+Γ2(cid:27) As a first design, we choose to investigate a silver
nanocubeadjacenttoasemiconductorslabwithelectron
Here, e is the electronic charge, f0 is the equilibrium andholeSchottkybarriersof2eV.Typicalsystemswith
n
Fermi-Dirac function evaluated at energy En, Γ is the suchbandoffsetsincludeAg/GaN,Ag/SiCandAg/TiO2
energy broadening of the n-th energy level, and the ma- [14]. TheAgcubehasapermittivityadoptedfromJohn-
trix element φnn′ =< n|φ(r)|n′ >, where φ(r) = zEz, son and Christy [15] with additional Drude damping γ
is the potential induced by the incident electromagnetic arising from the finite size effect [16]. The surrounding
field inside the metal cube. Using Eq. 1, it is possible to mediumisconsideredto haveaconstantrefractiveindex
estimate the number of carriers ∆N that possess energy of 1.5 (the effective index of vacuum/high-index semi-
E ≥(E +E ) through the summation: conductor half-spaces [17]). The electronic structure of
F b
silver is approximatedby a free-electrongas in the spec-
∆N(E +E )= δρ (2) tral range where no interband transitions occur (~ω <4
F b nn
En≥(XEF+Eb) eV) [15].
Figures 2a and 2b show the calculated distribution of
ThesecarriershavesufficientenergytocrosstheSchot- electrons in silver cubes upon illumination. Silver cubes
tky barrier, though they may not have the appropriate with edge lengths of 10 nm and 5 nm are considered un-
momentum for injection. For injection into the semicon- der illumination at three photon energies: 2.5, 2.9 and
ductor,hotcarriersmustpossessaminimummomentum 3.3 eV. As seen, illumination creates hot electrons with
(denoted by k ) in the z-direction. The injection effi- energies much larger than E . In Fig. 2a, the peak
b F
ciency η is defined as the fractionof energetic carriers near E corresponds to the collective oscillation of free
inj F
∆N possessing z-momentum k ≥k and is given by: electrons (i.e., surface plasmons). The distribution has
z b
3
ħω m ħω m a) 0.8
a) Ag 0 n b) Ag 5 n d=5 nm
1 d=10 nm
100 100 0.6
d) d) V)
e e e
δρ (normaliz1100--21 ħω22 ..=59 eeVV δρ (normaliz1100--21 ħω22 ..=59 eeVV η (E = 2 injb00..24
10-3 3.3 eV 10-3 3.3 eV

0 1 2 3 4 0 1 2 3 4 0
E-E (eV) E-E (eV) 2 2.5 3 3.5
c) F d) F Incident Photon Energy (eV)
70 30 k
E > E+2 eV)F34560000 CΔNabs E > E+2 eV)F11220505 CΔNabs b) kx qω kbarrierkz πk (/d)x1200 Fermi sphere barrier
ΔN (1200 ΔN ( 5 k 2.92 eV
0 0 δρ barrier0
2 2.5 3 3.5 2 2.5 3 3.5 1
Incident Photon Energy (eV) Incident Photon Energy (eV) 20
0.5
d)
FIG. 2. hot electron energy-spread in silver nanocubes with ̟/
edge-lengthsof10nm(a) and5nm(b). In(c)and(d),solid 0 k (x10
linesplot thepopulation of hotelectrons with energygreater −0.5 3.08 eV 3.40 eV
than 2 eV (∆N) in 10 nm (c) and 5 nm (d) silver cubes. 0
Dottedlinesrepresentthenormalizedabsorptioncross-section −1x 10-3 0 1k0 (π/d)20 0 1k0 (π/d)20
z z
of the silver nanocube.
FIG.3. a)Efficiencyofhotelectroninjectionovera2eVbar-
rier for silver nanocubes with edge lengths of 5 and 10 nm.
a plateau for slightly higher energies and a steep roll-off
The plot for the 5 nm cube includes three red circles which
beyondE−EF closeto the incidentphotonenergy. The correspond to the photon energies at which the hot electron
surfaceplasmonpeak isweakerfor 5nmparticles,asex- distribution snapshots in k-space are shown in panel (b). b)
pected, though the plateau is broader. This broadening Schematic representation of equilibrium Fermi sphere (blue
resultsfrommoreeffectivescatteringinsmallerparticles, circle) and its displacement upon illumination. Color maps
show the change in occupation probability δρ of the discrete
leading to more efficient hot carrier generation [11].
states. Thedashedquarter-circlesrepresenttheFermisphere
The dependence of hot carrier generation on incident
andthedashedstraightlinesrepresentthethresholdmomen-
photon wavelength is shown in Fig. 2c and 2d, which tum required for injection.
plot ∆N(E +2eV) as a function of photon energy for
F
10 and 5-nm silver nanocubes. The figures also plot the
normalized absorption of the metal cube (dashed line). tionwithlargerdisplacementsforhigherphotonenergies.
It is clear that ∆N follows the spectral dependence of Injected carriers are represented by the fraction of this
absorption, with localized peaks resulting from energy sphere exceeding k . As the incident photon en-
barrier
quantization. Sinceabsorptionpeaksattheplasmonres- ergy increases, state-filling begins first in the direction
onance,hotcarriergenerationalsopeaks atthe plasmon of polarization (z-axis) and then spreads into other di-
resonance. Further, since plasmon absorption becomes rections before filling the next energy level. Thus, the
spectrallybroaderforsmallercubes(owingtotheirlarger injection efficiency peaks whenever the hot electron dis-
Drude damping), ∆N also becomes spectrally broader. tributionbeginsfillingagivenenergylevel,sinceatthose
Fig. 3a plots the calculated injection efficiencies for energies, most hot electrons are distributed in the z di-
hot carriers from 5 and 10-nm Ag cubes into the semi- rection. Figure 3b also plots the calculated change in
conductor. As with hot carrier generation, the injection electron occupation probability δρ of each quantized en-
efficiency curves exhibit localized peaks arising from en- ergy level in k-space for a 5 nm silver cube illuminated
ergyquantization. Significantinjectiononlyoccurswhen with photons of energies 2.92, 3.08, and 3.4 eV. As the
the incident photon energy is greater than the Schottky photon energy increases from 2.92 to 3.08 eV, more hot
barrier (2 eV). As the incident photon energy increases, electrons with momentum greater than the barrier are
theenergyspreadofhotelectronsincreases,causingthem created, resulting in a rise of injection efficiency. Be-
to fill higher and higher energy levels. To better under- yond3.08eV,hotelectronsbeginspillingoverinto other
standthespectralfeaturesofinjection,Fig. 3billustrates directions along the edge of the Fermi sphere, resulting
the displacement of the Fermi sphere upon illumination. in a smaller fraction of hot electrons with the correct
The Fermi sphere displaces in the direction of polariza- momentum for injection. Accordingly, the injection ef-
4
a) 0.3 b) 0.3 2E >~Ω≥~ω. The emitter choice is not only dictated
b
d=5 nm
0.25 d=10 nm 0.25 by the desired upconverted photon energy, but also by
V) 0.2 UC 0.2 many other factors such as band alignment, quality of
e η
(E=2 Cb 0.01.51 max. 00.1.15 ttuhisoeinn.gmTeatlhateel-rrsnaeamntgiiveceoonpfdlwuascamtvoeorlneinincgttemhrsfaatcceaern,iaalasnlsd[o19eb]aesseeuxcothfeniandsteeTdgirbNay-,
U
η0.05 0.05 which can be designed to be transparent at the upcon-
0 0 vertedwavelength,therebyavoidingquenchingofupcon-
2 2.5 3 3.5 5 10 15
Incident Photon Energy (eV) Size (nm) verted photons. With many plasmonic materials avail-
ableandmanyefficientsemiconductorquantumemitters
FIG. 4. a) Calculated internal quantum efficiency of upcon- available,thisupconversionschemepromisesbroadband,
versionfor5nmand10nmsilvercubes. TheSchottkybarri- high-efficiency upconversion.
ers for electrons and holes are assumed to be 2 eV. b) Maxi-
Inconclusion,anovelstrategyforphotonupconversion
mumvalueofupconversioninternalquantumefficiencyinthe
spectral range 2–4 eV is plotted for silver cubes of different using the energy of hot carriers in plasmonic nanostruc-
sizes. The efficiency maxima occur in the range of 3–3.5 eV tures is presented. Trapping both hot electrons and hot
foralltheparticlesizes considered. Notethatthetheoretical holesinaquantumwellallowsphotonupconversionwhile
limit of highest upconversion quantum efficiency is 50%. maintaining linear operation and avoiding any need for
high-poweror coherentillumination. Theoreticalstudies
reveal that smaller metal nanoparticles generate and in-
ficiency peaks at 3.08 eV. Meanwhile, peaks at higher jecthot carriersmore efficiently,leading to upconversion
photon energies steadily increase in height as a greater efficiencies ashighas25%for5 nmsilvercubes. Further
portion of the Fermi sphere is cut by the kbarrier plane improvementsinthe efficiencyarepossiblebyemploying
when the Fermi sphere is further displaced. Note that materialsandgeometriesthatallowmoreefficientcarrier
smallercubes havelargerpeak injectionefficiencies since injection. Comparing this scheme to the state-of-the-art
they have greater energy quantization steps, increasing solid state upconverters, the proposed scheme is more
their separation in k-space. This trait allows increased efficient, tunable, and broadband.
filling of states along k before carriers spill over to the
z TheauthorsthankProf. MarkBrongersmaforfruitful
states in other directions. Importantly, a 5 nm silver
scientific discussions, all Dionne-group members, espe-
cube can achieve injection efficiency as high as 80%.
ciallyDianeWuforhelpinginpreparingthismanuscript.
Assuming all electrons and holes injected are trapped
Funding from a Department of Energy EERE Sunshot
in the quantum well, the ideal internal quantum effi-
grant under Grant number–DE-EE0005331 and from
ciency ofupconversionfor this system may be computed
Stanford’sGlobalClimateandEnergyProjectaregrate-
using Eq. 4. We assume the quantum well has unity
fully acknowledged.
quantum yield (η = 1) [18]. Figure 4a plots the cal-
well
culated upconversion quantum efficiency as a function
of incident photon energy (~ω) for 5 and 10 nm silver
cubes. Note that the upconversion efficiency peaks not
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