2-D Simulation of Quantum Effects in Small Semiconductor Devices Using Quantum Hydrodynamic Equations *

We discuss the basis of a set of quantum hydrodynamic equations and the use of this set of equations in the two-dimensional simulation of quantum effects in deep submicron semiconductor devices. The equations are obtained from the Wigner function equation-of-motion. Explicit quantum correction is built into these equations by using the quantum mechanical expression of the moments of the Wigner function, and its physical implication is clearly explained. These equations are then applied to numerical simulation of various small semiconductor devices, which demonstrate expected quantum effects, such as barrier penetration and repulsion. These effects modify the electron density distribution and current density distribution, and consequently cause a change of the total current flow by 10-15 per cent for the simulated HEMT devices. Our work suggests that the inclusion of quantum effects into the simulation of deep submicron and ultra-submicron semiconductor devices is necessary.


I. INTRODUCTION
ince the advent of the integrated circuit in the late 1950's, the number of devices contained on a single chip has approximately doubled every three years as a result of the tendency for semiconductor devices to become smaller.As devices become small, some physical effects (such as quantum effects) which are not important for large devices may change the device operation significantly.The physical ef- fects inherent in the operation of ultra-small devices are based on the fact that the critical length (e.g. the gate length or the depletion length) becomes so small that it approaches the coherence length of the electrons that provide the operation, which suggests that such small devices must be treated as quantum mechanical objects [1][2][3][4].The coherence length; or the inelastic mean free path, can be more than 1/xm at low temperatures and as much as 0.1/zm at room temperature in high-quality heterojunction * Work supported by the Army Research Office.
structures.This is much larger than the gate length (~20-25 nm) of the smallest transistors that have been made [5][6][7][8].Due to the quantum interference within the devices, as well as between the devices, these physical effects may greatly modify the opera- tion of a single device as well as an integrated circuit.It is very important to fully understand these quantum effects on the device and circuit operations.
The classical semiconductor transport theory is based on the Boltzmann transport equation (BTE).
Numerous analytical and numerical methods have been developed for solving this equation in various semiconductor problems [9].The Monte Carlo method provides the most accurate and detailed solution but is limited in practical engineering appli- cations by its computational expense [10].As an alternative, a reduced description of the BTE based upon moment equations has played a significant role in advanced device modeling [11][12][13][14][15].As device feature sizes are reduced to the ultra-submicron regime, and sometimes with a narrow quantum well structure feature, device simulation faces new chal- lenges.Even the hydrodynamic model, upon which the moment equations are based and used to inves- tigate non-stationary and hot electron dynamics through the distinction of the momentum and en- ergy relaxation times, must be improved.Some ef- forts have been made in including quantum effects in the simulation of quantum well devices [16].These generally are a combination of a classical descrip- tion (either drift-diffusion equation or Monte Carlo method) with a quantum treatment in one dimen- sion normal to the heterojunction interface.However, this does not appear to affect device performance beyond mobility modifications.As the device structures are made smaller, the 1-D treatment of the 2-D electron gas is no longer accurate, for the quantum well is not uniform along the channel and a single quantum well model is not valid.One im- provement uses a set of quantum moment equations developed from a Wigner function prototype [17,18], which preserves explicit quantum correc- tions as well as the classical hydrodynamic model features.
This paper is a review of our work on the model- ing of quantum hydrodynamic equations and simula- tion of quantum effects in small semiconductor de- vices (including our work in [19-21]).In section II, we describe the formulation of the quantum hydrodynamic equations based upon the Wigner function equation-of-motion; In section III, we discuss the physics of quantum effects in small semiconductor devices; In section IV, we present the numerical technique and structure model for the simulation; And in section V and VI, we discuss our simulation results for MESFET and HEMT structures; finally, we will give our summary and conclusion.

II. QUANTUM HYDRODYNAMIC EQUATIONS
In principle, large-scale devices can be modeled classically, with an accurate description given by the Boltzmann transport equation.This equation time- evolves a complete single-particle phase-space dis- tribution.However, the accurate simulation of ultra-small devices requires quantum effects such as tunneling and quantum repulsion (complementary to barrier penetration) to be included.A quantum phase-space distribution function, analogous to the Boltzmann distribution function, is useful for use in the existing mathematical methods for the classical theories [22].A full quantum description, at the single particle level, can fruitfully be based on the Wigner distribution function (WDF) [23] fw(x,p,t) (27rh) 3 f_oo dy X eip" y/ hp X + " X - (1)   a transformation of the density matrix p(x,x')= (*(x)O(x')) (* and , are field operators in a position representation), which is a natural general- ization of the classical phase-space distribution function.Here x is the space coordinate, p is the momentum coordinate, and h is Planck's constant (reduced by 2r).The WDF satisfies the collisionless time-evolution equation where the operator 0 fw(x, p, t) is 0 "fw(x,p,t) sin .P fw(x,p + P,t), (3)   which can be derived from Schr6dinger's equation or Liouville's equation and has a form similar to that of the BTE, but with quantum corrections built-in by including both static potential V(x) and momen- tum non-locsalities into the equation.The Wigner distribution function has been successfully used in simulation of a resonant tunneling diode in one dimension [24,25], but it is not expected to be directly used for multi-dimensional device simula- tion because of its expense in memory storage and computation time.For a device simulation with a higher-dimensional description, the practical alter- native is the reduced description of the Wigner distribution function, i.e. its moments, which are very useful because the lowest several moments represent the basic physical quantities such as density, momentum, and energy of a physical system.The equation of motion of the distribution function then results in the hydrodynamic equations.Following the same procedure as that for the classical BTE, we have (with a relaxation time approximation) Ot (PP) ] ,r m (p:)0 with 7" W (ln) f dPfw(x,p,t)p", (7)   where n is an integer, E is the electric field, m* is electron effective mass, "r m is the momentum relax- ation time, 'w is the energy relaxation time.For n 0, Eq. (7) gives the density.However, the lowest three moment equations above are formally identi- cal to their classical analog under the relaxation-time approximation and do not contain explicit quantum corrections, which are expected [26].The key step to preserve quantum corrections in the lowest three moment equations relies on the method of decou- piing the energy equation from higher-order mo- ment equations and the treatment of the tensor in the momentum equation.In order to get explicit quantum corrections into the hydrodynamic equations, several different methods have been proposed [17,[27][28][29][30].We adopt the method in [17].By writing the WDF in the following form fw(X,p,t) =-hT L dyeip'y/ t (8) The first moment carries the information of the current density m*nv (p) =-f dppfw(x,p,t) The energy density can be derived from the second moment (p2) f dpp2fw(x, p, t) -[q(x)VZq*(x) 27q*(x) Vq(x) + q*(x)Vq(x)].(11) The second moment can be expressed in terms of the zeroth and the first moments, by using the following identities [25] + V2n 2Vqtt(s) VO(x), (Vn) 2 (2i) 2 --(p)2 4n V*(x) V0(x), (13)   we have the various moments can be evaluated.We remark that, in this equation and the following text, , and , are treated as wave functions instead of field operators.By taking the zero moment we get the density This leads to the zero temperature energy density 2 In n (15) n (pO) f dPfw(x,p,t) ,t(x)O(x). ( 9 which consists of drift kinetic energy density and quantum potential energy density.Now we see ex- plicit quantum correction enters the second moment but not the lower moments.Next, we need to derive the expressions of the tensor term (pp) and the third moment (p3).From definition, the tensor term is (pp) f dpPPfw(x,p,t) (16)   where pp is a tensor.By expanding in detail, we have 3 (17)   i,j--1 where and j are unit vectors.By performing the integration we get the following results (pp where (p)(p) and VV are tensors.This is the zero temperature tensor with explicit quantum correction included.Similarly, the third moment is (p3 After some tedious work, one find the above equation can be written as

21)
This is the exact equation for the third moment at zero temperature with quantum correction included.So far, we have derived the expressions of the first four moments and a tensor moment (Eqs. (9), (10),( 14), (18) and ( 21)) for the WDF at zero tempera- ture.Except for the definition of the WDF Eq. (8), there is no further restriction imposed on the distribution function.In other words, the deriva- tion of the moments above is exact under the defini- tion of Eq. (8).The apparently missing terms in the moments are the temperature terms.We now turn to investigate how the temperature terms should be included in these moments.
At high temperature limit, quantum corrections are negligible, thus a classical distribution function could be used in describing a physical system.Under drifted-symmetric approximation [or f(x, p, t) f(x, [p (p)/n]2, t)] of the distribution function, the classical second, third and tensor moments can be written as follows (p2) 3nm, kBTe + (p)2/n (22) (pp) nm*kBTe(ii + jj + kk) + (p)(p)/n, (23) (p3) (pZ)(p)/n + 2m, kBTe(P ) + q, (24) where drodynamic equations q ((p (p)/n)3), (25) is zero under the drifted-symmetric approximation of the distribution function, k n is the Boltzmann constant, T e is the average electron effective tem- perature.Extensive discussions on the term q can be found in the literature [14, 15], which regard this term as heat flow.We leave this term in the form of (25), since we believe it may represent more than just the heat flow and could be characterized by other methods [29, 30].
Comparing Eqs. (14), ( 18 and ( 21)with these latter three equations, one finds that the first lacks thermal energy and the latter set misses quantum corrections.It is physically acceptable that a combi- nation of the quantum version and classical version of moments should give us a correct set of moments which can be used in describing a quantum mechan- ical system.As a matter of fact, conceptually, the thermal energy density can be defined as the total kinetic energy density minus the drift kinetic energy density and minus the quantum potential energy density -nqE V(nkBT) (30) , ) w 2m. (p)2 + -knT + Uq. (36)   where the factor of 3 account for three dimensions.
Thus by combining the quantum and classical ver- sions of the moments, we arrive the following mo- ments with explicit quantum corrections included 1 h 2 (112) 3nm*knTe + --(tl) 2 -n 72 In n, (27)   n (pp) (p)(p)/n + nm*kTe(ii (p3) (pZ)(p)/n + 2m, kTe(P ) + q 2( 4 2(p). ( Substituting these moments into the hydrodynamic equations (4-6), we get the following quantum hy- Equations (30)(31)(32)(33)(34)(35)(36) are our complete quantum hy- drodynamic equations.We point out here: 1) The closure for the equations (the decoupling to the higher moments hierarchy) is done as generally as in the classical case.2) The quantum correction terms are exact under the definition of the WDF in (6), although one may argue that the formalism is for a pure state and lacks the ensemble average, it is still suitable for a general physical system, since we have no restriction on the form of the wave function.
Furthermore, Ancona and Iafrate [31], using the expansion of the potential in terms of the ratio of the thermal wavelength to the characteristic length over which the potential varied, obtained the same form of (34)with a reduction factor of 1/3.Grubin et al. [32], working with the density matrix, achieved the same reduction factor."Gardner [33], again fol- lowing the potential expansion, gets exactly the same quantum correction terms as ours except for the factor of 1/3.While the validity of the potential expansion is questionable, we notice that if we have a reduction factor of 1/3, the equations can not return to zero temperature equations correctly. 3) No clear quantum corrections have been included explicitly into the scattering terms and the term q, and there is no simple method that can introduce a proper quantum correction for the scattering term with even moderate computational effort.Any at- tempt to evaluate the scattering terms quantum mechanically should return to Levinson's formalism [34], which is the Wigner-Weyl transformation of the interaction terms of the Hamiltonian.A further investigation of the quantum corrections and their origin is discussed in [35].
Comparing to classical equations, the quantum hydrodynamic equations need more computational effort, needless to say much more than the drift- diffusion equations.To investigate the impact of the quantum corrections on the simulation with a clear physical picture and moderate computation, we con- centrated on the correction term of the energy, as the modification of the energy directly changes the density distribution which is proportional to the factor of e -(v+w)/kBr, quantum penetration (and repulsion) can be observed.By keeping the major correction for the energy, we approximate the quantum hydrodynamic equations to a simpler set of equations (with temperature representation) the electron is large, the quantum correction has less effect.But as the temperature is lowered, the quantum correction will become dominant.With Poisson's equation VZv q--( where V is the electrical potential, q is the absolute electron charge, is the semiconductor permittivity, and No is the doping concentration, Eqs. (37-40)  are used in our numerical simulations.
Ill. ON THE CONCEPT OF QUANTUM CORRECTION IN SMALL DEVICES In order to demonstrate how the quantum correc- tion (8)works and what this correction means physi- cally, we sketch a single barrier with an approximate electron density distribution in Fig. 1.With the quantum correction included, the total energy can be written as where Eel is the classical total energy and Uq is the quantum correction energy.The density is proportional to the Boltzmann factor p : exp[-(E, + Uq)]. ( For the potential barrier model in Fig. l(a), without Uq, the classical total energy is constant in and outside the barrier, which results in a constant den- where E is the electric field, T O is the lattice temper- ature and Tq is 2 Tq T+ -B Uq. ( This set of equations preserves all classical features except the heat flow property (we leave this for future investigation), and gives explicit quantum cor- rections.As h goes to zero, the equations return to the full classical hydrodynamic equations.From Eq. (40), one observes that if the thermal energy of Eel E0 nq nc (b) Quantum penetration and repulsion by a potential sity distribution (n)outside the barrier, and a con- stant zero density inside the barrier.The energy discontinuity creates a density discontinuity at the interface of the barrier, which is quantum mechani- cally incorrect.By including quantum corrections, the quantum potential energy serves to smooth the actual potential, and modifies the total energy to a smooth transition at the interface, which results in smooth density (nq) change at the interface transi- tion.In Fig. l(b), we illustrate the modification of the density distribution by inclusion of the quantum corrections.We distinguish these quantum effects from those of transverse quantization of the electron in a MOSI'ET or HEMT channel, an effect which does not affect overall device performance [10, 16], and which requires the treatment of each discretized energy level individually.However, since with cer- tain ensemble statistics [28][29][30], the quantum correc- tions still take essentially the same form, we expect that certain summation effects of the transverse quantization levels may be included in the formula- tions.Besides the abrupt barrier example, the quantum correction arises from the change of the density (the density is reduced when the second derivative of the local log density is negative, and the density is enhanced when the second derivative of the local log density is positive), one can imagine that any- where a large density change occurs in a short distance, the actual density distribution will be much different from that of a classical picture, even the classical picture gives a continuous density distribu- tion as it occurs in small semiconductor devices.

IV. NUMERICAL IMPLEMENTATION
We use finite difference methods to discretize the quantum hydrodynamic equations and the Poisson's equation in a two-dimension structure (Fig. 2), which is suitable to any planar device structure.The differ- ence schemes used in the simulation are described Source Gate Drain FIGURE 2 Device structure for simulation.
as follows: a central difference is used for Poisson's equation.In the continuity equation, the gradient term V. (nv) is discretized by using a second-up- wind method (or donor cell method) [36], which possesses both conservative and transport properties.In the momentum equation, a central differ- ence is used for the term V(nknTq) at half-grid points, and a first-upwind method is used for the term (v.V)v.In the energy equation, two terms need to be discretized in space.These are v.VTq and V. (vTq).The temperature pressure term V.
(vTq) is in a conservative form, so the donor cell scheme is suitable for this term.The convective term v. VTq doesn't have a conservative property, and a first-upwind difference is implemented for this term.
A forward-time difference is adopted for the con- tinuity equation, and an integration and expansion method [15] is used for the momentum and energy equation.Both uniform and nonuniform meshes are used in our simulation.Grid sizes are chosen to satisfy the constraint of the extrinsic Debye length, and a time step is chosen by considering the Courant-Freidrichs-Lewy stability condition.The re- laxation times are computed from the velocity-field and energy-field relations [37] which are Monte Carlo simulation results for bulk material.For GaAs material, it is equivalent to convert a multi-valley system to an effective single valley model by this approach.While not strictly accurate, it is suitable for examining the impact of the quantum correc- tions, which is the aim of the present work.Dirichlet boundary conditions are applied to the contacts and Neumann boundary conditions are used where the perpendicular current flow is zero.An incomplete Cholesky conjugate gradient method [38][39][40] is used to solve Poisson's equation.Gauss-Seidel iteration is applied to the quantum moment equations.After the initial guess for the potential, density, temperature and velocity, the various equations are solved successively for the corresponding quantities.
All devices simulated here have the structure shown in Fig. 2.However, the interface for het- erostructure devices needs to be considered in more detail.For an ideal AIGaAs/GaAs interface, the transition of the conduction band from one material to another is abrupt, and characterized by a distinct band offset.At an equilibrium condition across the interface, due to the diffusion of electron, a field is generated that depletes the donors in the A1GaAs bulk and creates an inversion layer of electrons at the interface on the GaAs side.The potential distri- bution, coupled with the band offset, forms a barrier on the AIGaAs side and a well on the GaAs side.
Electrons in the quantum well are prevented from drifting into the AIGaAs by the band offset.Elec- trons can climb over the potential wall only if they have kinetic energy comparable to the conduction band discontinuity.From the interface structure, one may realize an obvious problem in using differ- ential equations in the simulation of this kind of device.The problem is that the partial differential equations can not normally handle the discontinuity, where an infinite field occurs, and this must be carefully handled by the method of discretization.The problem arises not in Poisson's equation, but in the moment equations themselves.The step in the potential can be a source of instability in the computation.While an abrupt change in the transition from one material to another may be ideal, there is a certain transition region to be expected [41].Thus, an assumption of a narrow transition region can be made, although the estimate of its extent is difficult to determine.The energy transport equation is the main one in the sense that the electron kinetic energy determines the transport of the electrons across the interface.The assumption of a small transition region does not create a significant error.
In our simulation, we assume a 0.3 volt potential drop across a 4 nanometer region at the interface for a AIGaAs/GaAs HEMT, and a 0.18 volt poten- tial drop across a 3 nanometer region for a SiGe device (in this case, a double heterojunction is treated).While this seems to be quite wide, it is regarded as a statistical average over the actual transition region and the wavefunction decay at this interface [42], an approach used extensively in staffs tical physics.However, a further reduce of the grad- ing width at the interface may be considered.
The conduction path in a quantum well device is from the source contact down to the two-dimen- sional electron gas, and then through the 2-D con- duction channel to the drain region.The source and drain regions are heavily doped, in general.Due to this, transport of electrons across the hetero-inter- face in the source/drain regions is probably by tunneling through a very thin potential barrier, in which effectively no discontinuity at all is experi- enced.For consistency with this concept, we assume that the interface discontinuity gradually disappears toward the contact regions.Although this is a con- ceptual problem, it does not affect the quantum effects that appear in the gate region and is a model of the "ohmic" contact.In order to give a clear picture of the effects of these assumptions, we plot the potential profile of a 24 nm gate length AIGaAs/GaAs HEMT device in Fig. 3 (we plot voltage rather than energy, so that the potential profile under the gate appears inverted), where we can see the transition of the potential at the inter- face and the reduced potential barrier height to- wards the source and drain regions.
The results we summarize here are for MESFET devices with gate lengths from 24 nm to 96 nm, AIGaAs/GaAs HEMT devices with gate lengths from 24 nm to 56 nm, and a modulation-doped Si0.TGe0.3/Si/Si0.TGe0.3SiGequantum-well device with a gate length of 0.18 /xm.For the MESFET, typical doping in the channel is 1.5 1018 cm -3, and a semi-insulating substrate is included.The FIGURE 3 The two dimensional potential profile of a HEMT.Lg 24 nm, V d V, 1.5 V.The interface is at 40 nm.doping in the HEMT AIGaAs is also 1.5 10 TM cm -3.Much higher doping (3.5 10 TM cm -3) is used for the modulation-doped SiGe device in the top Si0.7Ge0.3layer.The lattice temperature is taken to be 300 K in all cases.The total simulation area is 0.36 /zm 0.1 /zm for MESFETs and AlGaAs/GaAs HEMTs, and 1.0 /zm 0.095 for the modulation-doped SiGe device.The thick- ness of the AlGaAs layer is 39 nm, and the thickness ofthe top SiGe layer is 19 nm.The strained Si channel is 18 nm.

V. GENERAL DEVICE CHARACTERIZATION
As we mentioned before, the gate lengths of the device we simulate ranges from deep submicron to ultra-submicron, which allows us to understand the small device operation and the effects of the quantum corrections on the device characteristics.We discuss the general device characteristics in this sec- tion, and leave the quantum effects to the next section.
Plotted in Fig. 4 and Fig. 5 are the I-V character- istics of 24 nm gate length GaAs MESFET and   AIGaAs/GaAs HEMT, respectively.The gate volt- age runs from 0 V to -2.5 V, in steps of -0.5 volts.For both devices, the interface (channel to substrate for MESFET, and A1GaAs to GaAs for HEMT) is located 39 nm from the gate.The characteristics of the devices are quit normal, and saturation of the current is obtained.Pinchoff is reached in the MESFET at a gate voltage of -2.5 V, and is deter- mined by checking the density distribution in the Lg 0.024 l.tm 0 0 Vds (V) I-V characteristics of a 0.18 /zm SiGe HEMT channel.The remaining large current at pinchoff is due to substrate current as the electrons are pushed into the substrate.At least another 1.0 V of negative gate bias is required to eliminate this parasitic sub- strate current.The I-V characteristics for a 0.18 gate modulation doped SiGe HEMT device are il- lustrated in Fig. 6 and the gate biases are 0.7, 0.5, 0.2, and 0 volts, respectively.The small thickness of the top SiGe layer (18 nm) provides a normally- off device, since a Schottky barrier height of 0.9 V (Pt on Si) leads to an estimated depletion width of 18.4 nm.Good saturation with a drain conductance of 4.6 mS/mm at the gate voltage of 0.5 V is obtained.Approximately the same current level was found in a 0.25/zm device.Obviously, for this larger gate length device, the current pinchoff is much easier to achieve.
The transconductance is 450 mS/mm for the MESFET, 480 mS/mm for A1GaAs/GaAs HEMT, and 300 mS/mm for the SiGe device discussed above.As for the depletion mode AIGaAs/GaAs HEMT device, the transconductance behavior is slightly different from the general picture of a HEMT.Fig. 7 shows the transconductance versus gate voltage for the 24 nm gate HEMT, at a drain bias of 2.0 V, in which the transconductance de- creases linearly from about 480 mS/mm to some 150 mS/mm as gate bias becomes more negative.
For most experimental HEMT device operation, one expects that the current flow will be mainly confined to the 2-DF electron gas channel, and the transconductance should have a peak value as the gate bias is varied, with the transconductance also becoming smaller as the gate voltage becomes posi- tive, where the 2-D gas channel is fully open and the gate bias loses control of the current flow in the channel [43].For the thickness of the AIGaAs, and the doping density, used here, it is certain that there is significant current flow through the AIGaAs for any gate bias larger than -1.0 V for a 24 nm gate length device.Although the mobility in the AIGaAs is low compared to that in the 2-D gas channel, the size structure used here is such that the gate does not really lose control of the channel charge within the range of biases examined.Rather, in this simula- tion, as the gate voltage begins to lose control of the charge in the 2-D gas, the conduction through the A1GaAs increases sufficiently rapidly that the transconductance continues to increase.We would expect that the transconductance would eventually decrease at positive gate voltages.We investigated the effect of gate length on the transconductance for the MESFET and A1GaAs/ GaAs HEMT by changing the gate length (with the doped layer depth and the doping concentration fixed).For the MESFET, the transconductance is evaluated at a drain voltage of 2.0 V. Fig. 8 illus- trates the transconductance characteristics in the range of gate length from 24 nm to 96 nm, for two active layer depths (a 30 nm and 39 nm).The transconductance for a 30 nm has a maximum value of about 800 mS/mm at a gate length of 60 nm.As gate length decreases from 96 nm, the transconductance increases until the peak is reached and then decreases with further decrease of the gate length.This transconductance behavior can be ex- plained by velocity overshoot effects for the increas- ing (longer gate length) transconductance region and small aspect ratio effect for the decreasing (shorter gate length)transconductance region.The transconductance for a 39 nm decreases mono- tonically for the entire range as gate length de- creases.By comparing these two curves, we see that the small aspect ratio effect begins at larger gate length for thicker active depth devices, an expected physical result.For the AIGaAs/GaAs HEMT, the peak transconductance observed for gate lengths from 24 nm to 56 nm (the thickness of AIGaAs layer is 39 nm) is shown in Fig. 9.These are all evaluated at Vg =-0.5 V and Vd 2.0 V.The monotonic decrease in transconductance with decrease of the gate length is directly related to the small aspect ratio (Lg/a) [5, 7, 44].We note that the devices are not scaled; only L g is varied in the simulation.In all cases, Lg/a < 1, rather than 3-4 that is normally used in longer-channel HEMTs, and this fact alone is felt to lead to the decreasing transconductance.The transport property of electrons in the conduc- tion channel is critical for the device performance.To see how the velocity changes under the influence AIGaAs/GaAs HEMT device at V a 2.0 V.The longitudinal velocity in the channel of a GaAs =24nm, a=39nm, V a=2V,Vg.= -1.5V. of the potential barrier induced by the gate, we plot the longitudinal velocity as a function of position along the conduction channels in Fig. 10, 11 and 12, for a 24 nm gate GaAs MESFET, a 24 nm gate AIGaAs/GaAs HEMT and a SiGe HEMT, respec- tively.Velocity overshoot is obvious, as the peak velocities are much larger than either the saturation velocity or the peak velocity in the velocity-field relations for all cases.The reason for velocity over- shoot in SiGe device is due to the high mobility of the electron in the strained Si layer with carrier transfer out of the lower set of valleys and into the upper set of strain split valleys [45].The velocity overshoot is thought to be important to achieve the high transconductance for these devices.The first velocity peak in the plot of SiGe device is due to the model we used for the change of interface disconti- nuity (as describe in section III), although it is not practical, it does suggest that the structure can in- crease the electron velocity between source and Position (nm) FIGURE 11 The longitudinal velocity in the channel of a AI-GaAs/GaAs HEMT.Lg 24 nm, a 39 nm, V a 2 V, Vg -1.5 V.
Lg=O.18 ktm Vg=0.5 V HEMT.Lg 0.18 m, V d 1.5 V, Vg 0.5 V. gate, which in turn will raise the average velocity through the device and enhance the device performance.It is interesting that there is no similar velocity peak for the AIGaAs/GaAs HEMT close to the source, although the velocity in the later de- vice does increase faster than that in the GaAs MESFET, which implies that with the structure model the double heterojunction creates larger lon- gitudinal electrical field in SiGe device than the single heterojunction does in the AIGaAs/GaAs HEMT.For the GaAs MESFET and AIGaAs/GaAs HEMT devices, the peak velocity exceeds 10 8 cm/sec, which is surprisingly high (near the band structure limit).However, if one notices that the velocity is actually built up within a 60 nm distance, one may realize that quasi-ballistic transport (the Position (nm) FIGURE 13 The longitudinal velocity in the channel of a GaAs MESFET for different y positions (y is the direction into the device from the surface).Lg 24 nm, a 39 nm, V d 2 V, Vg -1.5 V.
electron experiences few scattering events during the fly) of the electron is quite possible as the length scale is comparable to the electron mean free path.We can estimate the velocity from Newton's law.For a relaxation time around 10 -13 s, and with an elec- tric field of 105 V/cm, an electron with the effective mass used here is able to reach a velocity of 108 cm/sec on this time and length scales.Thus, within the accuracy of the relaxation-time approxi- mation, large velocity overshoot is possible.We re- mark, however, that the actual simulations use energy-dependent relaxation times that fully match detailed ensemble Monte Carlo results; the previous argument is one of justification only.We also re- mark that these results have good validity, as it has been shown that hydrodynamic equations agree well with Monte Carlo results for transient response when the models are the same [46].
To see the relations among velocity, field and temperature, we take the MESFET as an example.
Since the MESFET has a wider conduction channel, it would be informative if we knew how the velocity changes for different transverse positions.To see this, we plot the longitudinal velocity as a function of position along the channel for different trans- verse positions in Fig. 13, for a 24 nm gate GaAs MESFET (although simulations have been done for many gate lengths and different devices, the details presented here are limited to this later value), for the bias condition of V d 2.0 V and Vg -1.5 V.
The active-layer-to-substrate interface is at 39 nm from the gate.Four velocity curves are plotted at here for different positions which have distances of 8, 6, 4, 2 nm from the interface, respectively.By inspecting the density distribution across the chan- nel, these positions cover the entire channel width on the active layer side.Corresponding longitudinal electric fields are shown in Fig. 14.For the axis direction we choose, the sign of all the field curves are negative.The peak field for each curve reaches 105 V/cm just a few nanometers beyond the gate metal edge on the drain side.The field is very low over a long path from the source to the gate and has a very sharp increase in the gate region.This is the key to the very high velocity overshoot, for the driving force increases quickly in a short distance.The decrease of the electric field near the drain is fast but the field remains relatively strong throughout the region from the gate to the drain.In Fig. 15, four curves of effective electron temperature are depicted which take corresponding space positions to the velocity curves under discussion.The temperature is close to the lattice temperature from the source to the center of the gate, then it rises quickly to above 3000 K over the next 60 nm.The low temperature in the source-to-gate region provides the key condition to the high velocity overshoot, since it accounts for less momentum scattering.By r'_,5.0103L y=35nm V 33..'.Position (nm) FIGURE 16 The velocity along the channel of a GaAs MES- FET for different gate length devices.Here, the velocity is plotted for y 39 nm.Position (nm) FIGURE 17 The electric field along the channel of a GaAs MESFET for different gate length devices.Here, the field is plotted for y 39.   checking the total energy at the velocity peak posi- tions, one finds that the values are about 0.38 eV, which roughly corresponds to the energy separation from L valley to F valley.This means that after the total energy exceeds the energy separation of the two valleys, more intervalley scattering occurs and the velocity is then reduced.The fact that the result is a somewhat higher energy than the actual A rL is though to be due to the use of a constant effective mass, which leads to an underestimation of the actual relaxation rates.
The longitudinal velocity profiles for MESFETs with several different gate lengths are shown in Fig. 16 (with 2 nm distance from the interface), with the same bias condition above.It can be seen that the peak velocity reached by the carriers is slightly higher in the 36 nm gate length device than in the 24 nm gate length device.This also corresponds to a higher peak electric field (Fig. 17).In essence, the peak velocities reached are limited by the short acceleration lengths in the short gate devices, a result predicted earlier [7, 47].In nearly all the cases, however, it is clear that the velocity rises almost linearly throughout the region under the gate, so that the overshoot region is essentially de- fined by the gate length of the device.

V. THE QUANTUM EFFECTS IN SMALL DEVICES
The effect of the quantum correction on the device characteristics can be found by comparing the com- puted results in the full model with those obtained when h 0, i.e., in the semiclassical hydrodynamic model.Here, we compare these differences in any quantity Q as dQ Qleuli model Qlno quantum terms- (44) As the gate length of the device is reduced to the length scale simulated here, we expect that quantum effects such as barrier repulsion and penetration can be observed.We first examine quantum effects on the density distribution.Fig. 18 shows a two-dimen- sional plot of the density difference between results obtained with and without the quantum potential correction for a 24 nm gate MESFET.The bias condition is 2.0 V on the drain and -2.5 V on the gate.The increase of the density on the inside of the gate depletion region edge and decrease of the density on the outside of the gate depletion region edge are evident.The same behavior appears at the interface of the active layer and substrate on the source end.This shows that barrier repulsion and penetration do occur.The modifications of the den- sity distribution due to the quantum effects are as large as 4 per cent in the channel and about 8 per cent at the interface of the active layer and substrate in the source side.The equivalent interpretation to the modification of the density distribution is that the quantum corrections serve as a quantum potential which acts to smooth the actual potential, especially at potential barrier edges, which in turn smoothes the electron density distribution.From Fig. 18, one could observe that two factors make the quantum effect important: one is a sharp potential change which also results in a sharp density change, another is low electron thermal energy.We can see that the quantum effect at the interface of the active Z Source FIGURE 18 A 2-D plot of density difference between the results of with and without quantum correction for a 24 nm gate GaAs MESFET device.Grid numbers from gate to substrate FIGURE 19 Density difference between the results of with and without quantum corrections across the channel for a 24 nm gate GaAs MESFET device.
without .q0 20 40 60 80 Distance from gate (nm) 100 FIGURE 20 Electron density distribution across the conduction channel as a function of distance from the gate contact for a AIGaAs/GaAs HEMT.
layer and substrate decreases from source to drain and disappear at the drain side where the electron effective temperature is high, which reflects the correct physical phenomena.
In order to have more clear observation, we check the density distribution across the conduction chan- nel under the gate for the three kind of devices.The electron density across the conduction channel for the 24 nm MESFET is illustrated in Fig. 19.Here, the total electron density and the density difference are plotted, under the same bias conditions as in Fig. 18.The interface of the active layer and sub- strate occurs at 39 nm from the gate.From the density difference curve, the density penetration to- ward the gate and substrate create two peaks, and repulsion from the gate and interface barriers re- suits in the valley.The net effect is that the electron density distribution across the channel is broadened, which corresponds to a smoothing of the actual potential by the effective quantum potential.Fol- lowing the same concept, one may expect that the same distribution behavior should be found in AIGaAs/GaAs HEMT and SiGe HEMT.However, this is not quite true.In Fig. 20 and 21, we plot the density distribution across the channel under the gate for a AIGaAs/GaAs HEMT and a SiGe HEMT, respectively.Because there is a larger den- sity change along the conduction channel (from source to drain) of the 2-D electron gas, the quan- tum effect in the channel behaves differently from that for the MESFET case.As can be seen in Fig. 20, there are two peaks across the channel, one in the A1GaAs side that reflects a parasitic MESFET effect, and the expected one in the 2-D gas.For the SiGe HEMT, the density in the top SiGe layer is essentially depleted.The density peak is reduced in the AIGaAs (also in the SiGe top layer) and in- creased in the quantum wells when the quantum corrections are included.Because of the double heterojunction, the SiGe device has a fatter density distribution.However, the major portion of the den- sity is still within about 10 nm for both cases, and the density peak is always closer to the gate.The effect of the quantum corrections on the two peaks can be explained as follows: (1) the quantum correc- tions normal to the interface would soften the inter- face potential, consequently raising the potential minima in the A1GaAs or the SiGe (a natural effect due effectively to quantization in this potential mini- mum), and lowering the density, and, (2) the quan- tum corrections along the channel increase the peak electron density in the quantum well (a reduction in the depleting effect of the gate potential).The total effect of the summation of these two corrections shows that the second dominates in the quantum well.In the..AIGaAs or the SiGe, the electron den- sity distribution under the gate (along the channel) is relatively fiat, when compared to the direction normal to the interface.The interface potential is broadened (towards the gate) and quantum correc- tions along the channel direction are small, and these effects result in a lower density peak.Due to the high electron density in the 2-D gas, the quantum correction along the channel direction is domi- nant, so the net effect is an increase of electron density, resulting in a fatter and higher peak.The relatively large change of the peak density in the A1GaAs is composed of both the local quantum correction and potential modification induced by the change of the electron density in the 2-D gas, which implies that real space transfer occurs.A detailed component analysis may be needed to fur- ther prove this behavior.Another obvious effect is that due to the higher density (from the higher doping in the SiGe) in the quantum well for the t ., ,,,ith q.c.
" ' , , ., withut q'c" 0 20 40 60 80 100 Distance from gate to substrate (nm) FIGURE 21 Electron density distribution across the conduction channel as a function of distance from the gate contact for a SiGe HEMT.
SiGe device, quantum effects introduce a larger density increase in the SiGe device than that in the A1GaAs/GaAs HEMT device.This large modifica- tion of the electron density in the SiGe device with relatively large gate length simulated here was not expected.However, by inspecting the density distri- bution along the channel, one can find that, al- though the gate is relatively long, a rapid density change occurs at the gate end close to the drain contact within a region much shorter than the gate length.In light of the quantum correction depend- ing on the density change, the modification of the density by the quantum effects is understandable, since the electron density is high and the density change occurs in a short distance.
The modification of the density distribution by the quantum effects in turn changes the current density distribution, and consequently the total cur- rent flow through the conduction channel.Although the total current change for the MESFET is quite small (the device is not a quantum device), the change for AIGaAs/GaAs HEMT and SiGe HEMT is appreciably large, which results in a 10 per cent and 15 per cent total current increase, as we plotted in Fig. 22 and Fig. 23, respectively, with the quantum correction included in the simulation.This certainly suggests the importance in including quantum cor- rections in small device simulation.
Although the transport of electrons is still domi- nated by classical transport, i.e., most of the elec- trons pass the potential barrier under the gate by gaining higher energy, the quantum effects do con- tribute significant changes.The increase of the total current, especially the increase of the peak electron density in the channel for the HEMTs and the current density increase toward the gate due to the density penetration into the gate barrier for MESFETs, suggests tunneling processes (through the depletion region induced by the gate) may occur in the device operation.However, there are two facts suggesting that the current and electron den- sity increase should not be interpreted as tunneling.It is well known that the tunneling current should exponentially decrease with an increase of the potential barrier.Thus, one expects that tunneling will become smaller as we increase the drain voltage or decrease (toward more negative values) the gate voltage.The depletion barrier will be widened in both cases, and will increase in amplitude, both of which should lead to a significant reduction in the tunneling current.This reduction is not observed in our present simulation.As an example, we plot the drain current against gate voltage in Fig. 24  Vd (V) FIGURE 22 Drain output characteristics for a 24 nm gate AIGaAs/GaAs HEMT at -1.5 V.These curves illustrate the effects studied.without q.c.
-2.5 -2 -1.5 -1 -0.5 0 0.5 vg (v)   Drain current variation with gate voltage for a 24 nm gate A1GaAs/GaAs HEMT at l/a 2.0 V. Two curves show the results with (dashed) and without (solid) quantum corrections included.Quantum effects are relatively insensitive to the gate bias, suggesting that the increase is not due to tunneling.Drain output characteristics for a 0.18 /xm gate FIGURE 23   SiGe HEMT at Vg 0.5 V.These curves illustrate the effects studied.
these effects is obviously insensitive to both gate and drain bias.This conclusion differs from early sugges- tions based upon the experiments [5].
As the physical quantities (density, velocity, tem- perature... ) are solved self consistently, any change of one quantity is related to other quantities.In Fig. 25, we plot the velocity difference between the velocities with and without the quantum corrections for various space positions for a 24 nm GaAs MESFET.It may be seen that the peak velocity achieved (the value near the drain edge of the gate) is lowered by the quantum potential.The physics in the velocity change is that the quantum potential modifies the electric field distribution, and thus results in the change of the electron temperature 24 nm gate AIGaAs/GaAs HEMT, for simulations both with and without the quantum corrections.As can be seen, the current increase due to quantum effects is relatively insensitive to the gate voltage.A similar property can be found in Fig. 22, in which the current increase due to quantum effects is rela- tively insensitive to the drain voltage.These results lead us to the conclusion that, if there is any tunnel- ing, its effect must be small.The reason the quantum effects are relatively insensitive to both the gate voltage and drain voltage is that the quantum effects that make major contribution to the increase of the total current primarily soften the gate depletion potential and give a higher electron density distribu- tion along the channel.The current increase due to FIGURE 25 Longitudinal velocity difference in the channel of a 24 nm gate GaAs MESFET for different y positions, a 39 nm, V d=2V,V= -1.5V.which subsequently causes the change of the relax- ation times and the velocity.The quantum effect on the effective electron temperature has a kind of anti-symmetry with respect to the gate center.This is shown in Fig. 26.As a result of the smoothing effect of the quantum potential on the actual poten- tial, the temperature is reduced around the source side of the gate.At the other side of the gate, due to the broadening of the electron accumulation hill, as for the curve at y 39 nm, the temperature distri- bution is .broadened to some extent.The major effect on this side is that the higher temperature is pushed towards the source direction.This leads to a reduction in the peak carrier velocity.On the other hand, there is a small increase of the velocity near the drain.This latter is a consequence of the fact that the electric field maintains a high value further from the gate.

VII. SUMMARY AND CONCLUSION
We present a set of 3-D quantum hydrodynamic equations developed from the Wigner function equation-of-motion.Explicit quantum corrections are build into these equations by using quantum mechanical expressions of the moments of the Wigner function.This set of equations returns to full classical hydrodynamic equations as h goes to zero, and retain exact form of the zero temperature equations correctly as temperature goes to zero.
Although caution needs to be taken for the lack of ensemble statistics in the derivations, the quantum correction forms are essentially the same as that obtained from other methods, and the use of the equation does give a correct physical effect.
The relation of the mathematical form of the quantum corrections to its physical implication is properly interpreted.The quantum potential energy is introduced from the change of the density distri- bution, which is also implicitly related to the actual potential profile.Quantum effects tend to smooth the actual potential and density changes.A clear picture is that it tends to reduce the peak and raise the valley of a density distribution.Due to these changes, device operation certainly will be affected.
We have performed a simulation of various small semiconductor devices by using this set of equations.The simulation reveals good device characteristics such as transconductance for all devices simulated here.Velocity overshoot observed in the simulation is important in achieving large transconductance for these devices.Our simulation predicts that very large velocity overshoot can be obtained for ultra-short channel devices, if a low field can be maintained from the source until very close to the gate, since under this condition, the electron could retain a low temperature and thus a higher mobility up to the gate center, where a very high electric field acceler- ates the electron to very high velocity.
As expected, quantum penetration and repulsion occurs in all simulated devices.The effect causes changes in both density distribution and total cur- rent flow.With the inclusion of quantum correc- tions, the density change in the 24 nm MESFET ranges from 4 per cent to 8 per cent, but the total current is essentially not changed, for the change of the density across the channel is averaged to small.The quantum effects become strong along the 2-D electron channel under the gate for HEMT devices.As the total quantum corrections consist of the effects along the channel direction and perpendicu- lar to the channel direction, the effect along the channel direction dominates under the gate, which is the result of higher density distribution away from the minimum under the gate.Due to the strong effect, the total current flow increases by 10 per cent for a 24 nm gate AIGaAs/GaAs HEMT and 15 per cent for a 0.18 /xm gate SiGe HEMT.These results suggest that the inclusion of quantum corrections in deep submicron and ultra-submicron devices is nec- essary.
FIGURE barrier.

5 FIGURE 7
FIGURE 7  Transconductance as a function of gate voltage for a 24 nm gate AIGaAs/GaAs HEMT device at V d 2.0 V.

FIGURE 9
FIGURE 9 Transconductance as a function of gate length FIGURE 10 MESFET.Lg FIGURE12 The longitudinal velocity in the channel of a SiGe

FIGURE 15
FIGURE15 Effective electron temperature in the channel of a GaAs MESFET for different y positions.Lg 24 nm, a 39nm, V a=2V,Vg= -1.5V. FIGURE24

FIGURE 26
FIGURE 26 Temperature difference in the channel of a 24 nm gate Gs MESFET for different y positions, a 2V,= -1.5V.