Quantum Dot Charge Sensor Physics in 2DEG Heterostructures

Transcription

1 Quantum Dot Charge Sensor Physics in 2DEG Heterostructures Morten Kjærgaard Spring 21 Abstract In this paper I introduce and discuss the details of the theory and numerical programs developed to understand the physics of a gate defined quantum dot as a charge sensor for the charge state of a nearby double quantum dot on 2DEG heterostructures. This paper is intended as extended material and background for the paper Fast sensing of double-dot charge arrangement and spin state with an rf sensor quantum dot C. Barthel,MK,J.Medford,M.Stopa, C. M.Marcus,M.P. Hanson, A.C.Gossard Physical Review B: Rapid Communications, 81, (21).

2 Contents 1 Quantum Dots for Quantum Computation 3 2 Spin qubit readout Measuring Charge State: Coulomb Blockade Measuring the Spin State: PauliBlockade Increasing the Charge Sensitivity 5 4 Sensitivity and Lever arms of Charge sensors 7 5 Numerically Calculating the Sensitivity SETE self-consistent DFT for φ(x,y,z)and ρ(x,y,z) gvqpcc QPC Conductance ivr SQDConductance Conclusion& Outlook 12 List of Figures 1 Schematic of double quantum dots in 2 dimensional electron gases Cartoon depicting the process of charge state readout of a quantum dot using a proximal quantumpoint contact Charge stability diagrams for a double quantum dot measured using adjacent quantum point contact Schematic showing the principle behind apauli spinblockade SEM micrograph of geometry used for sensing quantum dot measurements in Barthel etal Charge stability diagrams measured using a sensing quantum dot and a quantum point contact inthe geometry fromfigure Self consistently calculated electron density in the geometry used for sensing quantum dot experiments Self consistently calculated potentials for use in the conductance calculation through the QPC Examples of transverse 1D potential from which eigenvalues for the effective longitudinal 1D potentialis created Numerically calculated values of conductance through QPC for (,2) and (1,1) charge stateson aproximaldouble quantum dot Sensing quantum dot conductance calculated using SETE and ivr for two fixed charge states(1,1)and (,2)

3 2 SPIN QUBIT READOUT 1 Quantum Dots for Quantum Computation Gate defined quantum dots in semiconducting heterostructures are one of the most promising candidates for the realization of spin qubit quantum information processing units. By applying a voltage to a (typically Ti/Au) gate positioned above a 2 dimensional electron gas (2DEG), a local electrostatic minima can be created, allowing for local confinement (the quantum dot) of electrons. Figure 1 shows a schematic of the layers of a heterostructure, a typical gate-pattern for a doublequantum dot (DQD) and aresulting potential in a DQD. For the realizationof a quantum computation using a spin qubit three distinct steps are necessary for a complete computation : Load in of a qubit, manipulation of the qubit and readout of the resulting state. The work presentedinc.bartheletal.,discussedindetailhere,focusses onanew andfastertechnique forthe readout of a qubit. Figure 1: left) Schematic of the layers of a semiconducting heterostructure used to create the 2DEG. The V shape in the 2DEG is the potential, with the fermi level at approximately the middle of the of the well-height, allowing for bound states in the well. The black boxes signify the gates. middle) SEM micrograph of a typical double quantum dot (DQD). right) Schematic of the potential in the DQD, defined by applying voltages to the gates shown in micrograph. The nucleii in the substrate are also drawn. The effect ofthesewillnotbediscussedinthispaper. ModifiedfromJ.R.Pettaetal.,Science,25. 2 Spin qubit readout A readout of a quantum state on quantom dot(s) amounts to a) measuring the charge state (i.e. number of electrons on dot(s)), and b) readout of the spin state of the electron(s) on the dot(s). The technique for a) relies on Coulomb blockade, and the technique for b) relies on Pauli blockade. 2.1 Measuring Charge State: Coulomb Blockade The principle of charge readout relies on the characteristic, that if the charge state on a quantum dot is changed (i.e. going from N to N +1 electrons), the potential around the dot is changed. Namely, if a quantum point contact is sufficiently close to a quantum dot, the effective potential at the saddle point of the QPC will be dependent on the charge state of the quantum dot. This combined with the Landauer formula: g QPC = 2e h n T n, T n is transmisson fromn th mode (1) and Büttikers 1991 [1] result that transmission through a QPC is dependent on the potential at the saddle point, then tells us, that a graph of the conductance through the QPC as a function of the difference in voltage on source and drain gives us information about changes to the number ofelectronsonthequantumdot. Figure2isacartoonshowingthisprinciple,wherethequantum 3

4 2 SPIN QUBIT READOUT dot is assumed to be initially empty, i.e. N =. By tuning the source and drain voltage on the dot,theoccupancyofthedotwillchange,andtheresultingchange, δg QPC,inconductanceofthe QPC is thus an indication that the charge state of the quantum dot has changed. The optimal δg QPC is achieved when the QPC is tuned such that the conductance is right at the shoulder of the opening of a new level of the QPC itself. This is called the operating point for a QPC. The measurement of the charge state of a double quantum dot is conceptually equivalent to the g QPC N g QPC V sd,1 E V sd,2 δg QPC N = N = 1 N = 2 N = 1 N = 2 V sd,1 V sd,2 V sd V sd Figure 2: Cartoon depicting charge sensing using a proximal quantum point contact. left) Schematic drawing of quantum dot with N electrons and adjacent QPC for charge sensing. middle) The two diagrams show a simplified version of the quantum dot, as having asource/drain voltageand discretelevels,fortwo differentvalues ofv sd. Current flows from right to left. right) Cartoon showing the conductance through a QPC proximal to a quantum dot as a function of V sd. When V sd is such that a new level of the dot is allowed to be occupied, an electron will hop on, and the electrostatic potential will change, and hence, the conductance through the QPC will change (see eq.??). The conductance plot will be thermally broadened when T =.. technique for a single quantum dot outlined in figure 2. Using the notation (N L,N R ) for the occupancy on the(left,right) dot, figure 3 shows a schematic and data on the conductance through a proximal quantum point contact as a function of double dot occupancy. Diagrams like these are called charge stability diagrams. The name Coulomb Blockade stems from the inability of the electronstohopfromthe2degonto thedot,unless itis energiticallyfeasible,e.g. bytuning V sd. 2.2 Measuring the Spin State: Pauli Blockade The Pauli blockade relies on the fact that the overall wavefunction of a two electron state should be asymmetric. This enforces the requirement, that in states of (,2) types, the spin part of the wavefunction must be asymmetric, and hence be a singlet. By exploiting this, the spin state of a double quantum dot can be derived. Figure 4 outlines the process of spin state readout using Pauli blockade. In panel (a) the system is initialized in (,2) configuration (M point in figure 6), resulting in conductance g (,2) through adjacent charge sensor. In panel (b), at time t = t 1, (V L,V R ) is tuned so that system changes to (1,1) (S point in figure 6), resulting in g (1,1) through theqpc. Thesystemisallowedtoevolveforatime t 2 t 1 = t,afterwhichthe systemiseither in (1,1)S or (1,1)T (these states are coupled by the hyperfine interaction). To measure wether thesystemisasingletoratriplet,attime t = t 2 thesystemisdrivenbackintothem point. Ifthe system was in a triplet, (panel (c1)), the transition to (,2) is not allowed due to Pauli exclusion, and hence, the conductance through the QPC will not change. If the system was in a singlet (panel (c2)), the transition is allowed and hence the conductance of the QPC goes back to g (1,1). The cartoon is somewhat simplified, since it reallyis the composite state (1,1)S and (1,1)T that the systemoscillates between it is not just the isolated spinon the left handdot. 4

5 3 INCREASING THE CHARGE SENSITIVITY Figure 3: Coupled double dot generalization of charge sensing using adjacent quantum point contact. left) Schematic showing (left,right) occupancy of the double dot as a function of V L V g1 and V R V g2. Current only flows through the system at the triple-points, denoted by black circles. Adapted from A.C. Johnsons PhD Thesis, 25. right) Data from device in figure 1 with indication of the double dot occupancy, identified by establishingthe(,) occupancy,andtuningv L andv R whilstlookingfor δg QPC. Adapted from J.R. Petta et al, Science, 25. (a) g SQD (c1) (b) t < t 1 t = t 1 g (,2) g (1,1) g (,2) g (1,1) t 1 t 1 t 2 t 2 t t (c2) t > t 2 t > t2 g (,2) g (1,1) g (,2) g (1,1) t 1 t 1 t 2 t 2 t t Figure 4: Schematic showing principle behind Pauli blockade. See text for details. The blue spin will only return to (,2) configuration if the time spent in (1,1) (t 2 t 1 ) is equal to two full (or multiples thereof) spin flips. The step like charicature of the conductance will be thermally and tunneling broadened. 3 Increasing the Charge Sensitivity Charge sensing in 2DEG heterostructure is thus of fundamental importance since it gives information about charge occupancy, and even more important, about the spin state. In the framework of quantum computing the knowledge of spin state is equivalent to knowledge of the outcome of a quantum calculation (is the outcome a binary zero, e.g. a singlet, or is the outcome a binary 1, e.g. a triplet?). The work presented in Barthel et al. is a variant of the QPC charge readout scheme that have so far been employed. The QPC is replaced with a quantum dot, termed the sensing quantum dot (SQD). The SQD is tuned such that it is right at the opening of a new coulomb peak, and the conductance through the SQD is now measured as a function of the source and drainvoltage on the quantum dot. Figure 5 shows a micrographof the geometry of the system employed in the study in Barthel et al. along with the conductance through the QPC and SQD. In Barthel et al. we present charge stability diagrams equivalent to the ones in figure 3 obtained using the SQD technique. Figure 6 shows the charge stability diagrams and the increased value of δg SQD over δg QPC. Simply modeling the change of the double dot occu- 5

6 3 INCREASING THE CHARGE SENSITIVITY g (e 2 / h) QPC1 QPC2 Dot-Sensor Figure5: left)semmicrographofdevicesimilartotheoneusedformeasurementsinbartheletal. TheQPCformedatthe left hand sideofthedqdisusedforcomparisonwiththesqd throughout the rest of the discussion. right) Conductance through QPC1 (blue), QPC2 (black), and the SQD (red) as a function of the respective backgates (V Q1, V Q2 and V D ). The operatingpointforthe SQD isatthe shoulderofacoulombpeak. pancy as changing the effective potential in the proximal SQD and QPC would already lead one to the conclusion that δg SQD > δg QPC simply due to the differenceinslope of conductance plots in figure 5. It turns out that this, however, is not the complete story. This is due to the different effective lever arms of the SQD and QPC. The effective lever arm is the number that relates the change in some back gate voltage (or a change in double dot occupancy) to the actual change in the electrostatic potential at the saddle point of the QPC / electrostatic potential at the center of the dot. Numerical simulations allow us to estimate this effective lever arm, and the calculation of these,will bethe subject of the next section. g g g g (1,1) S (1,2) (1,1) (1,2) M (,1) (,2) (,1) (,2) (a) (b) (c) Detuning (mv) (d) Detuning (mv) Figure 6: left) Charge stability diagram of the double quantum dot measured using the sensing quantum dot in figure 5 tuned to the operating point. The line cut below clearly shows the difference δg SQD inconductance throughsqd when tuning V L and V R to allow the (1, 1) (, 2) transition. right) Equivalent data obtained using the quantum point contact labeled QPC1 in figure 5. In both charge stability diagrams the color scale shows the relativedifference g/ḡ,where g = g (1,1) g (,2) and ḡ = (g (1,1) +g (,2) )/2 6

7 5 NUMERICALLY CALCULATING THE SENSITIVITY 4 Sensitivity and Lever arms of Charge sensors InBarthelet al. we introduceameasureof the sensitivity, s, of aqpc or QDas follows: s QPC = g V QPC = g φ SP φ SP V QPC = α QPC φ SP V QPC s SQD = g V dot = g φ dot φ dot V dot = α SQD φ dot V dot, wherev is theback gatevoltage onqpc orqd, and φ is the electrostaticpotential atthe saddle point (QPC) or middle of dot (QD). The number α is the effectiveleverarm, that depends on position of nearby conductors that screen the interaction between the source of the voltage and the potential at the point of interest. The issue of sensitivity of the QPC vs a SQD can now be quantified by establishing the numerical value of the leverarms. Using the three numerical methods described below, we find a ratio of α SQD /α QPC 2, arising from screening from conductors, as calculated using self consistent density functional theory modeling of the geometry shown in figure 5. 5 Numerically Calculating the Sensitivity The numerical values for the QPC arefound as follows. Details of thesetecode instep 1canbe found in section 5.1 (and ref [2], and the details of the gvqpcc code in step 2 3 can be found in section Using self consistent density functional theory(within the local density and effective mass approximation), Poissons equation is solved in the full 3D geometry from experiment (including gate geometry, gate voltages, depth of 2DEG etc). This yields and effective two dimensional potential and density of the entire geometry. See figure 7 2. By cutting out the part of the 2D potential in the area around the QPC and solving the transverse Schrödinger equation in slices through this area, an effective 1D potential for the lowest subband can be created. 3. Using a WKB approximation to the transmission through this effective 1D potential, the transmission, and hence conductance(see eq. 1), can be found. Equivalently, for the SQD, the numerical values are found as follows (for details on step 1, see 5.1,for detailson step2see5.3). 1. Using the SETE code, the free energy, F SQD, as a function of the back gate voltage and the numberof electrons onthe dot N is evaluatedfor the full3dgeometry. 2. The conductance through the SQD is calculated using Beenakkers 1991 result [3], in the low tunneling regime,with F SQD (V SQD,N)as input. The simulations produced for the Barthel et al. paper were generated using three different programs. Two are developed by Mike Stopa, SETE [2], for calculating the potential and charge density of a gate-defined 2DEG heterostructure, and ivr [4], for evaluating the conductance through a quantum dot, using Beenakkers conductance formula [3]. The code gvqpcc is developed by MK, and uses a WKB approximation to the potential in a quantum point contact to evaluate the conductance. 7

8 5 NUMERICALLY CALCULATING THE SENSITIVITY 5.1 SETE self-consistent DFT for φ(x,y,z) and ρ(x,y,z) SETE, Single Electron Tunneling Elements, is a self consistent code for calculating the electronic structure of semiconductor quantum dots. The code uses density functional theory (within the effective mass and local density approximations) to solve the 3D Poisson equation 2 φ(x,y,z) = 4πǫ ρ(x,y,z) (2) κ andcalculatethefreeenergy, F,of quantumdotsystems. InthepaperSETEis usedtofinda)the effective 2D-potential of a quantum point contact at the depth of the 2 dimensional electron gas in anal x Ga 1 x As-GaAsheterostructureandb)the freeenergyofaquantumdot,bothofwhichare locatedadjacenttoalaterallydefineddoublequantumdot. Theresultofa)isusedingvQPCcand the resultof b) is usedinivr. SETE fully incorporates gate-geometry, donor concentration, depth of 2DEG and voltages on gates. After initializing the entire device, an inhomogeneous grid is set up on the gate-geometry, and the density is found at each lattice-site using either a quantum mechanical solution (in the quantum dots, or regions of interest) or a classical solution (in regions of less importance). For the quantum mechanical solution, we are in principle looking for solutions to the full 3D- Schrödinger equation [ ] h2 2m 2 +eφ(x,y,z) +V B (z) ψ(x,y,z) = Eψ(x,y,z), (3) where V B (z) is the band offset between the AlGaAs and the GaAs at the interface, and m is the effectivemass,m.67m. Equation(3)isintractable,andinstead,anadiabaticapproximation is assumed to reduce the 3D-potential to an effective 2D-potential. A 1D-Schrödinger equation is solved ateachlatticesite [ ] h2 2m 2 z +eφ xy (z) +V B (z) ξ xy n (z) = ǫ xy n ξ xy n (z), (4) where superscript x, y denotes discrete indices on the lattice. z direction is the growth-direction of the sample. The discrete 2D-potential is now interpreted as the continuous effective potential ǫ (x,y), where we assume only filling of the lowest subband. Assuming adiabadicity, i.e. x ξ xy (z) = y ξ xy (z) =, the 3D-density in the poisson equation can now be found by solving a 2D-Schrödinger [ h2 2m ( 2 x + 2 y from which the density is calculated as ] ) + ǫ (x,y) f (x,y) = Ef (x,y), (5) ρ QM (x,y,z) = e ψ(x,y,z) 2 = e f (x,y)ξ xy (z) 2. (6) The density is calculated classically in regions far from the quantum dot(s), using the Thomas- Fermi approximation, i.e. we assume the potential varies slowly on scale of 1/λ F. Under this assumption the density is found from ρ cl (x,y,z) = ǫ (x,y) µ ξ xy (z) 2, 2π sowe donot need tofind the full eigenstates,asin eq. 6 The classical and quantum mechanical densities are patched together to update the poisson equation, which is solved again, using the Bank-Rose method[5], to yield a new 3D-potential, and the procedure is iterated until convergence. We apply Dirichlet boundary conditions at lattice points in the gates, i.e. φ(x,y,z) = V g, and Neumann boundary conditions elsewhere, φ(x,y,z) =. Asample density, calculatedwithseteis depictedinfig. 7 8

9 5 NUMERICALLY CALCULATING THE SENSITIVITY Figure7: Self-consistently calculated 2D density ρ(x,y) = e f (x,y) 2, with gate-geometry superimposed. The color-scale bar is centered around cm 2, to accentuate the densities in the quantum point contact and sensor quantum dot. The intergral of the density in the region of the(left,right) quantum dot corresponds to a(1,1) electron configuration. 5.2 gvqpcc QPC Conductance This code calculates the conductance through a quantum point contact given the 2D potential landscape around the QPC. Assuming ballistic transport through the quantum point contact, the conductance is given by the classic Landauer result: G = 2e2 h n T n (7) where T n is the transmission from the n th mode through the local minima. Thus, the problem of conductance through a QPC is essentially reduced to the problem of finding the transmission coefficient through a QPC. The strategy used for this in gvqpcc is to reduce the 2D potential landscape around the QPC to an effective 1D potential, and then calculate the transmission through this. To this end, the code assumes adiabatic conditions for the system, such that a Wentzel Kramers Brillouin approximation to the transmission coefficient is applicable. Fromthe fullsetecalculatedpotential, φ(x,y)(leftpanelinfigure 8),only the partpertaining to the QPC is picked out (denoted φ QPC (x,y),see right panel infigure 8). This local 2Dpotential is y (.53 a *) x (.55 a *) Figure 8: left) Full potential landscape for the QPC DQD SQD setup calculated using. right) 3D picture of the potential landscape in the black box from the lefthand figure. This is an exampleofthe potential usedasinputingvqpcc now parametrized into 1D potentials in the transverse direction, denoted φ y (x). An example is 9

10 5 NUMERICALLY CALCULATING THE SENSITIVITY shown in the left panel of figure 9, which is exactly the 1D potential corresponding to y = 25 in figure 8. The Schrödinger equation for the lowest energy state in this potential reads H y=25 (x)ψ y=25 (x) = ǫ y=25 ψ y=25 (x). (8) This equation is solved numerically to find ǫ y=25. By looping over all 1D linecuts, φ y i(x) in φ QPC (x,y) and solving equation 8, an ensemble of eigenvalues, {ǫ y i } is found for a given gate voltage configuration on the entire geometry. The code then assumes a continuous limit of potential cuts, such that {ǫ y i } ǫ (y). A plot of a sample ǫ (y) calculated from φ QPC (x,y) is shown on the right panel of figure 9. The black circled eigenvalue is ǫ y=25, and Fermi energy at E =. Finally, the transmission is found through the effective 1D potential given by ǫ (y) 4.1 E / Ry* E (Ry*) x (.55 a *) y (.63a *) Figure9: left)anexampleofa1d linecutaty = 25inthe3Dplotfromfigure8,denoted φ y=25 (x). Thisisthepotentialforwhichthe lowesteigenstateisfoundineq. (8). right)thisgraph shows the result of looping over all values of y in the 3D potential in figure 8, finding the lowest eigenvalues, and plotting them as function of y, denoted ǫ (y). This is the potential from which the transmission using WKB is calculated. The marked eigenvalue is the lowest eigenvalue from the potential on the left. using a WKB approximation, [ T WKB = exp 2 ] dy 2m(ǫ (y) µ) (9) which is plugged into the Landauer equation, assuming transmission only from the lowest lying state: G(V QPC ) = 2e2 h TWKB (V QPC ). (1) In the paper we performed this calculation for values of gate voltages corresponding to the measured values in the experiment. By numerically fixing the charge density in the double quantum dot to (,2) and (1,1), and looping over gate voltages, we were able to quantify the difference in conductance due to charge rearrangement, as plotted in figure 1. For the QPC we find g /ḡ.1, roughly consistent with the experimental value (.3). The difference is primarily attributed to breakdown of the breakdown of WKB approximation close to the classical turning point. 5.3 ivr SQD Conductance This code calculates the conductance through a quantum dot using Beenakkers 1991 result [3]. Using a master equation approach, a linear-response theory calculation of the transport through 1

11 5 NUMERICALLY CALCULATING THE SENSITIVITY 1. g (e 2 /h) (b) g (,2) g (1,1) g.4 g (e 2 /h) V Q1 (mv) Figure 1: The conductance through a proximal quantum point contact as function of back gate voltage. The conductance is numerically calculated for two different, but fixed, charge densities on a proximal double quantum dot. The black line is the relative difference. Consistent with experiment the relative difference in conductance is largest at the operating point. a quantum dot yields the equation G(V g ) = e2 k B T {n i } P eq ({n i }) δ np,γ p f(f({n i + p},n+1,v g ) F({n i },N,V g ) µ), (11) p the first sum is on electron-configurations on the dot, and the second sum is over levels on the dot. P eq is the Gibbs distributionfor the configuration P eq ({n i }) = exp( βf({n i },N,V g ) µ) {ni } exp( βf({n i},n,v g ) µ). (12) The δ-function ensures that the on-dot level is empty, and γ p is the tunneling coupling of level p, which is simply set to 1 for all p. f is the fermi-function, µ is the chemical potential of the source drain, F is the free energy of the dot for a given electron configuration, {n i }, the total number of electrons, and the voltage of the plunger-gate. We truncate the summation in the partition function in the denominator of eq.(12), by only taking into account contributions from the occupation numbers N min1 and N min2 which minimizes the free energy. In the constant interaction approximation,forexample,the freeenergyof aquantum dot is givenby F = (en)2 2C QD-gate +enαv g, which is a parabola in N, so terms far from N min1 and N min2 will not contribute significantly. Thus, truncating the sum yields the Gibbs distribution used in ivr P eq ({n i } N ) = exp( βf({n i },N,V g ) µ) exp( βf({n i },N min1,v g ) µ) +exp( βf({n i },N min2,v g ) µ) (13) Theivrcodetakesasinput the gate voltage,the ondotelectronnumber andthe associatedfree energy, which is calculated by SETE, for N min1 and N min2. The sums are then evaluated to yield the corresponding conductance. A conductance calculated using ivr can be seen in figure??. As in the case of the QPC, we have calculated the conductance keeping the charge distribution on the double quantum dot fixed at either (,2) or (1,1). For the SQD we get an approximal value of g/ḡ 1.4,also comparable to experiment (.9). The primary cause of the discrepancy is the assumption that hγ k B T,wherewe experimentallyfind hγ k B T. 11

12 REFERENCES.5 (a) g(,2).4 g (e /h) 2 g (1,1) Δg Δg (e /h) Δ V (mv) D Figure 11: Sensing quantum dot conductance calculated using SETE and ivr for two fixed charge states (1,1) and (,2) 6 Conclusion In this paper the concept of charge sensing using an adjacent quantum point contact and an adjacent quantum dot was introduced. The theory of Coulomb blockade for charge measurements of quantum dots, and Pauli blockade for measurement of singlet/triplet states of double quantum dots were introduced. In the paper by Barthel et al. we used a quantum dot for sensing the charge state of a double quantum dot, which resulted in a factor of 3 increase in sensitivity. By self consistently modelling the device, we showed that this difference is primarily due to screening effects from the 2DEG in the QPC as opposed to the isolated sensing quantum dot. Using the Pauli blockade, we also show in Barthel et al. that the SQD allows for almost an order of magnitude reduction in the time needed for a singleshot spin state readout. References [1] M. Büttiker. Quantized transmission of a saddle-point constriction. Phys. Rev. B, 41(11): , Apr 199. [2] M. Stopa. Quantum dot self-consistent electronic structure and the coulomb blockade. Physical Review B, 54(19): , November [3] C. W. J. Beenakker. Theory of coulomb-blockade oscillations in the conductance of a quantum dot. Physical Review B, 44(4): , July [4] M. Stopa. Coulomb oscillation amplitudes and semiconductor quantum-dot self-consistent level structure. Phys. Rev. B, 48(24): , Dec [5] R. E. Bank and D. J. Rose. Global approximate newton methods. Numerische Mathematik, 37(2): ,