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Charge transport in low-dimensional nitride semiconductor heterostructures
Physica B 296 (2001) 264}270
Charge transport in low-dimensional nitride semiconductor heterostructures Aris Christou* Microelectronics Devices Laboratory, Department of Materials Science and Nuclear Engineering, University of Maryland, Building 090, College Park, MD 20742-2115, USA
Abstract Low-dimensional conducting channels in high electron mobility transistors (HEMTs) based on AlGaN/GaN heterostructures have been modeled and the results are described in this paper. The Schroedinger's equation and the Poisson's equation have been solved self-consistently in order to obtain a relationship between the sheet carrier density and the applied gate voltage. The relationship is treated using a non-linear exponential "t that enables a more accurate analysis of the transport saturation region compared to other models used hitherto. The current}voltage (I}<) characteristics have then been determined by a charge control analysis that utilizes the exponential charge}potential relationship. This identi"es the saturation point of channel conduction without parameters extracted from experiments. The intrinsic non-linearity of transport identi"ed by the simulation explains why such devices do not have a singular channel `pinch-o!a voltage. 2001 Elsevier Science B.V. All rights reserved. Keywords: Charge control; Transistors; Gallium nitride; Numerical simulation
1. Introduction A wide band-gap material, GaN, has great potential for applications in high-power, high-temperature devices as well as for optoelectronic devices. GaN-based p}n junction blue light-emitting diodes [1] have already been fabricated utilizing GaN and GaAlN heterostructures grown on sapphire substrates. Metal semiconductor "eld e!ect transistors (MESFETs) and high electron mobility transistors (HEMTs) based on AlGaN/GaN and AlN/GaN heterostructures have been reported by several authors [2}5]. The design and fabrication of such
* Corresponding author. Fax: #1-301-314-2029. E-mail address: [email protected] (A. Christou).
devices requires accurate models, which may be used to simulate device characteristics and to optimize their "nal con"guration. A number of authors have investigated several methods to solve the coupled Poisson's and Schroedinger's equations [6}8]. A comparison, based on accuracy and computational e$ciency, between the "nite di!erence method and the variational method (using the Rayleigh}Ritz technique) has been reported [9,10]. We have used the variational technique to simulate transport in AlGaN/GaN heterostructures in our work because of its accuracy and e$ciency. In such structures, electron transport takes place in a nearly twodimensional channel formed at the interface between the wider and narrower conduction band interfaces.
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In the present work, Poisson's and Schroedinger's equations are solved self-consistently to obtain the wave functions, the eigenenergies and the carrier distribution at the hetero-interface. Calculations are performed for di!erent gate biases to obtain a relationship between the carrier density and the applied gate voltage. The variational technique is the basis for the self-consistent solution of the equations and is described in the present work.
Fig. 1. A cross section of the AlGaN/GaN HEMT on a sapphire substrate.
2. Description of the charge control model The one-dimentional Schroedinger's equation is given by
R (x) K #<(x) (x)"E (x), ! K K K 2mH Rx
(1)
<(x)"!q(x)#< (x)#< (x),
(2)
where E is the eigenenergy for the mth subband, K the wave function corresponding to the K eigenenergy E , h the Planck's constant, m* the K electron e!ective mass, and q the electronic charge. The quantity (x) is the electrostatic potential, < is the conduction band discontinuity and < is the local exchange}correlation potential which accounts for the electron}electron interaction. The last term has been neglected, since from previous investigations [11], electron}electron interactions in inversion layers on polar materials are negligible. The Rayleigh}Ritz method has been applied to solve Eq. (1). The procedure for solving eigenvalue problems using the variational method has been previously reported [12]. This method determines the "nite set of eigenvalues and eigenfunctions. The basic functions that characterize the wave function are chosen so as to satisfy the boundary conditions (0)"0 and (a)"0, where a is the width of the channel. This is set equal to 10 nm suitable for the conventional AlGaN/GaN structure shown in Fig. 1, where X"0 is the location of the interface. The wave function corresponding to the mth subband is given by
, nx " a sin . K KL a L
(3)
Ideally, if N were in"nite, the solution would be exact. For a "nite large N, the eigenfunctions may still be accurately determined [13]. The coe$cients of the wave functions, a , the Ritz parameters, are KL determined by applying the variational principle to the eigenvalue problem, resulting in the following matrix equation: , (A !E)X "0, KL L L where the matrix elements are given by
(4)
n A "! KL LK 2mH a
mx nx 2 ? # <(x)sin sin (5) a a a and X "a corresponds to the mth eigenstate. L KL The solution of Eq. (4) results in the determination of the eigenenergies E and the wave functions K . An e$cient algorithm has been previously deK veloped to obtain the eigenvalues and the eigenfunctions [14]. The algorithm for solving the Schroedinger's equation was tested by comparing the eigenenergies for a rectangular and a triangular quantum well with the exact solutions for these quantum wells. The number of basis functions used was 30. The eigenenergies obtained by the solution of the Raleigh}Ritz method, when compared with the exact solutions obtained analytically for the square and the triangular potential well, gave an error less than 0.0001%, ensuring the validity of the algorithm developed to solve the Schroedinger's
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where is the semiconductor permittivity of the GaN channel layer, which is assumed to be position independent, N> is the ionized impurity donor concentration, given as 10/cm, the lowest unintentional background carrier concentration reported previously [15]. The quantity n is the two-dimensional (2D) electron gas concentration which is given by mHk¹ ln[1#exp((E !E )/k¹)] (x), n " $ K G (7)
Fig. 2. Wave functions within a square well for n"1 (*), n"2(;;) and n"3 (} ) }).
where k is the Boltzmann's constant, ¹ the absolute temperature, E the Fermi level, and E is the $ intrinsic energy level. A necessary step of the simulation is to relate the channel electron concentration with the external gate voltage. Poisson's equation is solved with the boundary condition which is established by the electric "eld at the interface, predetermined by the external bias. The electric "eld at the interface is given by d< !q(n !N>) , " (8) dx where n is the total channel charge density. The other boundary condition imposed is at the interface where the conduction band edge of the channel is "xed with respect to the conduction band edge of the spacer layer. Such a boundary condition establishes potential (0), at the interface, with respect to the Fermi level.
Fig. 3. Conduction band pro"le in the conducting channel.
equation. The wave functions for the square quantum well is shown in Fig. 2, while Fig. 3 shows the conduction band pro"le in the channel. The electrostatic potential in Eq. (1) is obtained by solving the Poisson's equation in the channel of the device shown in Fig. 1, which is given as follows: R q "! (N>(x)!n (x)), Rx
(6)
3. Self-consistent solution of Poisson's and Schroedinger's equation The parameters used in the calculation of the transport characteristics of the conducting channel of Fig. 1 are as follows. The composition of AlN in AlGaN was 20%. The thickness of the donor layer shown in Fig. 1 was set to be 30 nm. The thickness of the GaN bu!er layer was taken to be 0.3 m. A typical donor concentration of 5;10/cm was also used in the calculations. The electron e!ective mass for GaN was taken as 0.18m and its relative permittivity as 9.5. The energy gap of GaN was
A. Christou / Physica B 296 (2001) 264}270
taken as 3.4 eV and that of AlN, 6.3 eV. The parameters for AlGaN were extrapolated by a simple application of Vegard's law. Eqs. (1)}(6) are solved iteratively until a selfconsistent solution is obtained. For the "rst iteration, a triangular well approximation is used. The solution of Eq. (4) gives the eigenenergies and the corresponding wave functions for each eigenstate, from which the carrier distribution across the channel is obtained from Eq. (7). The solution of the Poisson's equation (Eq. (6)) gives the new electrostatic potential. In order to ensure smooth convergence, a linear combination of the new and old potential is substituted back into Schroedinger's equation while performing the iterative calculations as had been previously described [16]. The system is iterated until the di!erence in the potential <(x) within the same iteration is less than 0.1 meV. The process is repeated for di!erent values of the conduction band edge with respect to the Fermi level. A variation of the carrier concentration with respect to this energy di!erence, E !E , may be ! $ obtained showing that the Fermi level is present throughout the structure and its position is directly related to the applied gate bias. The solution of
Fig. 4. Variation of charge concentration with respect to the applied gate bias.
267
Poisson's equation in the donor layer will yield the required relationship [17]. Hence, the variation of channel charge density with the applied voltage has been obtained and is shown in Fig. 4. The charge}potential relationship in the channel is given by
< !< 2 n (x)"n 1!exp ! %1 1 <
,
(9)
where < is the applied gate to source bias, < the %1 2 threshold voltage, and n and < are parameters 1 obtained from the self-consistent solution. The above procedure for the AlGaN/GaN heterostructure resulted in n "4.7;10/cm and < " 1 0.1987. The exact determination of the channel charge is critical in the accurate calculation of channel transport characteristics, and will be applied extensively in the following sections.
4. Calculation of the intrinsic I}< characteristics The transport characteristics are obtained by performing the analysis in two conduction channel regions; the linear region in which the gradual channel approximation (GCA) is valid and the saturation region where it is not [13]. The linear region is characterized by an increase in the drain current with an applied drain-to-source voltage, and by a "eld E(x) along the channel which is much less than the transverse "eld and the saturation "eld E . The saturation region is where the drain voltage < reaches a saturation value. The im"1 2 portant step in simulating the characteristics is in calculating the physical boundary, or the saturation point, that separates the two regions. Once the saturation point is determined, the analysis may then be performed in the two regions separately. The following sections delineate the procedure for carrying out the charge control analysis, utilizing the charge}potential relationship, which is a function of a spatially varying channel potential and the applied drain voltage as given by the GCA of Ref. [13]:
<(y)!< %2 n (y)"n 1!exp ! 1 <
.
(10)
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Here, y is the distance measured from the source to the drain, <(y) is the potential along the channel and < "< !< . Eq. (10) is obtained by in%2 %1 2 cluding the channel potential shown in Eq. (9). 4.1. Linear region Using the GCA in the linear region [13], the transport current}voltage characteristics are obtained by solving the following equation: d<(y)/dy "q=n (y) , (11) 1#(1/2E )d<(y)/dy where 2E is the electric "eld at which the electron velocity saturates, the low "eld mobility of the electron which is assumed to be constant in the analysis, and = the e!ective width of the gate. The current I of Eq. (11), tacitly assumes the "1* electron velocity}"eld relationship characteristic of silicon MOSFETs which also holds true for all GaAs-based transistor. Such an analytic expression is superior to the linear two-piece model since it reproduces the soft saturation characteristics in a continuous, smooth fashion. In addition, this model does not su!er from the Tri"menko! relationship that underestimates the current in both the linear and the saturation regions [18]. Substituting Eq. (10) in Eq. (11), rearranging and integrating from y"0 to ¸, where ¸ is the gate length, yields the following expression for the drain current in the linear region: I
"1*
< 2=qn v 1 < #< exp ! %2 " " "1* < < #2< " * < ; 1!exp ! " (12) < where v is the saturation velocity, < the drain "1 to-source voltage and < "E ¸, where ¸ is the * 1 e!ective gate length. I
4.2. Saturation point The saturation point is determined by substituting < in Eq. (12), yielding a single equation for "1 2 the saturation point. We assume current continuity at the boundary between the linear region and the
saturation region:
2=qn v < 1 < " #< exp ! %2 "1 2 < #2< < "1 2 * < ; 1!exp ! "1 2 . (13) < The saturation current is related to the total channel charge concentration for an applied gate and drain bias. Hence, the second equation required to determine the saturation point is given by I
"1 2
I
"1 2
< !< %2 "qv =n 1!exp ! "1 2 1 <
.
(14) Solving Eqs. (13) and (14) simultaneously results in the following non-linear equation that has been solved iteratively by the Newton-Raphson method:
< (< #2< exp ! %2 !2< ) "1 2 * < #(< #2< !2< ) "1 2 * < !< %2 "0. (15) ;exp ! "1 2 < Eq. (15) is solved iteratively for varying gate biases until the saturation point is determined. The analysis is then carried out within the saturation region.
4.3. Saturation region In order to calculate the characteristics in the saturation region, the channel length modulation e!ect, must be taken into account. In this e!ort, an increase in < beyond the saturation voltage "1 moves the saturation point closer to the source. The modulation e!ects have been incorporated by dividing the channel spatially, such that in the region near the source within a distance, ¸!¸ from the source, the GCA is valid, and in the region ¸, the channel length modulation e!ect must be included. Substituting (< !< ) for < in Eq. (13) results * * * in the expression for the drain current in the saturation region where GCA is valid: I
"11
"I
"1 2 <
<
#2< "1 2 * , #2< 2< "1 2 * *
(16)
A. Christou / Physica B 296 (2001) 264}270
269
and where < "E ¸ is the potential across the * Q length ¸. The channel modulation length, ¸, is determined by the solution of the Poisson's equation in the region where the GCA is not valid within the saturation regime [19]. Since ¸;¸, < ;< and < !< is small, the following * * " "1 2 linear relationship results: < !< "1 2 . < " "1 *
(17)
The factor is determined directly from the current continuity condition for drain conductance at the saturation point given as follows:
RI RI "1* " "11 . (18) R< R< "1 4"1 4"1 2 "1 4"1 4"1 2 Hence, equating the drain conductances obtained from Eqs. (12) and (16) yields the necessary equation for drain current in the saturation regime:
Fig. 5. The simulated I}< characteristics for < "!3.5 (}), !3.1 ( ' ' ' ' ), !2.7 (;;;;), !2.3 ( ) ) ) ) ) and !1.9 (} ) } ) })V.
< #2< "11 . (19) "1 2 < #2< "1 2 * Therefore, the I}< characteristics may now be obtained by the solving Eqs. (12)}(19). The parameters used in the simulation are the following: the gate width was taken to be 150 m and the gate length to be 0.5 m. The saturation velocity was taken as 2.5;10 cm/s and the mobility was taken to be 350 cm/V/s. The saturation "eld was taken to be 1.5;10 V/cm. Applying these parameters in the charge control equations results in the determination of the transistor's transport characteristics as shown in Fig. 5. The drain current as a function of the applied drain-to-source voltage for di!erent gate biases ranging from !3.5 to 1.5 V is shown. Fig. 6 shows the variation of the transconductance as a function of gate voltage resulting from the simulation. I
"11
"I
Fig. 6. The AlGaN/GaN transfer characteristics de"ned as the transconductance versus gate voltage.
5. Summary and conclusions The I}< characteristics of an AIGaN HEMT have been simulated by a charge control model that takes into account the size quantization e!ects present in the interface. The simulation was achieved by self-consistently solving the Poisson's
and the Schroedinger's equations. We have used an exponential charge } potential relationship which more accurately describes the I}< characteristics of a HEMT by incorporating the e!ect of the device's non-linear conductance near its saturation region. The saturation region is of interest in many
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A. Christou / Physica B 296 (2001) 264}270
applications [20}22]. The exponential relationship, when used in the charge control model, determines the saturation point without utilizing parameter extractions from other data. The exponential relationship also explains the non-linear pinch-o! characteristics found in these devices.
Acknowledgements The authors acknowledge partial support from the NSF-University of Maryland, Materials Research Science and Engineering Center (MRSEC).
References [1] S. Nakamura, T. Mukai, M. Seroh, Jpn. J. Appl. Phys 30 (1991) 1998. [2] M. Asif Khan, J.N. Kuznia, A.R. Bhattarai, D.T. Olson, Appl. Phys. Lett. 62 (15) (1993) 1786. [3] S.C. Binari, Electrochem. Soc. Proc. 95-21 (1995) 693. [4] M. Asif Khan, J.N. Kuznia, D.T. Olson, M. Blasingame, L.F. Reitz, Appl. Phys. Lett. 2 (29) (1992) 2917. [5] J.S. Foresi, T.D. Moustakas, Appl. Phys. Lett. 62 (220) (1993) 2859. [6] K. Yokoyama, K. Hess, Phys. Rev. B 63 (1986) 938. [7] G. Halkias, A. Vegiri, G. Pananakakis, A. Christou, Solid State Electron. 35 (4) (1992) 459.
[8] A.M. Cruz Serra, H. Abreu Santos, J. Appl. Phys. 70 (5) (1991) 2734. [9] A. Abou-Elnour, K. Schuenemann, J. Appl. Phys. 74 (5) (1993) 3273. [10] C. Chang, H. Fetterman, IEEE Trans. Electron Dev. 34 (7) (1987) 1456. [11] T. Ando, J Phys. Soc. Japan 51 (12) (1988) 3900. [12] M.N.O. Sadiku, Numerical Techniques in Electromagnetics, CRS Press, Boca Raton, FL, 1992, pp. 255}302. [13] A. Abou-Elnour, K. Schuenemann, Solid State Electron. 37 (1) (1994) 27. [14] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1986, pp. 335}377. [15] L.B. Rowland, K. Doverspike, D.K. Gaskill, J.A. Freitas Jr., Proceedings of the Material Research Society Symposium, Vol. 339, 1994. [16] A. Abou-Elnour, K. Schuenemann, Proceedings of MTT Symposium, 1993, p. 1177. [17] H. Morkoc, H. Unlu, G. Ji, Principles, Technology of MODFETs, Wiley, New York, 1991, p. 143}151. [18] B.J. Moon, Y.H. Byun, K. Lee, M. Shur, IEEE Trans. Electron Devices 37 (4) (1990) 908. [19] L. Guan, A. Christou, G. Halkias, D. Barbe, IEEE Trans. Electron Devices 42 (4) (1995) 612. [20] E. Monroy, F. Calle, E. Munoz, F. Omnes, P. Gibart, J.A. Munoz, Appl. Phys. Lett. 73 (1998) 2146. [21] E. Alekseev, D. Pavlidis, Solid State Electron. 44 (6) (2000) 941. [22] S. Sheppard, S.T. Doverspike, W.L. Pribble, S.T. Allen, J. Palmour, W. Kehios, T.J. Jenkins, Electron Dev. Lett. 20 (1999) 161.
Charge transport in low-dimensional nitride semiconductor heterostructures Aris Christou* Microelectronics Devices Laboratory, Department of Materials Science and Nuclear Engineering, University of Maryland, Building 090, College Park, MD 20742-2115, USA
Abstract Low-dimensional conducting channels in high electron mobility transistors (HEMTs) based on AlGaN/GaN heterostructures have been modeled and the results are described in this paper. The Schroedinger's equation and the Poisson's equation have been solved self-consistently in order to obtain a relationship between the sheet carrier density and the applied gate voltage. The relationship is treated using a non-linear exponential "t that enables a more accurate analysis of the transport saturation region compared to other models used hitherto. The current}voltage (I}<) characteristics have then been determined by a charge control analysis that utilizes the exponential charge}potential relationship. This identi"es the saturation point of channel conduction without parameters extracted from experiments. The intrinsic non-linearity of transport identi"ed by the simulation explains why such devices do not have a singular channel `pinch-o!a voltage. 2001 Elsevier Science B.V. All rights reserved. Keywords: Charge control; Transistors; Gallium nitride; Numerical simulation
1. Introduction A wide band-gap material, GaN, has great potential for applications in high-power, high-temperature devices as well as for optoelectronic devices. GaN-based p}n junction blue light-emitting diodes [1] have already been fabricated utilizing GaN and GaAlN heterostructures grown on sapphire substrates. Metal semiconductor "eld e!ect transistors (MESFETs) and high electron mobility transistors (HEMTs) based on AlGaN/GaN and AlN/GaN heterostructures have been reported by several authors [2}5]. The design and fabrication of such
* Corresponding author. Fax: #1-301-314-2029. E-mail address: [email protected] (A. Christou).
devices requires accurate models, which may be used to simulate device characteristics and to optimize their "nal con"guration. A number of authors have investigated several methods to solve the coupled Poisson's and Schroedinger's equations [6}8]. A comparison, based on accuracy and computational e$ciency, between the "nite di!erence method and the variational method (using the Rayleigh}Ritz technique) has been reported [9,10]. We have used the variational technique to simulate transport in AlGaN/GaN heterostructures in our work because of its accuracy and e$ciency. In such structures, electron transport takes place in a nearly twodimensional channel formed at the interface between the wider and narrower conduction band interfaces.
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In the present work, Poisson's and Schroedinger's equations are solved self-consistently to obtain the wave functions, the eigenenergies and the carrier distribution at the hetero-interface. Calculations are performed for di!erent gate biases to obtain a relationship between the carrier density and the applied gate voltage. The variational technique is the basis for the self-consistent solution of the equations and is described in the present work.
Fig. 1. A cross section of the AlGaN/GaN HEMT on a sapphire substrate.
2. Description of the charge control model The one-dimentional Schroedinger's equation is given by
R (x) K #<(x) (x)"E (x), ! K K K 2mH Rx
(1)
<(x)"!q(x)#< (x)#< (x),
(2)
where E is the eigenenergy for the mth subband, K the wave function corresponding to the K eigenenergy E , h the Planck's constant, m* the K electron e!ective mass, and q the electronic charge. The quantity (x) is the electrostatic potential, < is the conduction band discontinuity and < is the local exchange}correlation potential which accounts for the electron}electron interaction. The last term has been neglected, since from previous investigations [11], electron}electron interactions in inversion layers on polar materials are negligible. The Rayleigh}Ritz method has been applied to solve Eq. (1). The procedure for solving eigenvalue problems using the variational method has been previously reported [12]. This method determines the "nite set of eigenvalues and eigenfunctions. The basic functions that characterize the wave function are chosen so as to satisfy the boundary conditions (0)"0 and (a)"0, where a is the width of the channel. This is set equal to 10 nm suitable for the conventional AlGaN/GaN structure shown in Fig. 1, where X"0 is the location of the interface. The wave function corresponding to the mth subband is given by
, nx " a sin . K KL a L
(3)
Ideally, if N were in"nite, the solution would be exact. For a "nite large N, the eigenfunctions may still be accurately determined [13]. The coe$cients of the wave functions, a , the Ritz parameters, are KL determined by applying the variational principle to the eigenvalue problem, resulting in the following matrix equation: , (A !E)X "0, KL L L where the matrix elements are given by
(4)
n A "! KL LK 2mH a
mx nx 2 ? # <(x)sin sin (5) a a a and X "a corresponds to the mth eigenstate. L KL The solution of Eq. (4) results in the determination of the eigenenergies E and the wave functions K . An e$cient algorithm has been previously deK veloped to obtain the eigenvalues and the eigenfunctions [14]. The algorithm for solving the Schroedinger's equation was tested by comparing the eigenenergies for a rectangular and a triangular quantum well with the exact solutions for these quantum wells. The number of basis functions used was 30. The eigenenergies obtained by the solution of the Raleigh}Ritz method, when compared with the exact solutions obtained analytically for the square and the triangular potential well, gave an error less than 0.0001%, ensuring the validity of the algorithm developed to solve the Schroedinger's
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A. Christou / Physica B 296 (2001) 264}270
where is the semiconductor permittivity of the GaN channel layer, which is assumed to be position independent, N> is the ionized impurity donor concentration, given as 10/cm, the lowest unintentional background carrier concentration reported previously [15]. The quantity n is the two-dimensional (2D) electron gas concentration which is given by mHk¹ ln[1#exp((E !E )/k¹)] (x), n " $ K G (7)
Fig. 2. Wave functions within a square well for n"1 (*), n"2(;;) and n"3 (} ) }).
where k is the Boltzmann's constant, ¹ the absolute temperature, E the Fermi level, and E is the $ intrinsic energy level. A necessary step of the simulation is to relate the channel electron concentration with the external gate voltage. Poisson's equation is solved with the boundary condition which is established by the electric "eld at the interface, predetermined by the external bias. The electric "eld at the interface is given by d< !q(n !N>) , " (8) dx where n is the total channel charge density. The other boundary condition imposed is at the interface where the conduction band edge of the channel is "xed with respect to the conduction band edge of the spacer layer. Such a boundary condition establishes potential (0), at the interface, with respect to the Fermi level.
Fig. 3. Conduction band pro"le in the conducting channel.
equation. The wave functions for the square quantum well is shown in Fig. 2, while Fig. 3 shows the conduction band pro"le in the channel. The electrostatic potential in Eq. (1) is obtained by solving the Poisson's equation in the channel of the device shown in Fig. 1, which is given as follows: R q "! (N>(x)!n (x)), Rx
(6)
3. Self-consistent solution of Poisson's and Schroedinger's equation The parameters used in the calculation of the transport characteristics of the conducting channel of Fig. 1 are as follows. The composition of AlN in AlGaN was 20%. The thickness of the donor layer shown in Fig. 1 was set to be 30 nm. The thickness of the GaN bu!er layer was taken to be 0.3 m. A typical donor concentration of 5;10/cm was also used in the calculations. The electron e!ective mass for GaN was taken as 0.18m and its relative permittivity as 9.5. The energy gap of GaN was
A. Christou / Physica B 296 (2001) 264}270
taken as 3.4 eV and that of AlN, 6.3 eV. The parameters for AlGaN were extrapolated by a simple application of Vegard's law. Eqs. (1)}(6) are solved iteratively until a selfconsistent solution is obtained. For the "rst iteration, a triangular well approximation is used. The solution of Eq. (4) gives the eigenenergies and the corresponding wave functions for each eigenstate, from which the carrier distribution across the channel is obtained from Eq. (7). The solution of the Poisson's equation (Eq. (6)) gives the new electrostatic potential. In order to ensure smooth convergence, a linear combination of the new and old potential is substituted back into Schroedinger's equation while performing the iterative calculations as had been previously described [16]. The system is iterated until the di!erence in the potential <(x) within the same iteration is less than 0.1 meV. The process is repeated for di!erent values of the conduction band edge with respect to the Fermi level. A variation of the carrier concentration with respect to this energy di!erence, E !E , may be ! $ obtained showing that the Fermi level is present throughout the structure and its position is directly related to the applied gate bias. The solution of
Fig. 4. Variation of charge concentration with respect to the applied gate bias.
267
Poisson's equation in the donor layer will yield the required relationship [17]. Hence, the variation of channel charge density with the applied voltage has been obtained and is shown in Fig. 4. The charge}potential relationship in the channel is given by
< !< 2 n (x)"n 1!exp ! %1 1 <
,
(9)
where < is the applied gate to source bias, < the %1 2 threshold voltage, and n and < are parameters 1 obtained from the self-consistent solution. The above procedure for the AlGaN/GaN heterostructure resulted in n "4.7;10/cm and < " 1 0.1987. The exact determination of the channel charge is critical in the accurate calculation of channel transport characteristics, and will be applied extensively in the following sections.
4. Calculation of the intrinsic I}< characteristics The transport characteristics are obtained by performing the analysis in two conduction channel regions; the linear region in which the gradual channel approximation (GCA) is valid and the saturation region where it is not [13]. The linear region is characterized by an increase in the drain current with an applied drain-to-source voltage, and by a "eld E(x) along the channel which is much less than the transverse "eld and the saturation "eld E . The saturation region is where the drain voltage < reaches a saturation value. The im"1 2 portant step in simulating the characteristics is in calculating the physical boundary, or the saturation point, that separates the two regions. Once the saturation point is determined, the analysis may then be performed in the two regions separately. The following sections delineate the procedure for carrying out the charge control analysis, utilizing the charge}potential relationship, which is a function of a spatially varying channel potential and the applied drain voltage as given by the GCA of Ref. [13]:
<(y)!< %2 n (y)"n 1!exp ! 1 <
.
(10)
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Here, y is the distance measured from the source to the drain, <(y) is the potential along the channel and < "< !< . Eq. (10) is obtained by in%2 %1 2 cluding the channel potential shown in Eq. (9). 4.1. Linear region Using the GCA in the linear region [13], the transport current}voltage characteristics are obtained by solving the following equation: d<(y)/dy "q=n (y) , (11) 1#(1/2E )d<(y)/dy where 2E is the electric "eld at which the electron velocity saturates, the low "eld mobility of the electron which is assumed to be constant in the analysis, and = the e!ective width of the gate. The current I of Eq. (11), tacitly assumes the "1* electron velocity}"eld relationship characteristic of silicon MOSFETs which also holds true for all GaAs-based transistor. Such an analytic expression is superior to the linear two-piece model since it reproduces the soft saturation characteristics in a continuous, smooth fashion. In addition, this model does not su!er from the Tri"menko! relationship that underestimates the current in both the linear and the saturation regions [18]. Substituting Eq. (10) in Eq. (11), rearranging and integrating from y"0 to ¸, where ¸ is the gate length, yields the following expression for the drain current in the linear region: I
"1*
< 2=qn v 1 < #< exp ! %2 " " "1* < < #2< " * < ; 1!exp ! " (12) < where v is the saturation velocity, < the drain "1 to-source voltage and < "E ¸, where ¸ is the * 1 e!ective gate length. I
4.2. Saturation point The saturation point is determined by substituting < in Eq. (12), yielding a single equation for "1 2 the saturation point. We assume current continuity at the boundary between the linear region and the
saturation region:
2=qn v < 1 < " #< exp ! %2 "1 2 < #2< < "1 2 * < ; 1!exp ! "1 2 . (13) < The saturation current is related to the total channel charge concentration for an applied gate and drain bias. Hence, the second equation required to determine the saturation point is given by I
"1 2
I
"1 2
< !< %2 "qv =n 1!exp ! "1 2 1 <
.
(14) Solving Eqs. (13) and (14) simultaneously results in the following non-linear equation that has been solved iteratively by the Newton-Raphson method:
< (< #2< exp ! %2 !2< ) "1 2 * < #(< #2< !2< ) "1 2 * < !< %2 "0. (15) ;exp ! "1 2 < Eq. (15) is solved iteratively for varying gate biases until the saturation point is determined. The analysis is then carried out within the saturation region.
4.3. Saturation region In order to calculate the characteristics in the saturation region, the channel length modulation e!ect, must be taken into account. In this e!ort, an increase in < beyond the saturation voltage "1 moves the saturation point closer to the source. The modulation e!ects have been incorporated by dividing the channel spatially, such that in the region near the source within a distance, ¸!¸ from the source, the GCA is valid, and in the region ¸, the channel length modulation e!ect must be included. Substituting (< !< ) for < in Eq. (13) results * * * in the expression for the drain current in the saturation region where GCA is valid: I
"11
"I
"1 2 <
<
#2< "1 2 * , #2< 2< "1 2 * *
(16)
A. Christou / Physica B 296 (2001) 264}270
269
and where < "E ¸ is the potential across the * Q length ¸. The channel modulation length, ¸, is determined by the solution of the Poisson's equation in the region where the GCA is not valid within the saturation regime [19]. Since ¸;¸, < ;< and < !< is small, the following * * " "1 2 linear relationship results: < !< "1 2 . < " "1 *
(17)
The factor is determined directly from the current continuity condition for drain conductance at the saturation point given as follows:
RI RI "1* " "11 . (18) R< R< "1 4"1 4"1 2 "1 4"1 4"1 2 Hence, equating the drain conductances obtained from Eqs. (12) and (16) yields the necessary equation for drain current in the saturation regime:
Fig. 5. The simulated I}< characteristics for < "!3.5 (}), !3.1 ( ' ' ' ' ), !2.7 (;;;;), !2.3 ( ) ) ) ) ) and !1.9 (} ) } ) })V.
< #2< "11 . (19) "1 2 < #2< "1 2 * Therefore, the I}< characteristics may now be obtained by the solving Eqs. (12)}(19). The parameters used in the simulation are the following: the gate width was taken to be 150 m and the gate length to be 0.5 m. The saturation velocity was taken as 2.5;10 cm/s and the mobility was taken to be 350 cm/V/s. The saturation "eld was taken to be 1.5;10 V/cm. Applying these parameters in the charge control equations results in the determination of the transistor's transport characteristics as shown in Fig. 5. The drain current as a function of the applied drain-to-source voltage for di!erent gate biases ranging from !3.5 to 1.5 V is shown. Fig. 6 shows the variation of the transconductance as a function of gate voltage resulting from the simulation. I
"11
"I
Fig. 6. The AlGaN/GaN transfer characteristics de"ned as the transconductance versus gate voltage.
5. Summary and conclusions The I}< characteristics of an AIGaN HEMT have been simulated by a charge control model that takes into account the size quantization e!ects present in the interface. The simulation was achieved by self-consistently solving the Poisson's
and the Schroedinger's equations. We have used an exponential charge } potential relationship which more accurately describes the I}< characteristics of a HEMT by incorporating the e!ect of the device's non-linear conductance near its saturation region. The saturation region is of interest in many
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A. Christou / Physica B 296 (2001) 264}270
applications [20}22]. The exponential relationship, when used in the charge control model, determines the saturation point without utilizing parameter extractions from other data. The exponential relationship also explains the non-linear pinch-o! characteristics found in these devices.
Acknowledgements The authors acknowledge partial support from the NSF-University of Maryland, Materials Research Science and Engineering Center (MRSEC).
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