Electric field gradient induced effects in GaAs/AlGaAs modulation-doped structures and high frequency sensing
D. Seliuta, A. Juozapavicius, V. Gruzinskis, S. Balakauskas, S. Asmontas, G. Valusis, W.-H. Chow, P. Steenson, P. Harrison, A. Lisauskas, H. G. Roskos, K. Koehler
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Electric field gradient induced effects in GaAs/AlGaAs modulation-doped structuresand high frequency sensing
D. Seliuta, A. Juozapaviˇcius, V. Gruˇzinskis, S. Balakauskas, S. Aˇsmontas, and G. Valuˇsis ‡ Semiconductor Physics Institute, A. Goˇstauto Street 11, LT-01108 Vilnius, Lithuania
W.-H. Chow § , P. Steenson, and P. Harrison Institute of Microwaves and Photonics, School of Electronic and ElectricalEngineering, University of Leeds, Leeds LS2 9JT, United Kingdom
A. Lisauskas and H. G. Roskos
Physikalisches Institut, Johann Wolfgang Goethe-Universit¨at,Max-von-Laue-Strasse 1, D-60438 Frankfurt/M, Germany
K. K¨ohler
Fraunhofer-Institut f¨ur Angewandte Festk¨orperphysik, Tullastrasse 72, D-79108 Freiburg, Germany
Electric field gradient effects induced by an asymmetrically in-plane shaped GaAs/AlGaAsmodulation-doped structures of various design are investigated within 4–300 K temperature range.It is demonstrated that current–voltage characteristics of such structures at low, 4–80 K, tempera-tures exhibit well-pronounced asymmetry arising due to a presence of two different gradients of theelectric field in a two dimensional electron gas. This phenomenon is caused by both, different accu-mulation of two-dimensional electrons due to asymmetrical shape of the structure and nonlocalityin the electron drift velocity. Experiments are illustrated by a phenomenological model and MonteCarlo simulation. Possible applications of the effect to detect electromagnetic radiation of GHz andTHz frequencies are discussed as well.
PACS numbers: 71.55.Eq, 73.61.-r, 73.23.Ad
I. INTRODUCTION
As it is known, in modulation-doped (or selectively-doped) structures free electrons, spatially separatedfrom ionized donors, are confined to a narrow potentialwell forming the so-called two-dimensional electron gas(2DEG) . Along the z axis, i. e. perpendicular to the2DEG plane, the confinement potential has a triangu-lar shape, and the spectrum of the 2DEG is composedof a sequence of quantum subbands. The shape of theconfining potential has influence only to the carrier tran-sitions between different subbands; meanwhile for exci-tations within each subband – so-called intraband exci-tations – this effect is very small.
P erpendicular to the z axis, i. e. in the 2DEG, or x − y plane, the carriersdisplay very high electron mobilities exceeding values of2 × cm /(V · s) at liquid nitrogen and 1 × cm /(V · s)at liquid helium temperatures. This fundamental prop-erty of these structures allowed to invent HEMTs – highelectron mobility transistors. The carrier transport in 2DEG layers at low electricfields is described by the Ohm’s law together with theEinstein relation for the diffusion coefficient, i.e. v = µ [ E − ( kT /e )(1 /n ) ∇ n ] , (1)where v is drift velocity, µ stands for electron mo-bility, E is the electric field and n is the free carrierconcentration; e is the electron charge, k and T de-note Boltzmann constant and lattice temperature, re-spectively. This so-called ”drift-diffusion equation” de- scribes the quasi-classical situation when the quantiza-tion of the electronic states-induced effects are not im-portant. On the other hand, it does not take into accountthe hot-electrons phenomena when the applied in-planeelectric field is strong enough to heat the electrons up tothe energy exceeding their equilibrium value. The electric field in modelling, as a rule, is assumedto be uniform. Although this assumption is very con-vienent for calculations, however, in many cases, in or-der to obtain the real picture of physical processes, it isnecessary to include spatial variation of the carrier con-centration or/and the electric field along the channel. Inparticular, when it becomes comparable with the rangeof a mean free path. Under these circumtances, effectssuch as velocity overshoot begin to predominate definingthus the performance of the device. Such an approachis very important, for instance, in understanding the be-havior of devices containing asymmetric channel profileproduced by a relevant doping. More specifically, in aquarter-micron n -type silicon metal-oxide semiconductorfield effect transistors (MOSFET) with asymmetric chan-nel profile, formed by the tilt-angle ion-implantation af-ter gate electron formation, allows one to achieve highcurrent drivability and hot-electron reliability. For ex-ample, in 0.1 µ m gate length asymmetric n -MOSFETstructures this technological innovation allows to attainhigher electron velocity in comparison with conventionaldevices of such type. In this article, we report on experimental and the-oretical investigation of the electric field gradient- induced effects due to asymmetrically in-plane shapedGaAs/AlGaAs modulation-doped structures. We showthat current–voltage characteristics of such structures atlow, 4–80 K, temperatures exhibit pronounced asym-metry. The physics behind is attributed to a two-dimensional bigradient effect which is equivalent to thephenomenon observed earlier in a bulk asymmetricaly-shaped semiconductors . We demonstrate that de-pending on the values of the in-plane electric fields andtheir gradients, the effect can reveal itself as a resultof different distribution of accumulating two-dimensionalelectrons due to the asymmetrical shape of the structure,and/or the exhibition of nonlocal drift velocity which be-comes pronounced in a different manner due to the pres-ence of two different gradients of the in-plane electricfield.The paper is organized as follows. In Sec. II we presentthe design of the GaAs/AlGaAs modulation-doped struc-tures, their electrical parameters and geometry features;we also describe briefly used measurement techniques.Section III reports on experimental results obtained inGaAs/AlGaAs modulation-doped structures of variousdesigns and at different lattice temperatures. Section IVis devoted to theoretical models and illustrates the con-cept of the electric field gradients-induced phenomenon– the bigradient effect with special emphasis on mani-festation of the electron drift velocity. In Sec. V pos-sible applications of the effect for the sensing of elec-tromagnetic radiation within GHz–THz frequencies aredisccused, while in Sec. VI features of three-dimensionalvs. two-dimentional effect are compared. Finally, con-clusions are given in Sec. VII.
II. SAMPLES AND MEASUREMENTTECHNIQUES
Two types of modulation-doped structures ofGaAs/Al . Ga . As (structure 2DEG-A) andGaAs/Al . Ga . As (structure 2DEG-B) were grownby molecular beam epitaxy technique. Their designparameters are given in the caption of Fig. 1.Electron sheet density, n , and low-field mobility, µ ,at different, 300 K, 77 K and 4.2 K, temperatures forstructure 2DEG-A are the following: 5.5 × cm − and4700 cm /(V · s), 1.9 × cm − and 190600 cm /(V · s),1.9 × cm − and 2 × cm /(V · s), respectively;while for structure 2DEG-B: 5.6 × cm − and8000 cm /(V · s), 4.3 × cm − and 116000 cm /(V · s),4.3 × cm − and 1 × cm /(V · s), respectively.The wafers were then processed into asymmetrically-shaped samples of different geometry which was changedvarying the width w of the neck. We have used the opticallithography for the samples with w = 12 µ m. The mezawith height, d , of 2 µ m in these samples was fabricatedby wet etching. In order to fabricate samples with anarrower neck, respectively 5 µ m, 2 µ m, and 1 µ m width,we have employed electron-beam lithography and shallow FIG. 1: The shape of the studied structures placed on semi-insulating substrate. Characteristic dimensions of the struc-ture are the following: L =500 µ m; D =250 µ m, A =50 µ m, W =100 µ m. White color denotes active part containing2DEG which is shown schematically as a black sheet; grey col-ored parts depict Ohmic contact areas of length C =100 µ m.Layer sequence of GaAs/AlGaAs modulation-doped struc-tures (from the top): structure 2DEG-A and structure 2DEG-B, respectively: 20 nm i -GaAs cap layer; 80 nm Si-doped,1 × cm − , layer of Al . Ga . As and 60 nm Si-doped,2 × cm − of Al . Ga . As; undoped spacers, 45 nmAl . Ga . As and 10 nm Al . Ga . As; 1000 nm and 600 nmof i -GaAs; smoothing superlattice – twenty and six periods of9 nm AlGaAs/1.5 nm-GaAs layers; 0.5 µ m and 0.6 µ m layerof i -GaAs; semi-insulating substrate. wet etching; the meza height in this case was 300 nm.The Ohmic contacts were produced by a rapid annealingprocedure of evaporated Au/Ge/Ni compound.The wired samples were then mounted into a two-stageclosed-cycle helium cryostat, and I − V characteristicswere measured within 4–300 K temperature range. Inmicrowave experiments performed at 10 GHz frequency,we have used magnetrons delivering pulses of 1.5–5 µ sduration with the repetition rate of 35–40 Hz; the sam-ples were then placed into rectangular waveguides, andthe microwave radiation-induced voltage arising over theends of the sample was measured by an oscilloscope; theexperiments were performed at room and liquid nitro-gen temperatures. In the terahertz (THz) range, 0.584–2.52 THz, the source of THz radiation was an optically-pumped molecular laser operating in the continuous waveregime; the illumination into the sample, located in theclosed-cycle helium cryostat behind the THz transparenthigh-pressure polythene window, was focused by spher-ical mirrors. As in the microwave case, the externalradiation-induced voltage arising over the ends of thesample was recorded by lock-in amplifier at a chopperfrequency of 187 Hz. III. EXPERIMENTAL RESULTS
Figure 2 shows the experimental results of I − V char-acteristics measured in structure 2DEG-A in the sam- A B m m m m Voltage (V) C u rr en t ( m A ) FIG. 2: (Color online) I − V characteristics of the asym-metrically shaped samples with different neck width at liq-uid helium temperatures. The inset depicts the measurementscheme: the contact B was always grounded, positive (filledsymbols) and negative (empty symbols) voltage was appliedto the contact A. ples with different neck widths recorded at liquid heliumtemperature. (The experimental data of the structure2DEG-B are equivalent and therefore are not presentedhere).As one can see, the salient feature of all the measured I − V characteristics is an asymmetry becoming well-procounced above 0.1 V. Moreover, the character of thecharacteristics strongly distingtive to the reduction of theneck width: For the sample with neck width of 12 µ m twoparts in the curve can be distinguished – the initial risingpart (below 0.3 V) which bends with voltage and above0.7 V transits to the second, saturated one. With thedecrease of the width w the saturated part transforms toa boosting region with highly expressed asymmetry. Acomplex behaviour is observed in the 5 µ m width sample:careful look can resolve initial rising part below 0.15 V,rudiments of saturated part within 0.15–0.6 V and al-ready emerging boosting region above 0.6 V with spreadinfluence on all the curve.It should be pointed out that the asymmetry isstrongly sensitive to the temperature variation. This isillustrated by the experimental data for 2 µ m neck sam-ple given in Fig. 3. It is seen that at low temperatures theasymmetry is rather large – if the coefficient of asymme-try of the I − V characteristics is defined as ( I f − I r ) /I f – at liquid helium temperature under 0.8 V it is 0.506,at 80 K its value is about 0.369, meanwhile at room tem-perature the asymmetry is nearly invisible. To resolvethe effect, the experimental data should be multiplied bya factor of one hundred. X 100 A sy mm e t r y ( m A ) Voltage (V) C u rr en t ( m A ) Voltage (V)
4K 60K 40K 80K
FIG. 3: (Color online) I − V characteristics of the asymmet-rically shaped sample vs. temperature. The neck width is2 µ m. Inset shows the change of asymmetry varying the lat-tice temperature. Measurement scheme is the same as in theprevious plot. IV. THEORETICAL APPROACH ANDPHYSICS BEHINDA. Electric-field gradient-induced effects: carrieraccumulation
To understand the physics behind the observed phe-nomenon, we have followed the same ideology as in thebulk case of asymmetrically-shaped structures: We ex-plore an asymmetrically shaped two-dimensional struc-ture by a solving a coupled system of one-dimensionalPoisson and current-flow equations neglecting quantumeffects : 1 S ( x ) ddx [ S ( x ) E ( x )] = − | e | ǫ ( n ( x ) − N d ) , (2) I = {| e | n ( x ) µ ( E ) E ( x ) + | e | D ( E ( x )) ddx [ n ( x )] } S ( x ) , (3)where x denotes the position along the symmetry axisof the asymmetrically-shaped sample, S ( x ) is the cross-section area of the sample, n ( x ) is the electron con-centration, N d is the donor concentration (used hereas a free fit parameter), E ( x ) denotes the electric fieldstrength, ǫ is the dielectric constant of the sample, D ( E )denotes the diffusion coefficient, and µ ( E ) is the elec-tron mobility. The diffusion coefficient may be set toa constant value as its dependence on the electric fieldhas no appreciable influence on the results. As it isknown, mobility µ ( E ) changes with the electric field: µ ( E ) = µ E c ( µ ) / [ E ( x ) + E c ( µ )], where E c = v c /µ ,and v c is the cut-off velocity equal to 10 cm/s.For a given value of the current I , one can employ an -5 0 5 10 15 m m neck A B R e l a t i v ee l e c t r onden s i t y Distance from the neck ( m m) Negative
A B E l e c t r i c f i e l d ( k V / c m ) A+, B -A -, B+
FIG. 4: (Color online) Calculated electric field and carrierdensity in the sample with 5 µ m neck width under I =0.03 mA at liquid helium temperature. iterative scheme to solve self-consistently the system andcalculate the voltage.Results of the electric field and relative carrier densitycalculations are given in Fig. 4. The model calculationsdo give two different gradients of the electric field inducedby the asymmetrical shape of the structure depicted inFig. 1. One must note further that the distribution de-pends on the current direction, and this difference is ap-preciable only within several microns from the neck: Theelectric field peaks are sharper in the case of the reversecurrent, when the A-contact of the sample is under apositive bias.Hence, the in-plane geometrical asymmetry inducestwo different gradients of the electric field in the sam-ple. As a consequence, different spatial accumulation oftwo-dimensional electrons in the vicinity of the neck canbe a reason for asymmetric I − V curves. This predictionis confirmed by the theoretically calculated I − V char-acteristics and their comparison with the experimentaldata presented in Fig. 5. As one can see, the measuredand calculated I − V curves in 12 µ m neck width samplesbehave similarly: in low electric fields, where Ohm’s lawis still valid, the current values are the same and do notdepend on the polarity of the applied voltage. However,with the increase of voltage – when sample is tuned toa strong field regime – the electrons become hot, theyaccumulate differently in the vicinity of the neck duethe asymmetrical shape of the structure. As a result,the asymmetry in the I − V characteristics is observed.Since this phenomenon relies on hot-carriers, it is reason-able that is temperature-sensitive: the decrease of latticetemperature will make the effect much more pronouncedsince the carrier heating at low temperatures is stronger.This is clearly seen in Fig. 3: the difference between theforward and the reverse-bias curves, for instance, at roomtemperature is approximately by three orders of magni- tude smaller than that at 4.2 K temperature in the givenvoltage range. The temperature effect is also confirmedby theoretical calculations (data are not given here).Therefore, the presence of two different electric fieldgradients in the hot two dimensional electron regime, al-lows one to explain the asymmetry of I − V characteris-tics at low temperatures. The phenomenon can be calledthus the two-dimensional bigradient effect.It is reasonable that the reduction of the neck shouldcause stronger asymmetry in I − V curves. Experimen-tally it is clearly illustrated in Fig. 2. However, if onecompares the behavior of the curves, for instance, for the5 µ m neck sample with a corresponding theoretical es-timates (Fig. 5), evident differences can be indentified: FIG. 5: Comparison of the measured (top plot) and calculated I − V (bottom plot) characteristics in the asymmetrically-shaped samples with different – 12 µ m and 5 µ m – neck widthsat liquid helium temperature. the theory does not follow the experiment anymore nei-ther in curve shape nor in asymmetry value. Therefore,in order to understand the origin of the behavior, a morecomplete theoretical model must be applied. B. Electric-field gradient-induced effects: Influenceof nonlocality in drift velocity
Althought the equation (1) is the backbone of elec-tronic device simulations, in the current situation oneneeds to be extended taking into account the electricfield gradient-induced term. The electron average veloc-ity can be then expressed using the extended/augmenteddrift-diffusion model v = v h (1 + δE dEdx ) − Dn dndx , (4)where v h is the drift velocity in a homogeneous electricfield, D denotes diffusion coefficient, δ depics the lengthcoefficient. The latter parameter is in the order of elec-tron mean free path and determines the severity of thevelocity overshoot effect which is pronounced when thevariation range of the electric field is equal to or shorterthan δ . Also, it is worth noting that in the given equationwe have assumed that nonlocality in diffusion coefficientis negligible and has no observable influence on the effect.A point of departure for analyzis is an estimate of amean free path ( λ ) of two dimensional electrons at liquidhelium temperatures. As electron mobility in the studiedstructures is of about 1 × cm /(V · s), this results to thevalue of λ = µe ( √ kT m ∗ ) ≈ µm, (5)where m ∗ =0.068 m e , the latter is the free electronmass. Meanings of other symbols are the same as inprevious formulae.The second step in considering the data is an estimateof the distribution of the electric field in the vicinity ofthe neck. Voltage value of 0.25 V was chosen as a refer-ence point, since experimentally observed asymmetry ofthe I − V characteristic starts to be strongly expressed.Theoretical calculations using the set of equations (2 –3) are given in Fig. 6. As one can see, the shape of the -0.5 0.0 0.5 1.0 1.5 U=0.25V E l e c t r i c f i e l d ( k V / c m ) Distance from neck ( m)
A- 12 m A+ 12 m A- 5 m A+ 5 m A- 2 m A+ 2 m
FIG. 6: (Color online) The dependence of the calculated elec-tric field distribution in the vicinity of the asymmetrically-shaped sample’s neck for two opposite currents at 4.2 K. Thebackground is shaded in the form of the sample. Note thedrastically increased asymmetry in the electric field distribu-tion with the reduction of the neck width. electric field distribution strongly varies with the re-duction of the neck width: it becomes sharper and the asymmetry becomes more pronounced. Futhermore, dis-tribution of the electric field is squeezed to the narrowerrange. If for 12 µ m width sample the electric field ex-tends over 1.7 µ m, in the 5 µ m and 2 µ m necks it isof about 1.2 µ m and 0.9 µ m, respectively. One can as-sume therefore that in 5 µ m neck samples the nonlocalitycan start to predominate since the change of the electricfield along the structure is comparable with the electronbalistic length.Calculations illustrating the effect of nonlocality inelectron drift velocity on I − V characteristics are given FIG. 7: Top plot: Calculated I − V -characteristics of 5 µ mneck sample using different length coefficient δ . Note the in-crease of the asymmetry with δ in the dependences. Bottomplot: Experimental results in 5 µ m neck sample given forcomparison. in Fig. 7. One can see that increase of δ has obvious ef-fect in 5 µ m neck sample – even small increase in lengthcoefficient (in the range of 0.1–0.2 µ m) stimulates theraise in the asymmetry. It is evident that calculationsfor the 5 µ m are very similar to the given experimentalresults. It is deserve to remark that in 12 µ m neck sam-ples such the influence is not observed (these theoreticaldata are omitted here).The inherent feature of the nonlocal effects is a mani-festation of the velocity overshoot. To evidence the lat-ter we have performed the Monte Carlo calculations ofthe drift velocity in a ”frozen” electric field taken froma solution of the coupled set of Poisson and current-flowequations (Eq. 2-3) as described previously. We havesimulated the drift of Monte Carlo particles along thex-direction in the given electric field distribution assum-ing that the particles enter the modelling region hav-ing the Boltzmann distribution at the lattice tempera-ture. The standard three-valley GaAs model is applied FIG. 8: (Color online) Monte Carlo calculations of the elec-tron drift velocity in ”frozen” electric field in 5 µ m neck sam-ple. Note oscilations in the drift velocity caused by opticalphonon emission. The vertical dashed line indicates the neckposition. Background is decorated schematically as the geo-metrical shape of the structure. in the simulation with an impurity scattering rate cor-responding to a 10 cm − donor concentration. Fig-ure 8 presents simulated electron velocity profiles in for-ward and reverse directions at the electric field profilesfor 5 µ m neck samples. For the sake of evidence, theelectric field profiles taken from Fig. 6 are given, too.As one can see, the velocity overshoot/undershoot andstrong nonlocality effects are clearly visible.The sharp peaks at the top of the velocity profiles arerelated to the locality in optical phonon emission rate ν op . Note that the velocity profile is asymmetric due tothe presence of the different electric field gradients andexhibits nearly constant value in the forward directionranging from 0.75 up to 3 µ m. The Monte Carlo simula-tion for electric field profiles from Fig. 8 gives the coeffi-cient of current asymmetry of 0.48. The factor stronglydepends on impurity scattering rate: it decreases as thescatering rate increases. The simulation with donor con-centrations 2 × , 1 × , and 5 × cm − gives thecoefficient values of 0.62, 0.48, and 0.25, respectively.To get close-up view in the physics behind, we have de-picted the velocity and ν op profiles in both, forward andreverse, directions (Fig. 9). Also, stationary velocity isadded to illustrate the overshoot/undershoot areas. It isobvious that the electron velocity minima coincide withthe ν op maxima forming thus ”comb” in electron distri-bution. The effect is equivalent to the the free-carriergrating formation phenomenon caused by locality of op-tical phonon emission as already demonstrated in n + nn + InN structures. -0.5 0.0 0.5 1.0-0.50.00.51.01.52.02.53.03.54.0 -0.50.00.51.01.52.02.53.03.54.0 stationary velocitydirection of electron flux A+ B- -0.5 0.0 0.5 1.0-0.50.00.51.01.52.02.53.03.54.0 -0.50.00.51.01.52.02.53.03.54.0 direction of electron flux stationary velocity
B+A-
Distance from neck ( m) Lo c a l e l e c t r on sc a tt. r a t e b y op t i c a l phonon s ( , s - ) E l e c t r on d r i ft v e l o c i t y ( m / s ) FIG. 9: (Color online) Monte Carlo calculations of drift veloc-ity and local scattering rate of electrons by optical phononsin ”frozen” electric field in 5 µ m neck sample. Note oppositephases in the velocity and the scattering rate. Backgroundschematically shows the geometrical shape of the structure. V. POSSIBLE APPLICATIONS OF THEEFFECT
Since the two-dimensional bigradient effect inducesasymmetry in I − V -characteristics, it is reasonable toconsider its possible applications in detection of the high-frequency electromagnetic radiation. To test this idea, wehave placed samples in a microwave field of 10 GHz fre-quency and recorded microwave-induced voltage arisingover the ends of the sample. The results for two sam-ples of different neck width recorded at room and liquidnitrogen temperatures are displayed in Fig. 10(a).One can see that with the increase of incident powerthe signal increases linearly in all the range of studiedpower. Thus, due to the asymmetry in I − V curvescaused by the two-dimensional bigradient effect the mi-crowave radiation can be detected . However, as es-timates show, the sensitivity at room temperature israther low. For instance, it is only about 12 mV/W and40 mV/W for the samples with neck width of 12 µ mand 2 µ m, respectively. Since the effect is related tothe electron heating, the decease in temperature shouldinduce the increase in the detected signal. The tempera-ture effect, however, is not as large as might be expectedfrom hot-electron physics where change of the signal isproportional to the carrier mobility. As one can see, at77 K, the detected voltage increases reaching the valuesof 40 mV/W (12 µ m neck sample) and 150 mV/W (2 µ mneck sample). In fact, the rise in only by a factor of about4, while the mobility increases about 14 times. We as- FIG. 10: (Color online) Detected signal versus incident powerat 10 GHz frequency in the samples fabricated from the struc-ture 2DEG-B. (a) – Data for two samples employing two-dimensional bigradient effect with neck sizes of 12 µ m and2 µ m at room and liquid nitrogen temperatures. (b) – Com-parison of the two-dimensional bigradient effect-based sam-ples with bow-ties diodes containing 2DEG layer. Geomet-rical dimensions and are the same for both type of samples.Data recorded in 12 µ m neck size samples at room and liq-uid nitrogen temperatures. Insets show measurement schemesand design differences. Note much higher sensitivity of the2DEG bow-ties diodes. sociate the outcome with a presence of ballistic electronswhich cross the effective length of the samples withoutscattering. It is evident that for implementation of theeffect into practical devices, such a sensitivity values arenot sufficient. Following the ideology of the bigradienteffect, one needs to produce much higher non-uniformityof the electric field in the sample’s neck vicinity. This canbe achieved easily by metallizing one of the asymmetricparts of the structure. This is evidenced by calculationsof the distribution of the electric field given in Fig. 11.It is seen that the non-uniformity of the electric fieldwith one part of the sample metallized is much strongerthan in the conventional asymmetrically-shaped struc-ture. It is therefore reasonable to expect significantlyhigher values of the sensitivity. The experimental illus-tration is given in Fig. 10(b) where the bigradient struc-tures are compared with the so-called 2DEG bow-tiesdiodes – structures where one of the parts of the sam-ple is covered by metal. One can see that these devicesare nearly in two orders of magnitude more sensitive com-pared to the structures of bigradient-design.Of particular interest for the direct applications arethe values of the voltage sensitivity at room tempera-ture. In order to increase the sensitivity of the diode, wehave chosen the design 2DEG-B (due to the higher valueof the sensitivity at room temperature) and reduced thesize of the apex down to 2 µ m keeping the other geomet-rical dimensions the same (Fig. 12). As one can see, inthis case the detected signals are significantly larger thanfor the 12 µ m neck diodes at 300 K, and the sensitivity BA x10B: Metal B: Semicond. C u rr en t ( m A ) Voltage (V)
A+, B- A-, B+ Asymmetry C oe ff. o f a sy mm e t r y -1 0 1024 E l .f i e l d ( k V / c m ) Distance ( m)
FIG. 11: Calculated effect of the metallization of the B-partof the asymmetrically shaped sample on I − V characteris-tics. Room temperature value of the electron mobility µ =4700 cm /(V · s) is used. The neck width is 5 µ m. The filledsquares denote the forward current I f , the empty squares -the reverse current I r . The coefficient of asymmetry is definedas ( I f − I r ) /I f . amounts to 2.5 V/W, which is nearly one order of magni-tude higher compared to the diodes of 12 µ m neck width,where it is about 0.32 V/W. The decrease in tempera-ture down to 77 K allows one to increase the sensitivityup to nearly 40 V/W, i.e. close to a factor of 14.5 whichcorresponds to the change of 2DEG mobility. The lattereffect – via thickness of spacer – is also nicely expressedin Fig. 12 were the detected signal in both devices, fab-ricated from 2DEG-A- and 2DEG-B-type structures iscompared.Additionally, it deserves mentioning that the furtherdecrease of the neck size down to 1 µ m – 800 nm, gives,however, no increase in the sensitivity (the data are notpresented here). We attribute it to weaker coupling ofthe microwave field into the structure with the reductionof the neck width.As to dynamical range, the devices can be used todetect rather different power levels. As already in-dicated above, the bigradient modulation-doped struc-tures at 300 K in microwaves exhibit sensitivity of about12 mV/W and 40 mV/W for the samples with neck widthof 12 µ m and 2 µ m, respectively, however, the sensingrange is below 1 W in a pulse mode, i.e. the resis-tance to overloading is small. Bulk silicon-based 12 µ mneck width structures are also not sensitive – only about0.1 V/W – but they can withstand up to 10 kW max-imum applied power at room temprature without anynegative outcome. Asymmetrically-shaped bulk GaAsstructures containing n − n + -junction were found to besuitable for broadband sensing within GHz–THz frequen-cies at 300 K. Finally, it is worth to remark on the operation of the bi- -4 -3 -2 -1 -1 line - linear functionas guide for eye
10 GHz - 12 m-A = 300 K- 12 m-A = 77 K- 12 m-B = 300 K- 12 m-B = 77 K - 2 m-B = 300 K- 2 m-B = 77K D e t e c t ed S i gna l ( m V ) Power (W)
FIG. 12: (Color online) Detected signal as a function of inci-dent power at 10 GHz frequency microwave field in 2DEGbow-ties diodes of different geometry and design. Datarecorded in 12 µ m and 2 µ m neck size samples based onstructures 2DEG-A and 2DEG-B at room and liquid nitro-gen temperatures. Linear function is also given as a guide foran eye. gradient structures within the THz range. One can notethat at 300 K the sensitivity at 0.763 THz in the samplewith neck width of 12 µ m it is about 13 mV/W, i.e. closeto the value in 10 GHz range. Narrowing the neck widthdown to 2 µ m, the sensitivity at 0.763 THz can increaseup to 17 mV/W. Within frequencies 1.41 THz, 1.63 THzand 2.52 THz the detection observed is rather poor. Weexplain it by a weak coupling of the incident THz fieldinto the structure. At helium temperatures, the sensitiv-ity reaches the value of 60 mV/W at 0.763 THz for the12 µ m neck width sample. VI. TWO-DIMENSIONAL VS. BULKBIGRADIENT EFFECT
The effect was originally discovered by measuring I − V characteristics in bulk semiconductors, n -Ge and n -Si, at room and liquid nitrogen temperatures. Inwhat follows we compare the two-dimensional bigradi-ent effect observed in asymmetrically in-plane shapedGaAs/AlGaAs modulation-doped structures with itsbulk equivalent desribed in details. The inherent fearure of both – the two-dimentional andbulk – effects is asymmetry in I − V characteristics. Dif-ferent material parameters and geometrical dimensionsof the samples determine rather distinctive voltage scaleand temperature conditions for experimental observa-tion. The effect of the dimensionallity we associate withunusual manifestation of the nonlocality in drift veloc-ity. Figure 13 presents a change of the relative current due to nonlocality inkinetic coefficients -0.020.000.020.040.060.08 bulk silicon results r e l a t i v e c hange o f c u rr en t Voltage (V)
77 K 300 K
X10 modulation-doped structure,5 m neck r e l a t i v e c hange o f c u rr en t, I/ I Voltage (V)
FIG. 13: Change of the relative current in 5 µ m neck sizesample at liquid helium and room temperatures. Insert de-picts the same plot for comparison in n − doped asymetrically-shaped silicon samples with specific resistance of 2.5 Ω · cm at300 K. The length of the sample is 23 mm, angles of the asym-metry are α =45 and α =13 . More details can be found inRef. 19. in 5 µ m neck size sample at 4.2 K and 300 K. For com-parison of the features, inset shows equivalent bulk datameasured in n -type silicon according to Ref. 19.It is seen that in the bulk semiconductor the nonlocal-ity is associated with the sign change in the asymmetryclearly observed within 1-10 V voltage scale at 77 K (seeinset). In contrast, in modulation-doped structure, nosuch peculiarity is observed. It is worth noting, however,that above 0.3 V the change of the relative current isnearly independent of voltage – as was indicated earlier(Fig. 7, above 0.25 V range, where asymmetry of I − V characteristics strongly increases) – we attribute it to theapperance of the nonlocal drift velocity. VII. CONCLUSIONS
Asymmetrically in-plane shaped GaAs/AlGaAsmodulation-doped structures of various design are stud-ied within 4–300 K temperature range. We have shownthat current–voltage characteristics of such structures atlow, 4–80 K, temperatures exhibits pronounced asym-metry which we explain by a two-dimensional bigradienteffect. This phenomenon can be decomposed into twoconstituents – carrier accumulation and nonlocality inthe electron drift velocity – each of which manifestsitself depending on the values of the in-plane electricfields and their gradients. Varying these parametersvia change of the sample neck we have inferred thatthe effect is induced by different accumulation oftwo-dimensional electrons due to asymmetrical shapeof the structure when the neck size down is 12 µ m;in 5 µ m neck size structures in electric fields close to3 kV/cm, the nonlocality of the electron drift velocitybecomes predominating causing thus further increaseof the asymmetry in I − V curves. The theoreticalmodels suggested conform well with the obtained ex-perimental results. Monte Carlo simulation has shownstrong nonlocal effects in the electron drift velocity andpresence of overshoot/undershoot regions depending onthe electric field and its gradient values. The featuresof the two-dimensional bigradient effect and its bulkequivalent are compared and possible applications ofthis phenomenon to detect electromagnetic radiation ofGHz and THz frequencies are discussed as well. Acknowledgments
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