ANALYTICAL MODEL OF ELECTRIC FIELD IN HETEROJUNCTION REGION OF HFET STRUCTURE

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1 Journal of Optoelectronics and Advanced Materials Vol. 7, No. 3, June 2005, p ANALYTICAL MODL OF LCTRIC FILD IN HTROJUNCTION RGION OF HFT STRUCTUR P. M. Luki *, R. M. Ramovi a, R. M. Šaši Faculty of Mechanical ngineering, University of Belgrade, Kraljice Marije 16, Belgrade, Seria and Montenegro a Faculty of lectrical ngineering, University of Belgrade, Bulevar kralja Aleksandra 73, Belgrade, Seria and Montenegro Faculty of Technology and Metallurgy, University of Belgrade, Karnegijeva 4, Belgrade, Seria and Montenegro A new analytical model of electric field in heterojunction region of heterostructure field effect traistor (HFT), is presented. lectric field dependences on surface deity of twodimeional electron gas and surface deity of ionized acceptors in uffer layer are included in developed model. The expression of the electric field inteity is given y using the new correction coefficient. The new coefficient emphasizes more precisely extremely asymmetric spreading of the electric field in the vicinity of a heterojunction. Proposed model is simple and it can e straightforwardly implemented. It can e applied to quite different types of HFTs. The results derived from simulatio ased on the exposed model are in very good agreement with the already known ones, availale in literature. (Received April 25, 2005; accepted May 26, 2005) Keywords: Heterostructure Field ffect Traistor (HFT), lectric Field, Analytical Model 1. Introduction Heterostructure electron devices, realized with thin crystalline layers of different semiconductor materials, are spearheading the drive toward faster, smaller and lower-power electronics [1]. The mentioned layers can e as thin as a several lattice parameters, thus quantum effects ecome dominant. In the Heterostructure Field ffect Traistor (HFT), quantum effects are respoile for confinement of carriers [1-6]. Block diagram of HFT structure is presented in Fig. 1. source gate drain 0 y n+ heavily doped donor layer spacer layer 2DG l heterojunction x unintentionally slightly doped uffer layer sustrate Fig. 1. Block diagram of HFT structure. * Corresponding author: plukic@mas.g.ac.yu

2 1612 P. M. Luki, R. M. Ramovi, R. M. Šaši Heavily doped donor layer is made of a semiconductor material with larger and gap, while unintentionally doped uffer layer is made of a semiconductor material with smaller and gap. As a coequence, a quantum well is formed at the uffer side of the heterojunction. lectro confined in a quantum well form a Two-Dimeional lectron Gas (2DG). Due to the spacer layer, which is etween the donor layer and the uffer layer, electro moility, or in fact the speed of HFT, is increased ecause scattering on donor impurities is decreased [1-6]. In the process of HFT electric field model developing, it is started from the theory of usual silicon electron devices modeling, taking into account all specifics which are referred to heterostructure devices. Bearing in mind significant unisotropic nature of HFT structure, it is usefull to coider separately vertikal v and lateral l component of the efective electric field (Fig. 1). 2. Model lectro in a HFT oserve real electric field as well as and structure of the device. To descrie this, quasi electric field concept is introduced. ffective vertical quasi-electric field v, which corresponds to the effective vertical quasi potential φ, is: ( x) ( x) dφ = 1. (1) qe dx In equation (1), q e >0 is the electron charge and x is the normal distance from the heterojunction (Fig. 1). If the quasi potential is modeled in the following form: φ( x) = ; x < 0 ( x) = φ ( 1 exp( a x) ); x 0 φ ( x) =, (2) φ 0 with a and φ 0 eing adjustale parameters, the model for the vertical effective quasi electric field will e: ( x) = a exp( a x) ; x 0 v φ 0. (3) From Mc Laurent s expaion of equation (3), y neglecting all ut the first two series terms, it is otained: ( x) = a ( 1 a x) ; x 0 v φ 0. (4) The first term in equation (4) is cotant, or etter to say it does not depend on distance from the heterojunction x. It mea that in the area close to the heterojunction (when x 0) vertical quasi electric field ecomes cotant. v = a φ 0 ; x 0. (5) Bearing in mind that HFT layers are very thin, it can e assumed that vertical component of the effective quasi electric field is cotant in the traistor channel. This coresponds to the assymetric triangular aproximation for the quantum well: φ(x)=+ for x<0 (in the donor, or in fact spacer layer) and φ(x)=q e Kx for x 0 (in the uffer layer), itead of real potential φ(x). That is the crudest approximation in this model, ut it tur out to e sufficient in CAD (computer aided design).

3 Analytical model of electric field in heterojunction region of HFT structure 1613 The expression for the vertical electric field can e determined starting from the Gauss Low, earing in mind that the carriers are confined in a very thin region, practically flat, located very close to the heterojunction [3-6]. Gauss Law suggests the following expression v (n s )=q e n s /ε (assuming that the field exists only on one side of the heterojunction), or v (n s )=q e n s /2ε (assuming that the field exists on the oth sides of the heterojunction), where n s is the 2DG surface deity of the carriers that are confined in the potential well and ε is the permittivity of the uffer semiconductor. With the correction for surface deity of ionized acceptors in the uffer layer n =x N, where N is ionised acceptors cocentration in the uffer layer, following model can e used for the vertical electric field: α. (6) ε The correction for surface deity of ionized acceptors in the uffer is important if this deity is to high or if 2DG surface deity is to low. In equation (6) adjusting coefficient α [1/2; 1]. Oviously, for α=1/2 the value of the electric field would e underestimated, and for α=1 the value of the electric field would e overestimated. The actual value falls etween these two limits. The vertical electric field is dominantly spread in the uffer layer, and can e expressed as:. (7) ε The expression of the electric field inteity in the potential well is given y a new correction coefficient α=0.88. The new coefficient emphasizes more precisely extremely asymmetric spreading of the electric field in the vicinity of a heterojunction, or in fact in the channel. As a coequence, this model appears to e more accurate than the other existing ones. Tale 1. Different models of the HFT vertical effective electric field otained for the case of trangular quantum well approximation. HFT vertical effective electric field model Comment Reference lectric field spreads Stern, Sarma [7] = qe ε only in the uffer Sen et al [8] lectric field spreads only in the uffer and Stern, Sarma [7] ε uffer dopants are taken into account lectric field spreads equaly on the oth sides of the 0.5 ε heterojunction, and uffer dopants are taken into account 0.75 ε Average solution This paper, ε improoved average solution Byun et al [9] Šaši, [10] Karm., Ramesh [11] Luki, [3] Ramovi, Luki [4]

4 1614 P. M. Luki, R. M. Ramovi, R. M. Šaši In Tale 1 different HFT vertical electric field models are presented. All solutio are ased on the Gauss Law, ut with different correctio which are taken into account for 2DG distriution and concentration of impurities in the uffer layer. The vertical electric field model given y the equation (7), as well as the other models in Tale 1 are taken as a cotant values in the channel (they are not distance dependent), ecause the shape of the potential-well is assumed to e triangular (parameters can e otained y using self-coistent procedure). For the nitride ased HFTs, polarisation should e introduced: v ~ pol + qe. (8) ε Lateral HFT effective electric field l is aproximately cotant except in the drain region, ut anyway that region is depleted. Following model is proposed: VDS l = L. (9) In equation (9), V DS is drain to source voltage and L is the channel length. Using equatio (7) and (9), general model for the HFT electric field can e written: n n V q s + = ortx DS e + ε L orty. (10) The lateral component of HFT electric field is much smaller than the vertical component, and is usually neglected. The exception is in the drain region. 3. Results In Tale 2 typical values of donor concentration in the donor layer N d, 2DG surface deity n s, unintentionally introduced acceptors concentration in the uffer layer N and surface deity of ionized aceptors in the uffer layer n, for the GaAs ased HFT, are shown. Tale 2. Typical values of impurity concentratio and carriers concentratio in GaAs HFT. Donor concentration in the donor layer N d [m -3 ] Surface deity of 2DG n s [m -2 ] Concentration of unintentionally introduced acceptors in the uffer layer N [m -3 ] Surface deity of ionized acceptors in the uffer layer n [m -2 ] By using the proposed model (7), simulatio were performed. Otained results for the electric field dependence on 2DG surface deity v (n s ) and for the electric field dependence on surface deity of ionized acceptors in the uffer layer v (n ) are presented in Fig. 2 and Fig. 3. In the simulation presented in Fig. 2, 2DG surface deity n s is varied in the range m m -2. For the surface deity of ionized acceptors in the uffer layer n is accepted m -2.

5 Analytical model of electric field in heterojunction region of HFT structure 1615 [V/m] 3.5 x 107 GaAs 3 InAs GaN x [m-2] Fig. 2. HFT vertical electric field v versus 2DG surface deity n s. In the simulation presented in Fig. 3, surface deity of ionized acceptors in the uffer layer n is varied in the range m m -2. For the 2DG surface deity n s is taken m -2. [V/m] 1.8 x 107 GaAs 1.7 InAs GaN n x [m-2] Fig. 3. HFT vertical electric field v versus surface deity of impurities in the uffer layer n. In oth cases, expected values for HFT electric field are otained. Minimal value for v (n s ) is aout 50 kv/cm for all three HFT types. Maximal value for v (n s ) is aout 180kV/cm for InAs ased HFT, aout 220 kv/cm for GaAs ased HFT and aout 300 kv/cm for GaN ased HFT. It should e noticed that for GaN HFT model (7) has to e expanded y introducing polarization term like in (8). Otained values for v (n ) are from aout 100 kv/cm to aout 110 kv/cm for InAs ased HFT, from approximately 120 kv/cm to approximately 135 kv/cm for GaAs ased HFT and from approximately 155 kv/cm to approximately 170 kv/cm for GaN ased HFT.

6 1616 P. M. Luki, R. M. Ramovi, R. M. Šaši It is clear that the concentration of unintentionally introduced impurities is for several orders of magnitude smaller than donor concentration. Thus, surface deity of ionized acceptors in the uffer layer is for the order of magnitude smaller than surface deity of carriers which are confined in a quantum well. It can e concluded that the correction n in the model is important only for threshold voltage operating regime, ecause n s is than approximately of the same order of magnitude like n. By using the proposed model for the electric field, HFT carrier moility dependence on 2DG surface deity was otained, itead of standard carrier moility dependence on elcetric field which can e find in literature. The GaAs ased HFT carrier moility µ dependence on surface carrier deity n s, at room temperature, is shown in Fig. 4. For low surface carrier deities, the carrier moility is cotant. For higher surface carrier deities, this moility decreases, ecause the frequency and inteity of collisio etween carriers increase. In this case (GaAs HFT at T=300 K) moility decrease starts for the surface deity concentratio n s = /m x10exp4 [cm2/vs] µ x10exp15[m-2] Fig. 4. HFT carrier moility µ dependence on 2DG surface deity n s. For a given temperature, the value of moility is found to e a function of an average vertical electric field only [5]. Moility is strongly influenced y scattering on ionized impurities. Surface state deity and the quantity of ionized impurities can not e easily exactly determined. This affects the accuracy of simulatio results. 4. Discussion There is no special physical explanation why this particular correction coefficient should e α=0.88, ut it is sure that with this coefficient electric field is modeled more accurate in comparison with the other similar models (given in Tale 1). We found out that the est fit for the value of the critical electric field in HFT is much higher than expected. The value accepted in literature is aout 4 kv/cm, and the est fit for this value is found to e 200 kv/cm. Our investigatio led to the conclusion that this value corresponded to the normal electric field and should express its influence on carriers moility in the channel. The point is that its value ( 0 = 200 kv/cm) far exceeds the value accepted in literature ( 0 = 4 kv/cm) which corresponds only to the lateral electric field. The way in wich electric field in HFT quantum well is descried is very important for solving Schrödinger s and Poisson s equatio. By using model (7) Schrödinger s equation could e analytically solved. igen energies and Fermi energy dependences on electric field could e otained explicitly. For accepted electric field dependence on 2DG surface deity, eigen energies

7 Analytical model of electric field in heterojunction region of HFT structure 1617 and Fermi energy dependences on 2DG surface deity could e determined. The value of the electric field at the spacer layer uffer layer interface is due to oundary condition used for solving Poisson s equation, and holds for donor layer. 5. Conclusio By using Gauss Law, HFT electric field model is developed. In this model electric field is supposed to e uniform. Thus, the model proposed in this paper holds for the case of electric field in the very thin area located close to the heterojunction, or etter to say in the channel. The model is extremely usefull and applicale, earing in mind that the channel is the most important region of HFT. Vertical electric field is significantly higher then lateral, thus lateral component can e neglected. This is not valid only in the drain region, ut anyway this region is depleted. The model is more accurate than the other existing, thanks to the new correction coefficient which emphasizes more precisely asymmetric electric field spreading in the HFT heterojunction region. Proposed model is applicale for different HFT types: for the GaAs ased devices which have een used for a long time and which are the est studied, as well as for the devices with strained lattices wich are less known. The model is also applicale to the newest GaN ased HFTs. The point is that the proposed model is very simple, and at the same time it descries complex HFT physics very good. This can e concluded ecause the results otained y using this model are in very good agreement with the expected and the known pulished ones. References [1] Patric Rolin, Ha Rohdin: High speed heterostructure devices, Camridge University Press, [2] Rifat Ramovi, Rajko Šaši : Analysis and Modeling of Small Dimeio Unipolar Traistors, (In Serian), Dinex, Belgrade, [3] Petar M Luki : Approximate Model of Quantum ffects in HMT Structures, (In Serian), Bulletin Vin a Ititute of Nuclear Sciences 8(1-4) (1-104), 70 (2003). [4] R. Ramovi, P. Luki : Surafce Deity Analytical Model of Two-Dimeional lectron Gas in HMT Structures, Materials Science Forum No , 27 (2004). [5] P. M. Luki, R. M. Ramovi, R. M. Šaši : HMT Carrier Moility Analytical Model, Materials Science Forum 494, 43 (2005). [6] P. M. Luki, R. M. Ramovi : Heterostructure Unipolar Traistor Donor Layer Process Modeling, (In Serian), Proceedings of the XLVIII Conference TRAN IV, pp. 113 a ak, Seria and Montenegro, June [7] F. Stern, S. D. Sarma: lectron energy levels in GaAs/Ga 1-x Al x As heterojunctios, Physical Review B, 30, 840 (1984). [8] S. Sen, M. K. Pandey, S. Haldar, R. S. Gupta: Temperature and aluminium composition dependent sheet carrier concentration at AlGaAs/GaAs interface, Journal of Physics D: Applied Physics 33, 18 (2000). [9] Y. H. Byun, K. Lee, M. Shur: Unified charge control model and suthreshold current in heterostructure Field ffect Traistors, I lectron Device Letters 11, 50 (1990). [10] R. Šaši : Analysis and Modeling of Physical Phenomena in GaAs Based Unipolar Traistors, (In Serian), PhD degree Dissertation, Faculty of lectrical ng., Belgrade, [11] S. Karmalkar, G. Ramesh, A simple yet compreheive unified physical model of the 2D electron gas in delta-doped and uniformly, I Traactio on lectron Devices 47, 11 (2000).