Photovoltaic effect in a gated two-dimensional electron gas in magnetic field

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by  Maria Lifshits

Photovoltaic effect in a gated two-dimensional electron gas in magnetic field
Maria Lifshits1,2 and Michel I. Dyakonov1
1Laboratoire de Physique Th´eorique et Astroparticules, Universit´e Montpellier II, CNRS, France
2 A.F. Ioffe Physico-Technical Institute, 194021, St. Petersburg, Russia
Thephotovoltaiceffectinducedbyterahertzradiation inagated two-dimensionalelectrongas in
magnetic field is considered theoretically. It is assumed that the incoming radiation creates an ac
voltagebetweenthesourceandgateandthatthegatelengthislongcomparedtothedampinglength
ofplasmawaves. InthepresenceofpronouncedShubnikov-deHaasoscillations,animportantsource
9
0 of non-linearity is the oscillating dependence of the mobility on the ac gate voltage. This results
0 in a photoresponse oscillating as a function of magnetic field, which is enhanced in the vicinity of
2 thecyclotronresonance,inaccordancewithrecentexperiments. Anothersmooth componentofthe
photovoltage, unrelated to SdHoscillations, has a maximum at cyclotron resonance.
n
a
J
The two-dimensional gated electron gas in a Field Ef- the channel by the plane capacitor formula:
8
fect Transistor can be used for generation [1] and detec-
1 en=CU. (2)
tion [2] of THz radiation, and both effects were demon-

strated experimentally [3, 4, 5, 6, 7]. Concerning the Here, e is the elementary charge, and C is the gate-to-
]
l detection,the ideais thatthe nonlinearpropertiesofthe channel capacitance per unit area. Eq. (2) is applicable
l
a electron fluid will lead to the rectification of the ac cur- ifthescaleofthevariationofthepotentialinthechannel
h
rent induced in the transistor channel by the incoming is large compared to the gate-to-channel separation.
-
s radiation. As a result, a photoresponse in the form of Recently, the first experiments on the photovoltaic ef-
e a dc voltage between source and drain appears, which fectatterahertzfrequenciesinagatedhighmobilitytwo-
m
is proportional to the radiation intensity (photovoltaic dimensional electron gas in a magnetic field were per-
. effect). Obviously some asymmetry between the source formed [8, 9]. The main new results are: (i) the pho-
t
a and drain is needed to induce such a voltage. toinduced dc drain-to-source voltage exhibits strong os-
m
Theremaybevariousreasonsofsuchasymmetry. One cillations as a function of magnetic field, similar to the
- ofthemisthedifferenceinthesourceanddrainboundary Shubnikov-deHaas(SdH)resistanceoscillations,and(ii)
d
conditions. Another one is the asymmetryin feeding the the oscillation amplitude strongly increases in the vicin-
n
o incoming radiation, which can be achieved either by us- ity of the cyclotron resonance.
c ing a special antenna,or by an asymmetric designof the In this Letter we consider theoretically the photo-
[ source and drain contacts with respect to the gate con- voltaic effect in a gated electron gas in a magnetic field

tact. Thus the radiation may predominantly create an assuming, as in Ref. [2], that the incoming radiation
1
v ac voltage between the source and the gate. Finally, the creates an ac voltage between the source and the drain.
2 asymmetry can naturally arise if a dc current is passed Further, in accordance with the experimental conditions
1 between source and drain, creating a depletion of the we assume that 1) the source-drain length, L, (x direc-
7 electron density on the drain side of the channel. tion) is greater than the plasma wave damping length,
2 Thephotoresponsecanbeeitherresonant,correspond- so that the plasma waves excited near the source do not
.
1 ing to the excitation of the descrete plasma oscillation reach the drain, and 2) the sample width, W, in the y
0 modes in the channel, or non resonant, if the plasma direction, is much greater than L, see Fig. 1. The first
9 oscillations are overdamped [2]. Both non-resonant [5] assumption means that the boundary conditions at the
0
and resonant [6, 7] detection were demonstrated experi- drain are irrelevant and, as far as plasma waves are con-
:
v mentally. A practically important case is that of a long cerned, the sample can be considered to be infinite in
i gate, such that the plasma waves excited by the incom- the x direction. The second one implies a quasi-Corbino
X
ingradiationatthesourcecannotreachthe drainsideof geometry(allvariablesdependonthexcoordinateonly).
r
a thechannelbecausetheirdampinglengthissmallerthan We explain the observed strongly oscillating photore-
the source-draindistance. Within the hydrodynamicap- sponseasbeingduetothe non-linearityoriginatingfrom
proach the following result for the photoinduced voltage the oscillating dependence of the mobility on the Fermi
was derived for this case [2]: energy, and hence on the ac part of the gate voltage.
The photovoltaic effect is due to a radiation-induced
1U2 2ωτ force G driving the electron current. Without mag-
U = af(ω), f(ω)=1+ , (1)
4U0 1+(ωτ)2 netic field,Gisobviouslydirectedinthexdirectionand
is compensated by the appearance of an electric field.
p
where ω is the radiation frequency, τ is the momentum In the presence of magnetic field the problem becomes
relaxationtime,U istheamplitudeoftheacmodulation more subtle, not only because in this case G has a y-
a
of the gate-to-source voltage by the incoming radiation component,butalsobecausethisradiation-inducedforce
and U0 is the static value of the gate-to-channel voltage becomes non-potential: curlG = 0. The non-potential
6
swing, U, which is related to the electron density, n, in part will drive an electric current along closed loops.
2
Ua Gate L mass, and γ = 1/τ. The parameter γ is an oscillating
function of the electron concentration (or gate voltage)
U
0 andmagnetic field, whichresults inthe SdHoscillations.
Eq. (4) is the Euler equation, taking account of the
Lorentz force and damping due to collisions. It differs
Source Drain
from the conventional Drude equation only by the con-
vective term (v )v. Equation (5) is the continuity
· ∇
equation rewritten with the use of Eq. (2).
y
The boundary condition at the source (x=0) is:
x
DU U(0,t)=U0+Uacosωt, (6)
where ω is the frequency of the incoming radiation, and
FIG. 1: Assumed design and geometry. The THz radiation
U is the amplitude ofthe radiation-inducedmodulation
produces an ac voltage Ua between the source and the gate a
of the gate-to-source voltage. For a long sample, the
inducingadcsource-drainvoltage∆U. Thegatewidth,W,is
boundary condition at the drain is
muchlargerthanthegatelengthL(quasi-Corbinogeometry)
v 0, U U0 for x . (7)
→ → →∞
Thesignificanceofthecyclotronresonanceforthepho-
tovoltaic effect is related to the well-known dispersion We will search for the solution of Eqs. (4) and (5) as
relation for plasma waves in a magnetic field [10]. For an expansion in powers of Ua:
gated two-dimensional electrons it reads:
v=v1+v2, U =U0+U1+U2. (8)
ω = ω2+s2k2, (3)
p c Here v1 and U1 are the ac components proportional to
whereωc isthecyclotronfrequency,sistheplasmawave Ua, which can be found by linearizing Eqs. (4, 5), v2
velocity,andkisthewavevector. Thus,theplasmawaves and U2 are the dc components, proportional to Ua2 (we
can propagate only if ωc < ω. In the opposite case the are not interested in the second harmonic terms U2).
wavevectorbecomes imaginary,so that the plasma oscil- It is convenient to introduce u = eU/m, u = e∼U /am,
a a
lations rapidly decay away from the source. The change andthe plasmawavevelocityin the absenceof magnetic
of regime when the magnetic field is driven through its field s=u10/2 =(eU0/m)1/2 [1].
resonant value will manifest itself in the photoresponse.
To the first order in U , we obtain:
a
Following Refs. [1, 2] and other theoretical work, we
will use the hydrodynamical approach because, like the ∂v1x ∂u1
Drude equation, it provides a relatively simple descrip- + +ωcv1y +γv1x =0, (9)
∂t ∂x
tion, compared to the full kinetic theory. However it
shouldbeunderstoodthatatlowtemperatures,atwhich
the experiments [8, 9] were done, this approach strictly ∂v1y
speaking is not justified, because the collisions between ∂t −ωcv1x+γv1y =0, (10)
electrons are strongly suppressed by the Pauli princi-
ple. Nevertheless, the qualitative physical results de-
rived from the kinetic equation and from the hydrody- ∂u1 +s2∂v1x =0, (11)
namic equations are usually similar, e.g. the properties ∂t ∂x
of plasma waves are identical in both approaches, pro-
where ω = eB/mc is the cyclotron frequency. The
c
videdthatthe plasmawavevelocitysisgreaterthanthe
boundary conditions follow from Eqs. (6, 7): u1(0,t) =
Fermi velocity, so that the Landau damping can be ne- uacos(ωt) and u1( ,t)=0,v1( ,t)=0.
glected [11]. For this reason, we leave the much more ∞ ∞
Searchingforthesolutions exp(ikx iωt),weobtain
complicated approach based on the kinetic equation for ∼ −
the dispersion equation for the plasma waves:
future studies.
The electrons in a gated 2D channel can be described s2 β2
2
by the following equations: k =1+iα , (12)
ω2 − 1+iα
∂v e e
+(v )v = U + B v γv, (4) where α=(ωτ)−1 and β =ω /ω is the magnetic field in
∂t ·∇ −m∇ mc × − c
units of its resonant value for a given ω. To ensure the
boundary condition at x the root with a positive
∂U +div(Uv)=0, (5) imaginary part of k shou→ld∞be chosen. If damping is
∂t
neglected (α = 0), this equation reduces to Eq. (3).
where v is the electron drift velocity, B is the magnetic The explicite expressions for u1, v1x, and v1y are easily
field along the z direction, m is the electron effective obtained from Eqs. (9-11).
3
In the second order in U , we find 2
a 6
du2 ∂v1x ′ 1
+ωcv2y+γv2x+ v1x +γ u1v1x =0, (13)
dx h ∂x i h i
4 2 1
ωcv2x+γv2y+ v1x∂v1y +γ′ u1v1y =0, (14) 3 3
− h ∂x i h i
1
2
2
dj 1
x =0, jx =v2x+ u1v1x , (15) 0 1 2 0 1 2
dx u0h i
FIG. 2: The functions f(β) (left) and g(β)(right) describing
where the angular brackets denote the time averaging respectively the smooth part and the envelope for the oscil-
over the period 2π/ω. Here we have expanded the func- lating part of the photovoltage. The values of theparameter
tion γ(u) to the first order in u1. The quantities γ α=(ωτ)−1: 1 - 0.2, 2 - 0.4, 3 - 0.8
′
and γ = dγ/du should be taken at u = u0. The
boundary conditions for Eqs. (13-15) are: u2(0) = 0,
v2x( )=v2y( )=0. The integration interval should be expanded to include
∞ ∞
From Eq. (15) we derive the obvious fact that j = the region where the current lines return backwards.
x
0 (j differs from the x component of the true current So far, we have no rigorousproof that this idea is cor-
x
densityonly byafactoren). Usingthis, andintroducing rect, however we have checked that both methods give
the y component of the current, j , by a relation similar similarqualitativeresults(butdiffer inthe exactformof
y
to Eq. (15), we can rewrite Eqs. (13, 14) as follows: the magnetic field dependence of the photovoltage).
As described above, we obtain ∆u=u2( ):
du2 ∞
ωcjy =Gx(x) , γjy =Gy(x), (16) ∞
− dx
∆u= G (x)dx. (19)
x
where the additional driving force G induced by the in- Z0
coming radiation is given by:
UsingEqs. (17,19)wefinallycalculatethedcphotovolt-
age∆U =m∆u/e,betweendrainandsourceinducedby
Gx = γ γ′ u1v1x +ωc u1v1y v1x∂v1x , (17) the incoming radiation:
(cid:18)u0 − (cid:19)h i u0h i−h ∂x i
1U2 dγ n
∆U = a f(β) g(β) . (20)
4 U0 (cid:20) − dnγ (cid:21)
γ ′ ωc ∂v1y
Gy = γ u1v1y u1v1x v1x . (18)
(cid:18)u0 − (cid:19)h i−u0h i−h ∂x i Here we have separated the photoreponse in a smooth
part and in an oscillating part. The second one, propor-
′′ ′′
Both G and G depend on x as exp( 2k x), where k
x y tionalto dγ/dn,is anoscillatingfunction ofgate voltage
−
is the imaginary part of the wavevector defined by Eq. or magnetic field dρ /dn, where ρ is the longitudi-
xx xx
(12), reflecting the decay of the plasma wave intensity nal resistivity of t∼he gated electron gas.
away from the source. Thus curlG=0.
Note,thateveniftheamplitudeoftheSdHoscillations
6
One could solve Eqs. (16) to obtain the photoinduced issmall,theparameter dρ /dn(n/ρ )canbelarge,so
∞ xx xx
voltage ∆u = 0 [Gx−(ωc/γ)Gy]dx and this would be that the oscillating cont|ribution|may dominate.
the correct resRult for the true Corbino geometry, where The frequency and magnetic field dependences of the
the currentj canfreely circulatearoundthe ring. How-
y photovoltage are described by the functions f(β) and
ever, we believe that this is not correct for a finite strip,
g(β), which are given by the following formulas [13]:
even if W >> L, because in this case the current j
y
induced by the non-potential part of the driving force, 1+F
f(β)=1+ , (21)
Gy(x), obviously must return back somewhere, forming √α2+F2
closed loops [12]. How exactly this will happen, is not
quite clear. In our model, the current loops are likely
to close through the source contact, however in reality 1+F F
g(β)= 1+ , (22)
the oppositely directed y-current will probably flow in 2 (cid:18) √α2+F2(cid:19)
the ungatedpartofthe channeladjacentto this contact.
Anyway, since the current jy integrated over x must be where F depends only on the ratio β = ωc/ω and the
zero (except near the extremities), we believe that the dimensionless parameter α=(ωτ)−1:
correctwayistointegratethefirstofequations(16)tak-
ingthisintoaccount,andtoignorethesecondone,which 1+α2 β2
F = − . (23)
is not applicable beyond the gated part of the channel. 1+α2+β2
4
the photovoltage to the region β 1.
∼
Todisplaytheoscillatingcontribution,wetakethepa-
rameter γ in the conventional form [14], which is valid
when the SdH oscillations are small:
0
.u χ π 2πE
.a ,e N= 6 γ =γ0(cid:20)1−4sinhχexp(cid:18)−ωcτq(cid:19)cos(cid:18) ~ωcF(cid:19)(cid:21), (24)
g
a
tlov where χ = 2π2kT/~ωc, τq is the “quantum” relaxation
o time, and E is the Fermi energy, which is proportional
to F
h 0 to the electron concentration n, and hence to the gate
P
voltage swing U.
WeintroducetheparameterN =E /~ω,whichisthe
F
number of Landau levels below the Fermi level at cy-
N= 2
clotron resonance. Figure 3 presents the oscillating part
of the photovoltage [the function (dγ/dn)(n/γ)g(β)]
−
0 1 2 3 for α = 0.1, χ = 0.7 (corresponding to T = 4K,
ω =2π 2.5THz), and ωτ =0.5, for two values of N.
Reduced magnetic field, · q
In spite of the crudeness of our model, which does not
accountforvariousfeaturesoftheexperimentalsituation
FIG. 3: Magnetic field dependence of the oscillating part of
(the unavoidable presence of ungated parts of the chan-
thephotovoltageforα=0.1,ωτq =0.5fortwovaluesofN =
EF/~ω. The vertical scale for the lower trace is expanded 4 nel, etc), our results show a good qualitative agreement
times with respect to theuppertrace. β =ωc/ω with the recent experimental findings [9].
In summary, we have calculated the photovoltage in-
duced in a gated electron gas by THz radiation in the
In the absence of magnetic field, β = 0, F = 1, and Eq. presenceofthemagneticfield. Asafunctionofmagnetic
(21) reduces to Eq. (1). field,thephotoresponsecontainsasmoothlyvaryingpart
Figure 2 shows the behavior ofthe functions f(β) and and an oscillating part proportional to the derivative of
g(β) for several values of the parameter α. One can see theSdHoscillationswithrespecttothegatevoltage. The
that for small values of α (or large ωτ) the smooth part smoothpartshowsanenhancementinthevicinityofthe
displays the cyclotron resonance with the unusual line- cyclotron resonance.
shape f(β) [(1 β)2+α2]−1/2. The envelope for the WeappreciatenumeroushelpfuldiscussionswithWoj-
∼ −
oscillating part, g(β) exhibits a fast decay beyond the ciechKnap,NinaDyakonova,MaciejSakovicz,St`ephane
cyclotronresonance(β >1), confining the oscillationsof Boubanga-Tombet,and Sergei Rumyantsev.
[1] M. Dyakonov and M. Shur, Phys. Rev. Lett. 71, 2465 [9] S. Boubanga-Tombet et al, Appl.Phys.Lett.tobe pub-
(1993) lished
[2] M. Dyakonov and M. Shur, IEEE Transactions on Elec- [10] K.W.Chiu andJ.J. Quinn,Phys.Rev.B 9, 4724 (1974)
tron Devices, 43, 380 (1996) [11] A.P.Dmitriev,V.Yu.Kachorovskii,andM.S.Shur,Appl.
[3] W. Knap, J. Lusakowski, T. Parenty, S. Bollaert, A. Phys. Lett. 79, 922 (2001)
CappyandM.S.Shur,Appl.Phys.Lett.84,2331(2004) [12] In a Hall transport experiment there is no significant
[4] N. Dyakonova, A. El Fatimy, J. Lusakowski, W. Knap, difference between the true Corbino geometry and the
M.I. Dyakonov, M.-A. Poisson, E. Morvan, S. Bollaert, quasi-Corbino case of a finite strip with W >> L. The
A. Shchepetov, Y. Roelens, Ch. Gaquiere, D. Theron, current jy exists everywhere, except the extremities of
and A. Cappy,Appl.Phys.Lett. 88, 141906 (2006) the sample at y = ±W/2, where the current lines exit
[5] W.Knap,V.Kachorovskii,Y.Deng,S.Rumyantsev,J.- and enter theleft and right contacts respectively. In our
Q. Lu, R. Gaska, M.S. Shur, G. Simin, X. Hu, M. Asif case,thecurrentlinesmustformclosedloops,whichmost
Khan, C.A. Saylor and L.C. Brunel, J. Appl. Phys. 91, probablywillpassthroughthesourcecontact,orthead-
9346 (2002) jacent tothis contact ungated part of thechannel
[6] W.Knap,Y.Deng,S.Rumyantsev,J-Q.Lu,M.S.Shur, [13] Similar results can beobtained within theDrudetheory
C.A.Saylor,andL.C.Brunel,Appl.Phys.Lett.43,3434 (neglectingtheconvectiveterm(v·∇)v).Theoscillating
(2002) part remains the same, while Eq. (21) aquires an ad-
[7] S.Kang,P.J.Burke,L.N.Pfeiffer,andK.W.West,Appl. ditional factor 1/2 in the second term, which does not
Phys.Lett. 89, 213512 (2006) modify thequalitative behavior of f(β)
[8] M. Sakowicz et al, Int.J. High Speed Electron. Syst.,to [14] P.T.Coleridge,R.Stoner,andR.Fletcher,Phys.Rev.B
bepublished;M.Sakowiczetal,Int.J.Mod.Phys.B,to 39, 1120 (1989)
bepublished