Thermionic Emission and Negative dI/dV in Photoactive Graphene Heterostructures

2.1 MB·English

by  J. F. Rodriguez-Nieva

Thermionic Emission and Negative dI/dV in Photoactive Graphene Heterostructures
J. F. Rodriguez-Nieva1, M. S. Dresselhaus1,2, L. S. Levitov1
1Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA and
2Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
(Dated: April 15, 2015)
Transport in photoactive graphene heterostructures, originating from the dynamics of photogen-
erated hot carriers, is governed by the processes of thermionic emission, electron-lattice thermal
imbalance and cooling. These processes give rise to interesting photoresponse effects, in particu-
5 lar negative differential resistance (NDR) arising in the hot-carrier regime. The NDR effect stems
1 from a strong dependence of electron-lattice cooling on the carrier density, which results in the
0 carrier temperature dropping precipitously upon increasing bias. The ON-OFF switching between
2 the NDR regime and the conventional cold emission regime, as well as the gate-controlled closed-
circuitcurrentthatispresentatzerobiasvoltage, canserveassignaturesofhot-carrierdominated
r
p transport.
A

4 Graphene, because of its unique characteristics, is of the atomically thin device renders the carrier density in
1
keeninterestforoptoelectronicsresearchinareassuchas graphene layers sensitive to the interlayer potential dif-

photodetection, solar cells and light-emitting devices [1– ference. These two effects combined together result in a
]
l 4]. Recentlyitwasemphasizedthatgraphenefeaturesan reduction of the electronic temperature and a suppres-
l
a unusual kind of photoresponse mediated by photogener- sion of thermionic current upon an increase of the bias
h
ated hot carriers. The hot-carrier regime originates from potential V . The NDR effect arises when this suppres-
- b
s thequenchingofelectron-latticecoolingwhenthesystem sion overwhelms the increase in the field-effect transport
e
is close to charge-neutrality [5, 6]. The hot carriers are under bias. Such an NDR mechanism manifests itself
m
exceptionally long-lived in graphene and can proliferate as an enhanced photo-current peaked at a bias potential
.
t across the entire system [7–9]. This behavior, which sets well below the onset of the conventional field-emission
a
graphene apart from other photoactive materials, leads regime,eV (cid:28)∆,where∆isthebarrierheight,asshown
m b
to a dramatic enhancement in photo-response. in Fig.1(c).
-
d The 2D character of electronic states, which are fully WenotethatthephotoactiveNDRarchitecturesarea
n exposed in materials such as graphene, can enable new class of their own, and are well suited for optoelectronic
o
device architectures. One system of high current in- applications. In particular, the fast response and in-situ
c
[ terest is stacked graphene-dielectric-graphene structures tunabilityofgraphenedevicesmakesthemidealasphoto-
[seeFig.1(a)]. Fabricatedwithatomicprecision,suchsys- active switches or light-detectors with high gain. The
2
tems can behave as field-effect transistors [10, 11], reso- NDR effect in photo-active devices, analyzed below, is
v
7 nant tunnel diodes [12, 13], photodetectors [14, 15]; they distinct from the one in traditional NDR devices, such
2 also provide a platform to explore the Coulomb Drag ef- as Gunn diodes [18] or resonant tunneling diodes [19,
9 fect [16] and the metal-insulator transition [17]. In the 20]whichrelyonnon-linearitiesundertheapplicationof
6 hot-carrierregime,proliferationofphotoexcitedelectron- large electric fields in the absence of light.
0
hole pairs can result in an enhanced thermionic emission Turningtothetechnicaldiscussion,wenotethat,while
.
1
ofhotcarriersoverthebarrier. Verticalcarrierextraction in general both electrons and holes can contribute to
0
in such structures is facilitated by short interlayer trans- thermionictransport, inpracticetransportisoftendom-
5
1 port lengths in the nanometer range, ultra-fast response inated by a single carrier type. In the case of hBN, the
v: times, and large active areas. Variability in properties of barrier heights are ∆el ∼ 3.5eV for electron transport
different 2D barrier materials (hBN, MoS , WSe , etc.) and∆ ∼1.3eVforholetransport[21]. Wecantherefore
i 2 2 h
X allows to tailor the photo-response characteristics to the treattheinterlayertransportinanhBN-basedsystemas
r different optoelectronic applications. dominated by a single carrier type (holes).
a
Here we predict that interlayer transport in graphene Below we focus on the behavior in wide-barrier struc-
heterostructuresoperatinginthehot-carrierregimeleads tures, where thermionic emission of thermally activated
to an unusual type of photoresponse, namely, a neg- carriersdominatesoverdirecttunneling. Thisisthecase
ative differential resistance (NDR). The mechanism of forhBNthicknessesexceeding4-5monolayers(d∼1nm)
this NDR response relies on the interplay of two effects. [11] at sufficiently high temperatures. Thermionic cur-
First, the phase space available for phonon scattering rents are described by a particularly simple model when
rapidly increases with doping, enhancing the electron- both graphene layers are at neutrality at Vb =0[22]:
lattice cooling and thereby altering the number of hot
carriers in the system. Second, the large capacitance of J(V ,T)=(g /eβ)e−β∆sinh(βeV /2), (1)
b 0 b
2
FIG. 2. The closed-circuit current J induced by optical
cc
pumping (P = 10µW/µm2) in the absence of voltage bias,
V = 0. In the transport regime dominated by hot carriers,
b
the value and polarity of J are sensitive to the carrier den-
cc
sities n , n in the graphene layers. Shown are experimental
1 2
schematics (a), and the dependence J vs. n and n (b).
cc 1 2
The four-fold pattern with multiple changes of the current
polarity arises due to the strong cooling power dependence
on carrier concentration (a non-linear color scale is used to
amplify the features of interest).
leads to NDR by the following mechanism. For electrons
in thermal equilibrium with the lattice, Eq.(1) predicts
a monotonic I − V dependence. A very different be-
FIG. 1. The Negative Differential Resistance (NDR) effect havior, which is key for NDR, arises under pumping.
in a photoactive heterostructure operated in the thermionic As pictured schematically in Fig.1(c) for three values
emission regime. Shown here are (a) device schematics, (b) T(1) > T(2) > T(3), in the hot-carrier regime the elec-
electronicbandstructurewiththequantitiesdiscussedinthe
tron temperature T becomes highly sensitive to V . The
text marked, and (c) the I −V dependence under optical b
temperature-bias coupling arises because of the large in-
pumping obtained from Eqs.(4)–(7). The bias voltage V
b
terlayer capacitance producing bias-dependent doping in
controls the electron cooling through electrostatic doping of
graphene layers. An enhancement in the cooling power upon the graphene layers. An increase in carrier density leads
increased V triggers carrier temperature dropping [marked toafasterelectron-latticecooling,whichreducesthermal
b
T(1,2,3) in (c)]; see also simulation results in Fig.4(a). The imbalance∆T =T−T ,withT thelatticetemperature.
0 0
resulting suppression of thermionic current leads to negative The dependence in Eq.(1) then predicts suppression of
dI/dV (inthegreyregion). Forlargerbiasvalues,transportis
thermionic emission. If strong enough, this suppression
dominated by field emission, yielding positive dI/dV outside
can lead to negative dI/dV. The NDR effect takes place
the grey region. Shown here are results for both graphene
in the grey region marked in Fig.1(c).
layers undoped at V = 0. Results for nonzero doping are
b
presented in Figs.2,3. The sensitivity of the electron-lattice cooling to car-
rierconcentrationprovidesasmokinggunfortheregime
dominatedbyhotcarriers,helpingdistinguishitfromthe
whereJ isthecurrentdensityperunitarea, β−1 =k T conventional resonant tunneling NDR mechanisms, such
B
as those discussed in Refs.[12, 13]. In that regard, we
and the g value is estimated in Eq.(8). This expression
0
mention that for the more general case of unequal car-
followsfromageneralmicroscopicmodelatnottoohigh
rier densities (n (cid:54)= n at V = 0), our analysis predicts
bias V , such that the effect of anti-symmetric doping 1 2 b
b
that the interlayer thermionic transport persists even at
induced by V (cid:54)= 0 is stronger than the corresponding
b
zero bias. As illustrated in Fig.2, this produces a closed-
change in the barrier skewness [see derivation and dis-
circuit current at V = 0 with a characteristic density
cussion in the paragraph before Eq.(11)]. For larger bias b
dependence: a four-fold pattern with multiple polarity
values, field corrections to the barrier potential become
changes. Such a pattern provides a characteristic signa-
important and must be accounted for; this is done in a
ture of hot-carrier dominated transport.
microscopic model developed below. We also note that
The strength of the hot-carrier effects, reflected in the
the electronic distribution is typically non-exponential
response of T to V [see Fig.4(a)], can be characterized
whenrelaxationisslow. Slowrelaxationwouldmakethe b
by the dimensionless quantity
distributiontailsmorepronounced,ultimatelyenhancing
the thermionic effects. ∆ dT
α=− . (2)
The steep dependence of J on T and Vb in Eq.(1) T d(eVb)
3
ThevalueαdependsmainlyonthepowerP pumpedinto A feature of our system which is key for NDR is the
theelectronicsystemandonthethicknessdofthebarrier large mutual capacitance of graphene layers, which cou-
via the capacitance effect. As we will see, α governs the plesV withthecarrierdensityandmakesthehot-carrier
b
N-shaped I−V dependence: the condition for NDR can properties of each layer tunable. This coupling acts as a
be stated as α>1/2. We will argue that values as large knob producing big changes in the hot-carrier photore-
as α∼25 can be reached under realistic conditions. sponse through modest changes of carrier concentration
Wenowproceedtointroduceourtransportmodel. For on the order δn ∼ 1012cm−2. Such bias-induced doping
thermionic transport over the barrier, as well as for tun- changes are routine in graphene/hBN systems [10].
neling through it, we adopt a quasi-elastic but momen- The bias-induced changes in carrier densities of
tum non-conserving approximation. Indeed, a number graphene layers, as well as the electric field E between
b
ofmomentumscatteringmechanismsattheinterfaceare the layers, can be described by a simple electrostatic
possible,suchasscatteringbydefects,intrinsicphonons, model. Weconsideradual-gateddevicewithfixedcharge
substratephonons,etc. Typicalenergyexchangeinthese densities n , n in the top and bottom gates, respec-
T B
processesissmallonthebarrierheightscale∆. Withthis tively. The neutrality condition relates the charge densi-
in mind, we use the (quasi-elastic) WKB model for the ties in the different regions of the device as
interlayer transition matrix element:
n +n +n +n =0, (5)
T 1 2 B
t((cid:15))=Γe−h¯1(cid:82)0x∗p(x)dx, p22(mx) =∆−eEbx−(cid:15), (3) wheren1,n2 arecarrierdensitiesonthegraphenelayers.
ThefieldE isrelatedtothesequantitiesthroughGauss’
for (cid:15)<∆, and t((cid:15))=Γ for (cid:15)>∆, where Γ is an energy- b
law:
independent prefactor which depends on the barrier ma-
terialproperties. InEq.(3),Eb istheelectricfieldwithin κEb =2πe(nT+n1−n2−nB), (6)
the barrier due to interlayer bias, see Eq.(7), m is the
electron effective mass in the dielectric, and x is the where κ is the dielectric constant of the barrier. A bias
∗
classical turning point for the skewed barrier potential, voltageV appliedbetweenthegraphenelayersresultsin
b
x = min[d,(∆−(cid:15))/eE ], see Fig. 1(b). Here, for the
∗ b
eV =µ −µ +eE d. (7)
sakeofsimplicity,weignoretheeffectofattractiontoim- b 1 2 b
age charges, described by a −1/|x|−1/|x−d| potential.
With doping-dependent µ , Eq.(7) accounts for the
1,2
For a large barrier width, this gives rise to the barrier
quantum capacitance effects. Here we will use the T =0
height Schottky dependence on the square root of E . (cid:112)
b expressionµ =sign(n )¯hv π|n |,whichprovidesagood
i i i
The effect is less dramatic for the not-so-large barrier
model over most of the relevant carrier density range.
widths analyzed below.
The three unknown variables n , n and E can now
1 2 b
Assuming an elastic but momentum non-conserving
be found by solving the three equations (5)-(7), once the
interlayer transport, the vertical current J(V ) can
b external variables n , n and V are fixed. Through-
T B b
be expressed through barrier transmission and carrier
out this work we focus on the symmetric case when
distribution[23]:
n = n = −n with no interlayer bias applied (which
T B 0
(cid:88)2π corresponds to both graphene layers at neutrality when
J(V )=e |t ((cid:15))|2D ((cid:15))D ((cid:15)˜)[f ((cid:15))−f ((cid:15)˜)], (4)
b ¯h ij 1 2 1 2 thegatesareuncharged). Inthiscase,Eqs.(5),(6)canbe
(cid:15),ij restatedasn =n +δn,n =n −δn,andκE =4πeδn.
1 0 2 0 b
where, due to the built-in field between layers, the en- Then, plugging these values into Eq.(7), the density im-
ergyforthequantitiesinlayer2isoffsetby(cid:15)˜=(cid:15)+eE d balanceδncanbeobtained. Thebuilt-infieldE matters
b b
[see Fig.1(b)]. Here J is the current per unit area, in two different ways: the electrostatic potential value
f1((cid:15)) = [eβ1((cid:15)−µ1) +1]−1, f2((cid:15)) = [eβ2((cid:15)−µ2) +1]−1 are eEbd enters the WKB model, Eq. (3), as well as in the
the Fermi distribution functions, the sum denotes inte- offset between D ((cid:15)) and D ((cid:15)) in Eq. (4).
1 2
gration over (cid:15) and summation over spins and valleys. The I −V dependence for the case n = 0 in shown
0
We use t ((cid:15)) = δ t((cid:15)) defined in Eq.(3), and the den- in Fig.1; for finite values n = n = −n it is shown
ij ij T B 0
sity of states per spin/valley D ((cid:15)) = |(cid:15)|/2π(¯hv)2 with in Fig.3. In our simulation, we first determine µ from
1,2 1,2
v ≈ 106m/s the carrier velocity. The temperatures es- Eqs.(5)-(7) as a function of V and n . Using the µ
b 0 1,2
tablished under pumping in unequally doped layers are values, the electronic temperatures T are determined
1,2
generally distinct, β (cid:54)= β , reflecting the cooling rates fromenergybalanceconsiderations,seeEq.(9)below. Fi-
1 2
density dependence. Also, importantly, the electrostatic nally, using the calculated values µ , T , and E , the
1,2 1,2 b
potential between layers, E d, is distinct from the bias currentisobtainedfromEq.(4). WeusehBNbarrierpa-
b
voltage V . This is so because the quantum capacitance rameters for numerical estimates, with a (hole) barrier
b
effects,prominentatsmallcarrierdensities,[24]makethe height ∆ ∼ 1.3eV [21], dielectric constant κ ∼ 5, thick-
quantitesµ andµ bias-dependent. Theseeffectswillbe ness d=6nm (∼ 20 monolayers). Room temperature is
1 2
investigated below. assumed, T =300K, unless stated otherwise.
0
4
The prefactor value Γ in Eq.(3) can be related to
the measured conductance. Physically, Γ accounts for
the processes in which tunneling couples to phonons or
defects. The rates for these processes, which typically
vary from interface to interface, can be estimated from
transport measurements. Linearizing Eq.(4) in eV for
b
k T (cid:28) ∆, and accounting for the thermionic contribu-
B
tion due to (cid:15) ≈ ∆, the zero-bias conductance per unit
area is
g e2
G= 0e−β∆, g =4πND (∆)D (∆)|Γ|2 , (8)
2 0 1 2 ¯h
where we take T = T . Here N = 4 is the spin/valley
1 2
degeneracy, and g is a prefactor in Eq. (1). The
0
activation T dependence is consistent with that mea-
sured for dark current [25]. Comparison with values FIG.3. MapofthecurrentJ vs. eVb/∆andzero-biasdoping
G∼10−7Ω−1µm−2 measuredatroomtemperature,and n0 (equal for both layers), for P =10µW/µm2. The current
peak positions track the V values where the emitter is at
∆ ∼ 0.4eV [25], yields values g ∼ 1Ω−1µm−2 and b
0 neutrality(whitedashedline). LocalminimaofJ vs. V are
Γ∼0.5eV˚A. b
marked with black dashed lines. The n = 0 slice, plotted
0
Nextwediscusshowhot-carriereffectsresultinacou- in Fig.1(c), is marked with a dotted line. The peak-to-valley
pling between the electronic temperature and the chem- ratio (PVR) vs. n is shown in the inset.
0
ical potential for each graphene layer. In the contin-
uous wave regime, the power P pumped into the elec-
tronic system is distributed among the electron and lat- so because acoustic phonon scattering is dominated by
ticedegreesoffreedom. Forsimplicity,wewilluseatwo- quasi-elastic scattering processes at the Fermi surface,
temperature model, describing electrons by a tempera- andalsobecauseofthestrongdependenceoftheelectron-
ture distinct from the lattice temperature, T >T , valid phonon coupling on the phonon energy. The resulting
0
when the carrier-carrier scattering rate is faster than the dependence on the chemical potential takes the form[5,
electron-lattice relaxation rate. Assuming spatially uni- 26]
form in-plane temperatures and chemical potentials, the
totalcoolingpowerP obeystheenergybalancecondition Pac ∝µ4(T −T0), Pdis ∝µ2(T3−T03)/kF(cid:96) (10)
P =Pac(µi,Ti)+Popt(µi,Ti)+Pdis(µi,Ti), (9) in the degenerate limit kBT (cid:28) µ. The factor 1/kF(cid:96),
where (cid:96) is the disorder mean free path, describes the
writtenseparatelyforeachlayeri=1,2. Hereweignored dependenceofP ondisorderstrength. Incontrast, the
dis
effects such as direct interlayer energy transfer as well contribution P is essentially µ-independent. Since the
opt
as the heat drained through the contacts. Equation (9) optical phonon energy in graphene is quite large, ¯hω ∼
0
accountsforthreecoolingpathwaysintrinsictographene, 0.2eV, the value P is quite small, behaving as P ∝
opt opt
mediated by acoustic and optical phonons (Pac, Popt) [5, exp(−¯hω0/T) for kBT,µ <∼ ¯hω0. The ratio Pac/Popt is
6], andthedisorder-assistedacousticphononmechanism therefore small near charge neutrality but can be order-
(P ) [26], see Supplement. The cooling rates in Eq.(9) oneforstronglydopedgraphenewithtypicaldopingn∼
dis
also depend on the lattice temperature T , however it 1013cm−2 [5]. The cooling power strong dependence on
0
suffices to treat T as a fixed parameter, since the heat µcantriggerthetemperaturedroppinguponanincrease
0
capacityofthelatticegreatlyexceedsthatoftheelectron in µ, i.e. dT/dµ<0, see Fig.4(a).
system. The transition between the hot-carrier dominated
The intralayer Joule heating is small and thus need regime, and the conventional field emission regime can
not be included in Eq.(9). Indeed, typical vertical cur- be controlled by the power pumped into the electronic
rentvaluesobtainedindevicesofactivearea∼1µm2and system. In our simulation we used the values for P typi-
underabiasV ∼1VdonotexceedafewnA[10,11,25]. cal of laboratory lasers below saturation [27] (a few mW
b
ThisyieldsJouleheatingsourceswhichareatleastthree per a µm-wide spot). We take P to represent the power
ordersofmagnitudesmallerthanthepowerspumpedop- absorbed in each graphene layer (2.3% of incident power
tically. The interlayer energy transfer as well as the heat [4]). The P values used are quoted in Figs.1,4 panels
drained through contacts can be ignored in the energy and in Figs.2,3 captions. As shown in Fig.1(c), a strictly
balance in Eq.(9) for similar reasons. monotonicI−V responseobtainedatP =0,transforms
The cooling power is a strong function of doping for into an N-shaped NDR dependence upon growing pump
the acoustic-phonon contributions P and P . This is power.
ac dis
5
The impact of doping n (taken to be equal for both To estimate α as a function of the model parameters,
0
layers at V =0) on the I −V dependence is illustrated it is convenient to factorize α by applying the chain rule
b
in Fig.3. The current peaks are shifted towards higher as α=α ·α , giving
T µ
V values upon n growing more negative [compare to
b 0
µdT ∆ dµ
Fig.1(c) which shows the n = 0 slice]. The peaks track
0 α (P)=− , α (d)= . (12)
the Vb values at which the emitter layer is bias-doped T T dµ µ eµdVb
to charge neutrality. This is to be expected since the
Here α depends only on the cooling pathways through
electron-lattice imbalance is maximal at neutrality. The T
Eq.(9), while α depends only on the barrier properties
peak-to-valley ratio (PVR) as high as ∼ 5 can be ob- µ
throughthequantumcapacitanceeffectofEq.(7). Below
tained (see Fig. 3 inset). For n > 0, in contrast, the
0 we use Eq.(12) to estimate α and show that the NDR
emitter layer is never at charge neutrality for any value
condition α>1/2 can be readily met.
of V , resulting in the N-shaped dependence fading out.
b Toestimateα weanalyzethedegenerateregimeµ(cid:29)
TheNDReffectissuppressedunderahighbiaspoten- T
k T, where the doping-dependent contributions P and
tial when field emission of carriers with energies below B ac
P dominate over the roughly doping-independent P
the barrier height overwhelms thermionic emission. As dis opt
[see Fig.4(a) inset]. In this regime, the cooling power
showninFig.1,thehighbiasregioneV /∆>1ischarac-
b behaves as P = γ µa(Tb −Tb), with a = 4, b = 1 for
terized by J monotonically growing with increasing V . i i 0
b acoustic phonon cooling, and a = 2, b = 3 for disorder-
This behavior arises because lowering the barrier facili-
assisted cooling (here γ are constants that depend on
tates tunneling and also because growing carrier density i
the cooling pathways, see Supplement). We assume, for
results in a faster cooling, thereby reducing the electron-
simplicity, that a single cooling pathway dominates over
lattice thermal imbalance [see Fig.1(c) inset].
other pathways. Then Eq.(9) yields
Next,weproceedtoderiveasimplecriterionforNDR.
We will focus on the fully-neutral case n = 0 (both
0 α =(a/b)[1−(T /T)b] (13)
T 0
graphene layers undoped at V = 0) pictured in Fig.1.
b
In this case, due to symmetry, we have T = T = T
1 2 This gives 0 < α < a/b with the low and high val-
T
and n = −n for any eV . Further, assuming a small
1 2 b ues corresponding to T ≈ T and T (cid:29) T , respectively.
0 0
bias and/or a not-too-wide barrier, we can approximate
Thecrossoverbetweenthesevaluesoccursatathreshold
the bias-induced chemical potentials as µ1,2 ≈ ±eVb/2. pump power P ∼ 0.5µW/µm2 that marks the onset of
∗
This simple relation is valid for eV (cid:28) κ(h¯v)2/e2d, cor-
b the hot-carrier regime under typical experimental condi-
responding to the last term in Eq.(7) much smaller than
tions. We define P as the value for P at typical carrier
∗
µ1 − µ2. Lastly, accounting for the dominant role of densitiesn∼1012cm−2,T =300K,andk (cid:96)=100such
0 F
thermionic emission, we model transmission as a step
that (T −T )/T = 0.1. This yields the above P value.
0 0 ∗
function, |t((cid:15))|2 ≈ Γ2θ((cid:15)−∆). We integrate in Eq.(4)
Maximum α values found from Eq.(13) are 0.6 and 4
T
over energies (cid:15) ≥ ∆ (cid:29) max[k T, V ], approximating
B b for the P and P pathways, respectively.
dis ac
D ((cid:15)) as a constant and the Fermi distribution tail as
1,2 Next,weestimateα asafunctionofthebarrierwidth.
µ
e−β((cid:15)−µ). This yields Eq.(1) to leading order in V /∆
b From Eq.(7), specializing to the case n =0, we find
0
and k T/∆ with the prefactor g given in Eq.(8). While
B 0
the validity of Eq.(1) is limited to d which are not too ∆/2µ 6.5
α = ∼ . (14)
smallandalsonottoolarge,wefindthatitpredictsNDR µ 1+(4e2/κ¯hv)k d 1+0.3d[nm]
F
in the parameter range close to that found from the full
microscopic model used to produce Figs.1–4. where k = µ/¯hv. Here we have used the hBN barrier
F
The criterion for NDR can be derived by taking the value∆∼1.3eV andµ∼0.1eV fortypicalbias-induced
derivativedJ/dV inEq.(1)andsettingittozero,giving doping. This gives limiting values α (d(cid:28)d )≈6.5 and
b µ ∗
α (d (cid:29) d ) = 0, with the crossover value d ∼ 20nm.
(cid:18) (cid:19) µ ∗ ∗
1 x 1
1+ β∆ tanhx− β∆ = 2α, x=βeVb/2, (11) Fromtheaboveweseethatthequantityα=αT·αµ can
reach values as high as α∼25.
where α is the quantity −(∆/eT)dT/dV introduced While the NDR criterion α > 1/2 is insensitive to
b
above,describingthecarriertemperaturedependencevs. the cooling mechanism (so long as it is a strong func-
V . The α value controls the NDR effect. Maximizing tion of carrier density, as discussed above), the form of
b
the left-hand side in x we find the value f(λ) = λ1/2 − the I − V dependence may reflect the cooling mecha-
(λ−1)tanh−1λ−1/2 parameterizedwithλ=1+(β∆)−1, nism specifics. This is illustrated in Fig.4(b) for the
√
which is attained at x∗ = sinh−1 β∆. It is straightfor- Pdis mechanism. In particular, we consider the bias Vpk
ward to check that f(λ)≤1 for all λ≥1. This gives the where the current peaks. From Eq.(11) we estimate
NDR condition α > 1/2. Below we use this condition, eVpk ≈ x∗kBT. When the Pdis mechanism dominates
derived for n0 = 0, as an approximation for the more (kF(cid:96) < 103) and T (cid:29) T0, a power-law relation is ob-
general case of n (cid:54)=0. tained: V ∝ (cid:2)k (cid:96)P/µ2(cid:3)1/3. Similar arguments lead
0 pk F
6
ties make the NDR effect potentially useful for design-
ing new types of optical switches and photodetectors.
Our estimates show that the NDR regime, facilitated by
graphene’suniqueopticalandthermalproperties,canbe
readily accessed in wide-barrier heterostructures.
ACKNOWLEDGMENTS
We thank N. Gabor and J. C. W. Song for useful
discussions. This work was supported as part of the
Center for Excitonics, an Energy Frontier Research Cen-
ter funded by the U.S. Department of Energy, Office of
Science, Basic Energy Sciences under Award No. de-
sc0001088(LL),andbytheNationalScienceFoundation
Grant NSF/DMR1004147 (JFRN and MSD).
[1] Bonaccorso, F.; Sun, Z.; Hasan, T.; Ferrari, A. C. Nat
Photon 2010, 4, 611
[2] Bao, Q.; Loh, K. P. ACS Nano 2012, 6, 3677–3694
[3] Xia, F.; Mueller, T.; Lin, Y.-m.; Valdes-Garcia, A.;
Avouris, P. Nat Nano 2009, 4, 839–843
[4] Nair, R. R.; Blake, P.; Grigorenko, A. N.;
Novoselov, K. S.; Booth, T. J.; Stauber, T.; Peres, N.
FIG.4. (a)ElectronictemperatureT vs. biasVb fordifferent M. R.; Geim, A. K. Science 2008, 320, 1308
pump power values P. The inset shows the dominant cool- [5] Bistritzer, R.; MacDonald, A. H. Phys. Rev. Lett. 2009,
ingpathwaysfordifferentchemicalpotentialandpumppower 102, 206410
values [for T0 = 0K, kF(cid:96) = 100]. (b) Current J vs. Vb for [6] Tse, W.-K.; Das Sarma, S. Phys. Rev. B 2009, 79,
different kF(cid:96) values, and P = 10µW/µm2. The bias poten- 235406
tialatwhichcurrentpeaks(Vpk)andthepeak-to-valleyratio [7] Xu,X.;Gabor,N.M.;Alden,J.S.;vanderZande,A.M.;
(PVR) are sensitive to the amounts of disorder (see inset). McEuen, P. L. Nano Lett. 2010, 10, 562-566
Results shown in this figure (except the inset of panel a) are [8] Lemme, M. C.; Koppens, F. H. L.; Falk, A. L.; Rud-
obtained for the system undoped at Vb =0, as in Fig.1. ner,M.S.;Park,H.;Levitov,L.S.;Marcus,C.M.,Nano
Lett. 2011, 11, 4134-4137
[9] Gabor, N. M.; Song, J. C. W.; Ma, Q.; Nair, N. L.;
toadisorder-controlledpeak-to-valleyratio(PVR).This Taychatanapat, T.; Watanabe, K.; Taniguchi, T.; Levi-
behavior is illustrated in Fig.4(b) inset. tov,L.S.;Jarillo-Herrero,P.Science 2011,334,648-652
[10] Britnell, L.; Gorbachev, R. V.; Jalil, R.; Belle, B. D.;
We note that the NDR features may be somewhat
Schedin, F.; Mishchenko, A.; Georgiou, T.; Katsnel-
smeared out by statistical fluctuations induced by disor-
son, M. I.; Eaves, L.; Morozov, S. V.; Peres, N.
derorinhomogeneities. However,wedonotexpectthese M. R.; Leist, J.; Geim, A. K.; Novoselov, K. S.; Pono-
effectstodestroyNDR.Indeed, opticalheatingoccursin marenko, L. A. Science 2012, 335, 947–950
µm-wideareasandatypicalcarrierdensityis1012cm−2. [11] Britnell, L.; Gorbachev, R. V.; Jalil, R.; Belle, B. D.;
At the same time, charge inhomogeneity lengthscales in Schedin, F.; Katsnelson, M. I.; Eaves, L.; Moro-
zov, S. V.; Mayorov, A. S.; Peres, N. M. R.; Cas-
graphene/hBNareafewtensofnm,whereastypicalden-
tro Neto, A. H.; Leist, J.; Geim, A. K.; Pono-
sity fluctuations are as low as ∼1011cm−2 [28, 29].
marenko, L. A.; Novoselov, K. S. Nano Letters 2012,
Summing up, vertically-stacked graphene heterostruc- 12, 1707–1710
tures afford a platform to realize and explore a range of [12] Britnell, L.; Gorbachev, R. V.; Geim, A. K.; Pono-
interesting optoelectronic phenomena due to photogen- marenko, L. A.; Mishchenko, A.; Greenaway, M. T.;
erated hot carriers. One such phenomenon is the light- Fromhold, T. M.; Novoselov, K. S.; Eaves, L. Nat Com-
mun 2013, 4,1794
induced NDR effect discussed above, manifesting itself
[13] Mishchenko, A.; Tu, J. S.; Cao, Y.; Gorbachev, R. V.;
through the I −V dependence, acquiring an N-shaped
Wallbank, J. R.; Greenaway, M. T.; Morozov, V. E.;
character under optical pumping. Vertical heterostruc-
Morozov, S. V.; Zhu, M. J.; Wong, S. L.; Withers, F.;
tures use the full graphene area as a photoactive region, Woods, C. R.; Kim, Y-J.; Watanabe, K.; Taniguchi, T.;
and possess a large degree of tunability. These proper- Vdovin, E. E.; Makarovsky, O.; Fromhold, T. M.;
7
Fal’ko, V. I.; Geim, A. K.; Eaves, L.; Novoselov, K. S.; SUPPLEMENT: ELECTRONIC COOLING
Nat Nano 2014, 9, 808 PATHWAYS
[14] Britnell, L.; Ribeiro, R. M.; Eckmann, A.; Jalil, R.;
Belle, B. D.; Mishchenko, A.; Kim, Y.-J.; Gor-
Herewesummarizethemainresultsonelectron-lattice
bachev, R. V.; Georgiou, T.; Morozov, S. V.; Grig-
cooling in graphene, following Refs. [5, 6, 26]. Elec-
orenko,A.N.;Geim,A.K.;Casiraghi,C.;Neto,A.H.C.;
Novoselov, K. S. Science 2013, 340, 1311–1314 tron cooling in graphene is usually assumed to be dom-
[15] Yu, W. J.; Liu, Y.; Zhou, H.; Yin, A.; Li, Z.; Huang, Y.; inated by three main mechanisms: acoustic and opti-
Duan, X. Nat Nano 2013, 8, 952 cal phonon emission [5, 6], and disorder-assisted acous-
[16] Gorbachev, R. V.; Geim, A. K.; Katsnelson, M. I.; tic phonon emission (“supercollisions”)[26]. We use the
Novoselov, K. S.; Tudorovskiy, T.; Grigorieva, I. V.; two-temperature model describing the electron and lat-
MacDonald, A. H.; Morozov, S. V.; Watanabe, K.;
ticesubsystemsbytwodifferenttemperatures,T andT .
Taniguchi, T.; Ponomarenko, L. A. Nat Phys 2012, 8, 0
This model is valid when the electronic system is ther-
896–901
[17] Ponomarenko, L. A.; Geim, A. K.; Zhukov, A. A.; malized quickly due to fast carrier scattering, whereas
Jalil, R.; Morozov, S. V.; Novoselov, K. S.; Grig- theelectron-latticecoolingoccursonalongertimescale.
orieva, I. V.; Hill, E. H.; Cheianov, V. V.; Fal’ko, V. I.; The contribution to cooling power due to acoustic
Watanabe,K.;Taniguchi,T.;Gorbachev,R.V.NatPhys phonons, obtained for pristine graphene, is given by [5]
2011, 7, 958
[18] Butcher,P.N.Reports on Progress in Physics 1967,30, (cid:90) ∞
97 Pac =γac(T −T0) dνν34[f(ν)+1−f(−ν)], (15)
[19] Tsu,R.;Esaki,L.AppliedPhysicsLetters 1973,22,562– 0
564
where f((cid:15)) = [eβ((cid:15)−µ) + 1]−1 and the quantity in the
[20] Sollner,T.C.L.G.;Goodhue,W.D.;Tannenwald,P.E.;
Parker,C.D.;Peck,D.D.AppliedPhysicsLetters 1983, prefactor γac = 8h¯πDρ(2h¯kvB)6 depends on the electron-phonon
43, 588–590 coupling strength [5]. Here D is the deformation poten-
[21] Kharche,N.;Nayak,S.K.NanoLetters 2011,11,5274–
tial, ρ is the mass density of the graphene monolayer.
5278
An explicit dependence on chemical potential µ can be
[22] Our expression for thermionic current density, Eq.(1),
obtained in the degenerate limit βµ (cid:29) 1. In this case,
featuresadependenceontheexternalbiasandapower-
law temperature dependence distinct from that known the integral in Eq.(15) yields
for the 3D case, where Jth ∝ T2e−∆/kBT. This reflects
thedifferenceinthetransportprocessesattheinterface. Pac =γacµ4(T −T0). (16)
Conventional thermionic emission (3D) is described as
the flux of free particles with energies (cid:15) > ∆ in the di- This contribution to cooling, due to its strong depen-
rection normal to the surface. In contrast, in our case dence on µ, becomes very small near the Dirac point.
transport depends on elastic scattering mechanisms at Disorder-assisted acoustic-phonon cooling originates
the graphene/hBN interface generating an out-of-plane
from electron-phonon scattering in the presence of dis-
current, as described in the text.
order, such that part of phonon momentum is absorbed
[23] Wolf,E.L.Principles of electron tunneling spectroscopy,
by disorder. Evaluated for a short-range disorder model,
Second Edition; Oxford University Press: New York,
2012. this mechanism yields cooling power [26]
[24] Luryi, S. Appl. Phys. Lett. 1988, 52, 501–503
[25] Georgiou, T.; Jalil, R.; Belle, B. D.; Britnell, L.; Gor- 2D2k3
P =γ µ2(T3−T3), γ = B , (17)
bachev,R.V.; Morozov,S.V.; Kim,Y.-J.; Gholinia,A.; dis dis 0 dis ρs2¯h(¯hv)4k l
F
Haigh, S. J.; Makarovsky, O.; Eaves, L.; Pono-
marenko, L. A.; Geim, A. K.; Novoselov, K. S.; where k l is the dimensionless disorder mean free path
F
Mishchenko, A. Nat Nano 2013, 8, 100–103
parameter, s is the speed of sound, and the degenerate
[26] Song, J. C. W.; Reizer, M. Y.; Levitov, L. S. Phys. Rev.
limit βµ (cid:29) 1 is assumed. The quadratic dependence of
Lett. 2012, 109, 106602
P on µ means that this contribution can win over P
[27] Sun, Z.; Hasan, T.; Torrisi, F.; Popa, D.; Privitera, G.; dis ac
Wang, F.; Bonaccorso, F.; Basko, D. M.; Ferrari, A. C. near the Dirac point.
ACS Nano 2010, 4, 803–810 Using values of D ≈ 20eV, s ≈ 2 · 104m/s, ρ ≈
[28] Zhang, Y.; Brar, V. W.; Girit, C.; Zettl, A.; Crom- 7.6·10−11kg/cm2,weestimateγ ∼0.5µW/µm2K(eV)4
ac
mie, M. F. Nat Phys 2009, 5, 722-726 and γ ∼5·10−4µW/(eV)2K3µm2.
dis
[29] Xue, J.; Sanchez-Yamagishi, J.; Bulmash, D.;
Lastly, the cooling power for the optical phonon path-
Jacquod, P.; Deshpande, A.; Watanabe, K.;
way equals [5]
Taniguchi, T.; Jarillo-Herrero, P.; LeRoy, B. J. Nat
Mater 2011, 10, 282-285
¯h(h¯ω )3
P = 0 [N (ω )−N (ω )]F(T,µ), (18)
opt 4πρa4(¯hv)2 el 0 ph 0
whereaflatopticalphonondispersionwith¯hω =0.2eV
0
is assumed, and a = 1.42˚A is the interatomic distance.
8
The quantities Nel(ω0) and Nph(ω0) represent the Bose large at temperatures kBT >∼ 0.2h¯ω0. In addition, even
distribution [eβh¯ω0 −1]−1 evaluated at the electron and at low temperatures, the effect of Popt can be important
latticetemperature,T andT ,respectively. Thequantity neartheDiracpoint,whereothercontributionsaresmall
0
F(T,µ) is a dimensionless integral since P ∝µ4, P ∝µ2.
ac dis
(cid:90) ∞ Using the above results we estimate the carrier densi-
F(T,µ)= dx|x(x−1)|[f(h¯ω (x−1))−f(¯hω x)], ties that mark the onset of the hot-carrier regime under
0 0
−∞ typical experimental conditions. Reaching these carrier
(19)
densities by adjusting by the gate potential and voltage
with f((cid:15)) defined above directly after Eq.(15). For weak
bias triggers the NDR regime (see main text). Below we
doping, µ (cid:28) ¯hω , and k T (cid:28) ¯hω , we can approximate
0 B 0 assume, for concreteness, that electron-lattice cooling is
f((cid:15)) by a step function θ(−(cid:15)). Integration in Eq.(19)
dominatedbythedisorder-assistedmechanism. Takinga
then yields F(T,µ)≈1/6, giving
typical pump power value P ∼ 10µW/µm2, equating it
¯h(h¯ω )3 toPdis,andusingvalueskFl∼100andT =2T0 ∼600K,
Popt =γopt[Nel(ω0)−Nph(ω0)], γopt = 24πρa4(0¯hv)2. Eq.(17) yields typical carrier densities on the order of
n∼1012cm−2. Similarvaluesareobtainedfortheacous-
(20)
tic phonon mechanism in the absence of disorder. The
At temperatures T, T (cid:28) ¯hω this gives a simple expo-
0 0
hot-carrier transport regime is therefore realized when
nential dependence:
the carrier density is n<∼1012cm−2, whereas field emis-
Popt =γopt(cid:2)e−βh¯ω0 −e−β0h¯ω0(cid:3). (21) sion dominates for n >∼ 1012cm−2. Such doping values
areroutinelyobtainedingraphenesystemsthroughelec-
The exponential temperature dependence in P makes trostatic doping, indicating that the densities required
opt
this contribution small at low temperatures. However, for NDR are easily reachable under realistic experimen-
thevalueP growsrapidlywithtempertaure,becoming tal conditions.
opt