Analytical Modeling and Simulation of SiGe MOS Gate HEMT

Transcription

1 Univerity of Tenneee, Knoxville Trace: Tenneee Reearch and Creative Exchange Mater Thee Graduate School Analytical Modeling and Simulation of SiGe MOS Gate HEMT Mohmmad Tanvir Alam Univerity of Tenneee - Knoxville Recommended Citation Alam, Mohmmad Tanvir, "Analytical Modeling and Simulation of SiGe MOS Gate HEMT. " Mater' Thei, Univerity of Tenneee, Thi Thei i brought to you for free and open acce by the Graduate School at Trace: Tenneee Reearch and Creative Exchange. It ha been accepted for incluion in Mater Thee by an authorized adminitrator of Trace: Tenneee Reearch and Creative Exchange. For more information, pleae contact trace@utk.edu.

2 To the Graduate Council: I am ubmitting herewith a thei written by Mohmmad Tanvir Alam entitled "Analytical Modeling and Simulation of SiGe MOS Gate HEMT." I have examined the final electronic copy of thi thei for form and content and recommend that it be accepted in partial fulfillment of the requirement for the degree of Mater of Science, with a major in Electrical Engineering. We have read thi thei and recommend it acceptance: Donald W. Bouldin, Benjamin J. Blalock (Original ignature are on file with official tudent record.) Syed K. Ilam, Major Profeor Accepted for the Council: Dixie L. Thompon Vice Provot and Dean of the Graduate School

3 To the Graduate Council: I am ubmitting herewith a thei written by Mohmmad Tanvir Alam entitled Analytical Modeling and Simulation of SiGe MOS Gate HEMT. I have examined the final electronic copy of thi thei for form and content and recommend that it be accepted in partial fulfillment of the requirement for the degree of Mater of Science, with a major in Electrical Engineering. We have read thi thei and recommend it acceptance: Syed K. Ilam Major Profeor Donald W. Bouldin Benjamin J. Blalock Accepted for the Council: Anne Mayhew Vice Chancellor and Dean of Graduate Studie (Original ignature are on file with official tudent record.

4 Analytical Modeling and Simulation of SiGe MOS Gate HEMT A Thei Preented For the Mater of Science Degree The Univerity of Tenneee, Knoxville Mohmmad Tanvir Alam Augut 005

5 Dedication Thi work i dedicated to my parent and my iter. Without their love, upport and encouragement, I could not come to thi point of my life. ii

6 Acknowledgement At firt, I would like to thank my upervior Dr. Syed K. Ilam for providing me the opportunity to work with him. Hi contant inpiration, guidance and comment have helped me in my reearch a well a in coure work. I would like to thank Dr. Donald Bouldin and Dr. Benjamin Blalock for their guidance and help. I would like to convey my pecial thank to Dr. Hanno Weitering of Phyic department for doing an excellent job in teaching quantum mechanic and helping me in my reearch work. I would like to thank all my group member for their continuou upport. I want to pecially thank Dr. Haanuzzaman for hi valuable advice, Mohammad Adeeb and Mohammad. Rahman for their help in coure work and reearch, Hung Nguyen and Rajagopal Vijayaraghavan for their help on different occaion. Matt Diney deerve pecial thank for helping me olve ome problem with Medici TM. Finally, I want to thank all my friend and family for helping and encouraging me at good time a well a at bad time. iii

7 Abtract Thi thei preent work on analytical modeling and imulation of SiGe MOS gate HEMT. A modified model for the threhold voltage of the MOS-gate HEMT i preented. An expreion to calculate accurately minimum gate voltage V Gmin of p- channel MOS-gate HEMT i derived. Uing the modified expreion, V THp and V Gmin are calculated. Current-voltage characteritic, tranconductance and cutoff frequency are calculated and plotted uing the modified model and the reult are compared with the reult obtained from an exiting model. The effect of different device and material parameter variation on V THp and V Gmin are alo invetigated. An analytical temperature model for the MOS-gate HEMT i propoed. The temperature variation of threhold voltage, current-voltage characteritic and tranconductance are imulated uing the analytical model. A model for a delta-doped MOS-gate HEMT i propoed which i valid for any width of the delta-doped layer. Effect of different device parameter on V THp and V tl have been invetigated uing thi model. In addition, device imulator Medici TM ha been ued to imulate the delta-doped and regular-doped MOS-gate HEMT. The reult obtained from the Medici TM imulation are compared to ome of the reult obtained from the analytical model. iv

8 Chapter Introduction Background Limitation of Silicon-Baed MOSFET with Short Channel HEMT: A Light of Hope MOS-Gate HEMT SiGe MOS-Gate HEMT Propoed Work Outline of the Thei... 7 Chapter Literature Review HEMT What i HEMT GaA/AlGaA HEMT SiGe MOS- gate HEMT Propertie of SiGe and Si/SiGe Heterotructure Development of SiGe MOS-Gate HEMT Chapter 3... Modified Model for MOS-Gate SiGe HEMT Exiting Model for the MOS-Gate HEMT The threhold voltage expreion Calculation of V THp Calculation of V Gmin The current-voltage characteritic The tranconductance (g m ) and cutoff frequency (f T ) The Modified Model The threhold voltage expreion Calculation of threhold voltage uing modified model Calculation of V Gmin Current-voltage characteritic Tranconductance (g m ) and cutoff frequency (f T ) Effect of Device and Material Parameter on V THp and V Gmin Effect of doping concentration Effect of pacer layer width Effect of ilicon layer Width Effect of oxide thickne Effect of mole fraction in Si x Ge 1-x Concluion... 6 Chapter The Temperature Model Effect of Temperature on Threhold Voltage Temperature dependence of doping concentration v

9 4.1. Temperature dependence of metal work function (ϕ m ) Temperature dependence of emiconductor electron affinity (χ ) Temperature dependence of bandgap Temperature dependence of E v Temperature dependence of E fo Calculation of temperature dependence of V THp Effect of Temperature on Hole Mobility (µ h ) The Current-Voltage Characteritic Tranconductance Concluion Chapter Analytical Modeling of Delta-Doped HEMT The Exiting Threhold Voltage Model The threhold voltage expreion Limitation of the exiting model The Modified Threhold Voltage Model Need for the modified model The threhold voltage expreion Calculation of threhold voltage The Expreion for V tl Effect of d 1 and δ on V THp and V tl Effect of d Effect of δ Concluion Chapter Medici TM Simulation of MOS-Gate HEMT Brief Introduction To Medici TM V THp and V Gmin for the MOS-gate HEMT Current Voltage Characteritic of the MOS-Gate HEMT Doping Concentration Dependence of V THp The Temperature Variation of the Threhold Voltage The Temperature Dependence of the I D -V D Characteritic V THp and V tl for Delta-Doped MOS-Gate HEMT Dicuion Chapter Concluion Summary of the Work Future Work Direction Reference vi

10 VITA vii

11 Lit of Table Table.1: Baic parameter of bulk Si 1-y Ge y, at 300 K Table 3. 1: Device and material parameter ued to calculate V THp at 300 K Table 3. : Device and material parameter for the calculation of V THp uing modified model Table 4. 1: The hole hall mobility at different temperature a etimated from the experimental plot Table 5. 1: Device and material parameter ued to calculate V THp of the delta-doped MOS-gate HEMT... 8 viii

12 Lit of Figure Figure 1. 1: The firt HEMT, fabricated by Fujitu Corp in Japan [3]... 4 Figure. 1: The cro-ectional view of a GaA/AlGaA n-channel HEMT... 1 Figure. : The cro-ectional view of a p-channel SiGe MOS-gate HEMT Figure. 3: The cro-ectional view of an n-channel SiGe MOS-gate HEMT Figure 3. 1: Cro-ectional chematic of a p-channel SiGe MOS-gate HEMT... 3 Figure 3. : Device chematic, charge ditribution and energy band diagram of the p- channel MOS-gate HEMT Figure 3. 3: (a) The I Dat -V G and (b) the I D -V D characteritic of the p-channel SiGe MOS-gate HEMT, uing the analytical model of Jain et al [16] Figure 3. 4: (a) The g m v. V G and (b) the f T v. L plot of the p-channel SiGe MOS-gate HEMT, uing the analytical model of Jain et al [16] Figure 3. 5: Device chematic, charge ditribution and energy band diagram of the p- channel MOS-gate HEMT, ued in the modified model Figure 3. 6: (a) The I Dat -V G and (b) the I D -V D characteritic, uing exiting and modified model... 5 Figure 3. 7: (a) The g m v. V G and (b) The f T v. L plot, uing exiting and modified model Figure 3. 8: (a) N v. V THp plot and (b) N v. V Gmin plot of the p-channel Figure 3. 9: (a) d i v. V THp plot and (b) d i v. V Gmin plot of the p-channel Figure 3. 10: (a) d v. V THp plot and (b) d v. V Gmin plot of the p-channel Figure 3. 11: (a) t ox v. V THp plot and (b) t ox v. V Gmin plot of the p-channel Figure 3. 1: (a) Mole fraction x v. V THp plot and (b) mole fraction x v. V Gmin plot of the p-channel SiGe MOS-gate HEMT Figure 4. 1: Temperature variation of the threhold voltage Figure 4. : Temperature variation of (a) the I Dat -V G and (b) the I D -V D characteritic.73 Figure 4. 3: g m v. V G plot of different temperature Figure 5. 1: The device tructure of the delta-doped MOS-gate SiGe HEMT Figure 5. : The device chematic, charge ditribution and band diagram ued to derive V THp expreion Figure 5. 3: (a) d 1 v. V THp plot and (b) d 1 v. V tl plot of the p-channel delta-doped Figure 5. 4: (a) δ v. V THp plot and (b) δ v. V tl plot of the p-channel delta-doped Figure 6. 1: Medici TM imulation reult ued to determine V THp and V Gmin Figure 6. : (a) Comparion of I Dat -V G plot obtained from analytical model and Medici Simulation and (b) I D -V D plot from Medici TM Figure 6. 3: I D -V G characteritic of the p-channel SiGe MOS-gate for different upply layer doping concentration ix

13 Figure 6. 4: Temperature Variation of V THp meaured from I D -V G plot at different temperature Figure 6. 5: Comparion of I D -V D characteritic at different temperature, obtained from Medici TM imulation Figure 6. 6: Medici TM imulation reult ued to determine V THp and V tl for the p-channel delta-doped SiGe MOS-gate HEMT x

14 Chapter 1 Introduction 1.1 Background In recent year the increaing need for high-peed communication and computation ha puhed ilicon-baed MOSFET technology to the edge. To meet the ever-increaing peed requirement, the gate length of the MOSFET ha been continually reduced. The gate length of the MOSFET ha reached uch mall dimenion that everal detrimental effect are taking place in the MOSFET. The overall behavior of the hort channel MOSFET i quite different from the long channel MOSFET. In thi ituation, cientit and engineer are contantly looking for new material a well a new device tructure that can replace ilicon-baed MOSFET and cope up with the peed requirement. 1. Limitation of Silicon-Baed MOSFET with Short Channel The hort channel device i ditinguihed from a long channel device by the following criteria [1]: 1. In hort channel device, the drain current no longer varie a 1/(gate length).. The threhold voltage decreae. 3. The gate voltage required to reduce the ub-threhold current become larger. 1

15 Variou effect occur due to the hrinking of the gate length and oxide thickne, which are known a hort channel effect. The gate length at which the hort channel effect tart to take place depend on the device dimenion, doping and other device parameter. Some of the hort channel effect are decribed below: Hot Carrier: A the gate length of the hort channel MOSFET i very mall, the critical electric field i reached for mall drain-ource voltage. Velocity aturation occur above the critical electric field. The carrier traveling at the aturation velocity v are called the hot carrier. Thee hot carrier can get trapped in the gate oxide and change the threhold voltage of the MOSFET. Alo, they can tunnel through the gate oxide (a the gate oxide of hort channel MOSFET i thin) and caue gate leakage current. Drain- Induced Barrier Lowering (DIBL): For an n-type MOSFET, the poitive potential at the drain terminal attract electron under the gate oxide. In hort channel device, thee electron caue lowering of the potential barrier between the drain and the ource. Thu, a the drain-ource voltage increae, potential barrier in the channel decreae. Eventually thi reduction in the potential barrier caue current flow between drain and ource even at a gate voltage lower than the threhold voltage. Thi current i called ubthrehold current. Oxide Breakdown: Maximum electric field acro a MOSFET gate oxide hould be limited to 7MV/cm []. Thu, to prevent poible oxide breakdown and enure

16 long-term operation, the gate bia i limited by a maximum value. The oxide thickne in a hort channel MOSFET i uually mall. A a reult, the maximum permiible gate voltage i mall for a hort channel MOSFET. Relationhip between drain current and gate voltage: In long channel MOSFET, the drain current in aturation region i proportional to the quare of the gateource voltage. For the hort channel, thi theory doe not apply. The aturation drain current varie linearly with the gate-ource voltage. Thi effect i the reult of the velocity aturation of the channel carrier. 1.3 HEMT: A Light of Hope High Electron Mobility Tranitor (HEMT) wa invented in the beginning of The HEMT i baically a high-peed heterojunction device, which ue two-dimenional electron or hole ga trapped in a quantum well a the carrier. The tructure of the HEMT i uch that the carrier in the channel are far from the ionized dopant atom. Thu, the carrier encounter mall Coulombic interaction with the ionized dopant atom and the carrier mobility i increaed. The formation of the quantum well in the HEMT retrict the carrier movement to only two dimenion. Thi i another reaon why the carrier in the channel acquire high mobility. Figure 1.1 [3] how the firt HEMT fabricated by Fujitu Corp, Japan. Thi i 3

17 Figure 1. 1: The firt HEMT, fabricated by Fujitu Corp in Japan [3]. a Schottky-gate GaA/AlGaA HEMT. The band diagram and device cro ection are alo hown in the figure. 1.4 MOS-Gate HEMT Integration of the HEMT into the CMOS microelectronic circuit i a very important tep that can enable u to ue the excellent propertie of the HEMT. Reearch effort are going on to make thi integration poible. One of the difficultie of integration i the fact that the traditional HEMT ue a Schottky gate. On the other hand, the CMOS circuit ue oxide-gate device uch a the MOSFET. To olve thi problem, reearch effort have been driven to deign and fabricate an oxide-gate HEMT. Mot of the HEMT fabricated in the early day ued GaA and AlGaA. But GaA or AlGaA do not have a native oxide. On the other hand, SiO poee attractive propertie that made it uitable for 4

18 microelectronic integration. To utilize SiO a the gate oxide of the HEMT device, Si baed material technology i incorporated in the deign of the MOS-gate HEMT. A a reult, the MOS-gate HEMT ha been fabricated by growing SiO on top of the Si/SiGe heterotructure [4]. 1.5 SiGe MOS-Gate HEMT The SiGe MOS-gate HEMT demontrate quite a few advantage over the traditional Schottky-Gate HEMT. Some of the advantage are lited below: The MOS-gate HEMT ha high input reitance like the MOSFET. The MOS-gate HEMT i capable of accepting a wider range of gate voltage wing than the Schottky-gate HEMT. For example, an n-channel Schottky-gate HEMT can accept only negative gate voltage. On the other hand, an n-channel MOS-gate HEMT can accept both poitive and negative gate bia. The gate voltage operating range of a MOS-gate HEMT can be controlled by varying the thickne of the oxide layer. Although fabrication of a MOS-gate HEMT i one tep toward the integration of a HEMT into maintream CMOS integrated circuit, more reearch challenge remain. The HEMT ha multiple layer and the number of layer i more than that of a conventional 5

19 MOSFET. That make it unuitable for the conventional CMOS proce, which deal with only a few layer. Alo, there are till ome problem aociated with the growth of oxide in the MOS-gate SiGe HEMT. Thee will be dicued in the next chapter. 1.6 Propoed Work In recent year, the HEMT ha been ubjected to lot of reearch effort in variou material ytem including SiGe. SiGe HEMT i one of the hot topic. The SiGe HEMT can be broadly claified into two group: the Schottky-gate SiGe HEMT and the MOSgate SiGe HEMT. Due to ome fabrication difficultie of the MOS-gate HEMT, more reearch effort have been devoted to Schottky-gate HEMT. But effort have been made to olve the fabrication problem of the MOS-gate HEMT and reearcher have come up with ome poible olution. Quite a few reearch group around the world are working on the SiGe MOS gate HEMT. Mot of the work concentrate on fabrication and experimental characterization of MOS gate HEMT. Not much work ha been devoted to the analytical modeling and imulation of thee device. A reliable device model i very important in device deign and in predicting the behavior of an exiting device. The analytical model can be ued a the firt tool in device deign, which i able to provide a rough, if not very accurate, idea about the behavior of the device to be deigned. A a econd tep of the deign proce, numerical imulation oftware uch a Medici TM, Davinci TM, Atla TM, etc. can be ued 6

20 Analytical modeling of the MOS-gate HEMT i the main cope of thi reearch work. Alo, device imulator Medici TM ha been ued to compare the reult obtained from thee analytical model. The main reearch goal of thi thei are to: (1) propoe a modified analytical model for calculating the threhold voltage V TH and the minimum gate voltage V Gmin of a p-channel MOS-gate HEMT, () calculate the current-voltage characteritic uing the modified model, (3) examine the effect of different device and material parameter on V TH and V Gmin, uing the modified analytical model, (4) introduce an analytical model to predict the temperature dependency of V TH and the current-voltage characteritic of the MOS-gate HEMT, (5) propoe a modified threhold voltage model for the delta-doped MOS-gate HEMT (6) derive the expreion for the minimum gate voltage V tl of delta-doped MOS-gate HEMT, (7) examine the effect of ome device parameter on V TH and V tl of the delta-doped HEMT, (8) perform Medici TM imulation of the MOS-gate HEMT to invetigate device behavior a well a compare the reult obtained from the analytical model to the reult of the Medici TM imulation. The p-channel MOS-gate SiGe HEMT ha been ued in the analytical model and the Medici TM imulation. But the analytical model can be extended to other MOS-gate HEMT with imilar tructure including the n-channel MOS-gate SiGe HEMT. 1.7 Outline of the Thei Chapter preent a review of the literature that include: HEMT, SiGe material propertie and recent work on SiGe MOS-gate HEMT. 7

21 Chapter 3 tart with the dicuion of the exiting model of the MOS-gate HEMT. Then, a modified model i propoed, and V THp, V Gmin, current voltage characteritic, tranconductance and cutoff frequency are calculated uing the modified model. The reult obtained from two model are compared. The lat part of thi chapter examine the effect of different device and material parameter on V THp and V Gmin. A imple temperature model for the MOS-gate HEMT i introduced in Chapter 4. The effect of temperature variation on threhold voltage, current voltage characteritic and tranconductance of the MOS-gate HEMT are invetigated uing the temperature model. A modified threhold voltage model for the delta-doped MOS-gate HEMT ha been propoed in Chapter 5 that can be applicable to any width of delta-doped layer a well a any ditance of the layer from the oxide interface. An expreion of the minimum gate voltage V tl for the delta-doped HEMT ha been derived and it value i calculated uing a p-channel SiGe delta-doped MOS-gate HEMT. The effect of doping concentration, ditance of the delta-doped layer from oxide interface and width of the delta-doped layer on V THp and V tl have been examined uing the model. Chapter 6 conit of the Medici TM imulation of the regular-doped and the delta-doped SiGe MOS gate HEMT. Thee imulation have been ued to explore the device behavior a well a the imulation reult have been compared to ome of the reult obtained from the analytical model. 8

22 Chapter 7 ummarize the work done in the thei. Alo, it include the dicuion of the future work direction that may help to improve the analytical model. 9

23 Chapter Literature Review.1 HEMT.1.1 What i HEMT HEMT (High Electron Mobility Tranitor) i a heterojunction device that ue the carrier in DEG or DHG for current tranport. It i one of the fatet olid-tate device ever reported. The name HEMT wa propoed by Fujitu Corp. Japan. Some other name of the HEMT, propoed by different laboratorie are: the MODFET (Modulation-Doped FET) (Univ. of Illinoi), the TEGFET (Two-Dimenional Electron-Ga FET) (Thomon CSF), the SDHT (Selectively-Doped Heterojunction Tranitor) (Bell Lab.) and the HFET (Heterojunction FET)..1. GaA/AlGaA HEMT The firt HEMT wa fabricated uing the compound emiconductor GaA and AlGaA [3]. Later, HEMT have been fabricated uing different material ytem uch a AlInA/GaInA [5], Si/SiGe [6], etc. AlGaA ha almot perfect lattice match with GaA for all the poible mole fraction of Al. Thu, a high quality epitaxial layer of AlGaA can be grown on GaA ubtrate. 10

24 Figure.1 how a typical tructure of an n-channel GaA/AlGaA HEMT. There exit dicontinuity in both the conduction band and the valence band at the GaA/AlGaA heterointerface. Electron diffue from the highly doped AlGaA region to the undoped GaA region and they are trapped in the almot triangular haped quantum well formed in the interface at the GaA ide. The motion of thee electron are retricted to two dimenion a they are not free to move in the direction normal to the heterointerface. The width of the quantum well being very thin, the electron energie in the well are quantized. The two lowet ubband are motly occupied and only they are included in the calculation of electron denity. A the electron motion are confined to only two direction, the mobility of the electron i certainly higher than that of the bulk material. Another reaon behind the mobility enhancement i the ditance of the electron in the channel from the ionized dopant impuritie [7]. The electron in the channel encounter much le Coulomb cattering than the electron in a typical MOSFET channel. Although at room temperature, the mobility i limited by optical phonon cattering, it can attain very high value at low temperature. The electron mobility at 100 K can be a high a twenty time of the mobility at room temperature [8]. For normal operation, the upply layer i totally depleted due to the gate voltage and the depletion at the AlGaA/GaA heterotructure. Thu, when voltage i applied between drain and ource, conduction can only take place through the DEG channel formed in the GaA layer. There exit two limit of the applied gate bia: the threhold voltage V TH 11

25 Source Gate Drain n + GaA n + GaA n AlGaA Supply Layer Intrinic AlGaA Spacer Layer DEG Channel n - Ge (Channel) Undoped GaA Buffer Layer Semi-Inualting GaA Subtrate Semi-Inulating GaA ubtrate Figure. 1: The cro-ectional view of a GaA/AlGaA n-channel HEMT. 1

26 and the minimum gate voltage V Gmin (for p-channel HEMT) or the maximum gate voltage V Gmax (for n-channel HEMT). The threhold voltage i defined a the gate voltage, which caue total depletion in the channel. Thu, above (for p-channel HEMT) or below (for n- channel HEMT) the gate voltage value of V TH, the HEMT i turned off. Again, at the gate voltage value of V Gmax or V Gmin, the upply layer i not anymore fully depleted. A a reult, conduction parallel to the channel take place in the AlGaA region. Thi parallel conduction reduce the gate tranconductance. For normal operation, the gate bia of the HEMT i kept within the range: V Gmin <V G <V TH (for p-channel HEMT) or V TH <V G <V Gmax (for n-channel HEMT)..1.3 SiGe MOS- gate HEMT A dicued in the previou chapter, the realization of the MOS-gate HEMT i a ignificant leap toward the integration of HEMT into maintream CMOS monolithic integrated circuit. SiO, being a teted oxide in integrated circuit, ha naturally become the choice for the gate oxide of the MOS-gate HEMT. A SiO i the native oxide of ilicon, the MOS gate HEMT ha been realized in the SiGe material ytem [4]. Figure. how a p-channel MOS-gate HEMT while Figure.3 how an n-channel MOS-gate HEMT. The operation of the MOS-gate HEMT i very imilar to that of the Schottky-gate (metal-gate) HEMT. The ilicon layer at the top of the SiGe upply layer i needed for growing SiO on the top. The SiGe buffer layer between the ubtrate and the channel i called the virtual ubtrate. 13

27 Gate Source Drain SiO Si p Si 0.5 Ge 0.5 Supply Layer p+ Ge p+ Ge Intrinic Si 0.5 Ge 0.5 Spacer Layer n - Ge (Channel) n Si 0.5 Ge 0.5 Virtual Subtrate Ge Subtrate Ge Subtrate Figure. : The cro-ectional view of a p-channel SiGe MOS-gate HEMT. 14

28 Gate Source Drain SiO Si n + Si n Si 0.5 Ge 0.5 Suuply Layer n + Si Intrinic Si 0.5 Ge 0.5 Spacer Layer p - Si (Channel) p Si 0.5 Ge 0.5 Virtual Subtrate Ge Subtrate Si Subtrate Figure. 3: The cro-ectional view of an n-channel SiGe MOS-gate HEMT. 15

29 The virtual ubtrate play a very important role in the trained SiGe material ytem. The thick and train relaxed layer of SiGe virtual ubtrate i grown on the ubtrate (Si or Ge). Then, (for p-channel HEMT hown in figure.) thin trained layer of Ge and Si 0.5 Ge. 05 i grown commenurately on top of the buffer layer [4]. The train on the Ge and Si 0.5 Ge. 05 layer and mot of the material propertie of the trained Ge and Si 0.5 Ge. 05 layer depend on the x value of the Si 1-x Ge x virtual ubtrate. Thu, compoition of the virtual ubtrate control the ize of the band dicontinuity and the quantum well formed in the heterointerface. Alo, the virtual ubtrate help better confinement of the carrier in the quantum well.. Propertie of SiGe and Si/SiGe Heterotructure SiGe material propertie and the behavior of Si/SiGe heterotructure play a very important role in the performance of the SiGe MOS-gate HEMT. In thi ection, ome of the important material propertie of bulk and trained SiGe, and Si/SiGe heterotructure will be briefly dicued. The band tructure of Si 1-x Ge x i imilar to that of Si for x<0.85. For x>0.85, the material demontrate Ge-like character with a conduction band L minima [9]. Thu, the bandgap (the ditance between the minimum of the conduction band and the maximum of the valence band) of the Si 1-x Ge x varie with compoition. Alo, when the heterotructure between SiGe and Si or Ge i formed, the bandgap i changed due to train. 16

30 The conduction band or the valence band offet i an important parameter to be conidered in a heterotructure. In the p-channel HEMT, large valance band offet i expected. On the other hand, large conduction band offet i required in an n-channel device. Not many reult are available for the Si/SiGe conduction band offet. But, lot of work ha been reported on theoretical and experimental tudie of the valence band offet in thi ytem [10] [11]. Obviouly, the band tructure and thu the band offet are crytallographic orientation dependent. For <100> SiGe heterotructure, in the cae of the trained Si 1-x Ge x grown on the relaxed Si 1-y Ge y virtual ubtrate, an average valence band offet expreion propoed by Rieger and Vogl [10] i given by, = ( y)( x )...1 E v, av y Thi valance band offet i calculated conidering only the hydrotatic train. Conideration of the effect of the uniaxial train component will provide more accurate reult. A mentioned before, the bandgap of SiGe depend on the train reulted from the peudomorphic growth. Alo, the bandgap i a function of temperature. For untrained Si 1-y Ge y, the bandgap at 4. K i given by [1], E g ( y,4.k) = y y, y < To predict the bandgap of the Si 1-y Ge y at higher temperature, Si-like bandgap-temperature relation i ued [13], 17

31 T E g ( y, T ) = Eg ( y,0k)....3 T Auming that, the bandgap at 0 K i almot equal to that at 4. K, T E g ( y, T ) = Eg ( y,4.k)..4 T Equation.4 can be ued to etimate the bandgap of the untrained Si 1-y Ge y for any compoition maller than 0.85 and at any temperature. For the trained Si 1-x Ge x, the bandgap can be etimated [14] uing the following equation, E g ( x,4.k) = x x...5 To etimate the temperature dependence of the bandgap of the trained Si 1-x Ge x, equation.4 (y replaced by x) can be ued. Table.1 how ome important parameter of bulk Si 1-y Ge y material at 300 K [15]..3 Development of SiGe MOS-Gate HEMT Although there till exit ome difficulty in the fabrication proce of SiGe MOS-gate HEMT, due to it attractive feature compared to the Schottky-gate HEMT, quite a few reearch group around the world are working on it. Some of the reearch reult reported in recent year will be dicued in thi ection. 18

32 Table.1: Baic parameter of bulk Si 1-y Ge y, at 300 K. Parameter Name Value Dielectric Contant y Effective Electron Ma (in unit of m 0 ) ~0.9 for y<0.85, ~1.59 for y>0.85 Electron Affinity (ev) y Lattice Contant (A ) y+0.07y Electron Mobility (cm /V-) y, 0 y < 0. 3 Hole Mobility (cm /V-) y, 0 y < 0. 3 Electron Diffuion Coefficient (cm /) 36-11y, 0 y < 0. 3 Hole Diffuion Coefficient (cm /) 1-y, 0 y < 0. 3 Electron Thermal Velocity (cm/).4x10 7, y< X10 7, y>0.85 Hole Thermal Velocity (cm/) y 19

33 In 1991, Murakami et al [4] reported the fabrication and characterization of traincontrolled SiGe delta-doped MOS-gate HEMT. Very high hole mobility ha been obtained in the DHG formed in the train-controlled Ge channel. Field effect mobility a high a 9500 cm /V- i etimated from the meaurement conducted at 77 K. In 199, Jain et al [16] propoed an analytical model for the SiGe MOS-gate HEMT. In thi work, a threhold voltage expreion for the MOS-gate HEMT ha been derived and the threhold voltage ha been calculated for a p-channel SiGe MOS-gate HEMT. Current-voltage characteritic, tranconductance and cutoff frequency for the HEMT are alo calculated uing thi model along with Chang-Fetterman extended model [17]. In a work publihed in 000, Lu et al [18] have reported the fabrication and characterization of a 0.1 µm gate length p-type SiGe Schottky-gate HEMT and a MOSgate HEMT. In their work, they have pointed out two important problem in the fabrication of SiGe MOS-gate HEMT: (1) high denity of interface tate of oxide formed on SiGe and () Ge egregation caued by the high temperature oxide proceing. They have compared two SiGe HEMT of imilar dimenion: one with MOS gate and the other with Schottky gate. The comparion how that the MOS-gate HEMT ha a wider gate voltage wing and lower leakage current than it counterpart. V. Gapari et al [19] have invetigated the effect of temperature on the tranfer characteritic of an n-channel Si/SiGe MOS-gate HEMT. They have experimentally meaured the extrinic tranconductance for different gate voltage at different 0

34 temperature ranging from 180 K to 300 K. It ha been found that at low temperature the off current i reduced and the ubthrehold lope become teeper. Their experimental reult have been compared with Medici TM imulation reult. Thi work wa publihed in 004. Velàzquez et al [0] have reported the deign of nearly body effect free Si/SiGe n- channel HEMT. They have numerically tudied the effect of doping of the etback layer and deigned the doping profile in uch a way that the effect of the ubtrate bia i ubtantially reduced. They have alo ued Medici TM and compared the imulation reult with reult obtained from the experiment. 1

35 Chapter 3 Modified Model for MOS-Gate SiGe HEMT In thi chapter, different apect of the analytical modeling of the SiGe MOS-gate HEMT will be dicued. Motly, the analye and dicuion will be limited to the p-channel MOS-gate HEMT. The analyi of the n-channel MOS-gate HEMT can be performed in a imilar fahion. At firt, the analytical model propoed by Jain et al [16] will be dicued. Then, the modified model including the effect of the ilicon cap layer will be propoed and the reult obtained from the two model will be compared. Lat part of thi chapter conit of the imulation reult and analye of the effect of the change of different device and material parameter uing the modified model. 3.1 Exiting Model for the MOS-Gate HEMT The threhold voltage expreion A threhold voltage expreion for the MOS-gate HEMT ha been reported in [16]. In thi work, a p-channel SiGe MOS-gate HEMT ha been ued to derive the threhold voltage equation. The tructure of the device i hown in Figure 3.1. Thi tructure combine a conventional MOSFET with the HEMT tructure. The p-channel SiGe MOSgate HEMT conit of a Si 0.5 Ge 0.5 (p-type) upply layer, an intrinic Si 0.5 Ge 0.5 pacer

36 Gate Source Drain SiO Si p + Ge p Si 0.5 Ge 0.5 Supply Layer p + Ge Intrinic Si 0.5 Ge 0.5 Spacer Layer n - Ge (Channel) n Si 0.5 Ge 0.5 Virtual Subtrate Ge Subtrate Ge Subtrate Figure 3. 1: Cro-ectional chematic of a p-channel SiGe MOS-gate HEMT. 3

37 layer, an n - -Ge channel, a Si 0.5 Ge 0.75 virtual ubtrate and a Ge ubtrate. Alo, a thin layer of ilicon i ued on the top of the upply layer, which i called ilicon cap. The purpoe of thi layer i to grow the SiO layer on the top. On the top of the gate oxide, the gate metal i depoited, which i Aluminum in thi cae. The threhold voltage expreion in [16] ha been derived without conidering the effect of the ilicon cap layer. The device tructure, charge ditribution and energy band diagram ued in the derivation i hown in Figure 3.. In thi derivation, d d, the width of the upply layer i defined a the ditance from the oxide/ilicon interface to the upply layer/pacer layer interface. The charge accumulated in the metal gate due to the applied gate voltage V G i compenated by part of the charge in the depletion layer of the Si 0.5 Ge 0.5 upply layer. Thi charge i termed a Q d. The ret of the charge in the depletion layer i compenated by the charge in the two dimenional hole ga (DEG) and a mall charge due to the depletion layer formed in the Ge channel. The derivation tart with the equation, V G Q D = VFB + + ψ.3.1 Cox where, V G i the applied gate voltage, V FB i the flatband voltage ψ i the urface potential, C ox i the oxide capacitance. 4

38 Figure 3. : Device chematic, charge ditribution and energy band diagram of the p-channel MOS-gate HEMT. 5

39 Uing the depletion approximation, the expreion for ψ and Q d are found a, qn ψ =.3. ε W dep Q d = qn W dep..3.3 where, N i the doping denity in the Si 0.5 Ge 0.5 upply layer, ε i the permittivity of the Si 0.5 Ge 0.5 upply layer = ε 0, ε 0 i the permittivity of the free pace. The expreion for V FB i given by, Qox V FB = φm.3.4 C ox ϕ m i the difference between the metal and the emiconductor (Si 0.5 Ge 0.5 ) work function, which i given by, E g E E v f φ m = φ m [ χ + ( Vd )].3.5 q q q where, χ i the electron affinity of the emiconductor, Φ m i the metal work function, E v i the valence band dicontinuity at the Si 0.5 Ge 0.5 /Ge Interface, V d i the potential difference between the maximum of the valence band and the bottom of the valence band dicontinuity, E g i the band gap of Si 0.5 Ge 0.5, E f i the Fermi level meaured for the top of the valence band dicontinuity. 6

40 Again, V d i expreed in term of V d = ρ ρ a, q q ρ d i ε N ε Two other ueful relation ued in the derivation are, ρ x m = + N d i and, W dep = ( d + d ) x d i m Uing equation 3.1 through 3.8, the final charge control equation can be derived. The charge control equation i given by, C ρ = q eq [ VG VTHp] 3.9 where, 1 C eq dd + di + d =, ε d = aε. q V THp i the threhold voltage. The expreion of V THp i given by, Eg V THp = φm χ q E + q v E q f 0 Q C ox ox qn d + C ox d qn d + ε d.3.10 The expreion for the n-channel MOS-gate HEMT can be derived in imilar fahion and i given by, 7

41 V = φ THn m E χ q c E + q f0 Q C ox ox qn d C ox d qn ε d d Calculation of V THp The threhold voltage for the MOS-gate HEMT hown in figure 3.1 will be calculated now. The device and material parameter ued to calculate the threhold voltage i lited in table 3.1. All the parameter are for 300 K. Some of the parameter value are not ame a the one ued in the work of Jain et al [16]. One of the reaon behind the difference i the fact that the calculation performed in [16] were for 77 K intead of 300 K. The electron affinity of Si 0.5 Ge 0.5 i calculated uing the following relation [15], χ = x..3.1 where, x i the mole fraction in Si 1-x Ge x. The bandgap of the trained Si 0.5 Ge 0.5 at 300 K i calculated uing the following equation [1] E g ( x,4.k) + 396x = x T E g ( x, T ) = E g ( x,0k) T Here, it i aumed that the band gap at 0 K i almot ame a the bandgap at 4 K. 8

42 Table 3. 1: Device and material parameter ued to calculate V THp at 300 K. Parameter Value Unit Φ m 4.10 EV χ 4.05 ev q 1.6X10-19 Coulomb E g ev E v 0.1 ev E f ev Q ox 1X10 11 Coulomb/cm t ox 50X10-7 cm a 0.15X10-1 V-cm - d d 30X10-7 cm d i 4X10-7 cm ε ε 0 ε X10-14 F/cm ε oxide 3.90 ε 0 N 5X10 17 cm -3 9

43 E f0 i zero at 300 K. The value of ε i determined uing the following relation, ε = x The calculation of the valence band dicontinuity require pecial attention. One might be tempted to ue the Anderon model [1] to calculate E v uing following equation, E v = E g E c Ec = χ1 χ.3.17 E g = E E g1 g.3.18 But the Anderon model i overimplified and it often fail to match the experimental data, mainly becaue it doe not include the effect of the urface tate, which i pretty dominant in mot heterotructure. The device under invetigation in thi thei conit of a trained SiGe/Ge interface peudomorphically grown on a relaxed SiGe buffer, the ue of the Anderon model would be inappropriate. Intead, the following relation derived by People and Bean [11] can be ued, E v = x( y) where, y i the mole fraction of the virtual ubtrate Si 1-y Ge y. y i 0.75 for the p- channel MOS-gate HEMT tructure ued in the thei. Uing the parameter pecified, the threhold voltage value i found a, V THp = V 30

44 It will be hown later that thi threhold voltage value i an overetimation Calculation of V Gmin The minimum gate voltage V Gmin i another important quantity of the HEMT. Unlike the MOSFET, the HEMT ha two boundarie for it gate bia. For p-channel HEMT, the gate voltage mut be le than the threhold voltage V THp to keep it ON. For gate voltage equal to or greater than V THp, the total channel i depleted and thu, there i no free carrier to conduct current flow from the ource to the drain. Again, the gate voltage ha to be greater than the minimum gate voltage V Gmin for normal operation. At gate voltage equal to or lower than V Gmin, the SiGe pacer layer i not fully depleted. Thu, there exit a conducting path in the SiGe upply layer parallel to the Ge channel. The device tranconductance goe down due to thi parallel conduction. Thu, thi parallel conduction i not expected for the normal HEMT operation. A a reult, HEMT operation i uually retricted within the gate voltage range: V Gmin <V G < V THp. The V Gmin value can be calculated uing the following relation, qρ o V G min = + Ceq V THp..3.0 where, ρ o i the equilibrium channel charge In the work of Jain el al [16], the value of ρ o ha been aumed to be 10 1 cm -. Uing thi value the minimum gate voltage V Gmin i calculated a, V Gmin = V 31

45 In the later part of thi chapter, a way to calculate more accurate value of ρ o will be dicued. Accurate etimation of ρ o help to get a better etimate of the value of V Gmin The current-voltage characteritic The baic analytical model developed by Chang and Fetterman [] ha been ued to calculate the current-voltage characteritic of the HEMT. There are everal model available for calculating the current-voltage characteritic of the HEMT. The Chang- Fetterman model [] ha been ued a it ha been hown to be effective to match the experimental data. The baic Chang-Fetterman model ha been derived for the n-channel HEMT. Thu, a et of modified equation will be ued here, which are valid for p-channel HEMT. The drain current I D i given by, I D [ µξ x = G0 VG VTHp V ( x)][1 exp( )] v where, G =C 0 eqv Z, µ = The hole mobility in the trained Ge channel, v = the hole aturation velocity in the Ge channel, ξ x = The electric field in the direction from ource to drain, Z= the gate-width of the HEMT. 3

46 Now, conidering the fact that ξ x = I t dv ( x) dx, equation 3.1 can be written a, L dt v LG0 =.3. t ln(1 + t) µ D to I D where, t ( x) =, G ( V V V ( )) 0 G THP x t 0 = G 0 ( V G I V D THP I D, R ) S t L = G ( V V I D V + I R 0 G THP D D d, ) R = Source Reitance, R d = Drain Reitance. Again, Saturation current I Dat i given by, I D at to 1 dt v LG0 =. 3.3 t ln(1 + t) µ where, I Dat t o =. G ( V V I R ) 0 G THp Dat Uing Equation 3.3, the I Dat v. V G plot i obtained which i hown in Figure 3.3 (a). A expected, the aturation current varie almot linearly with the gate voltage. A the gate voltage approache the threhold voltage ( volt in thi cae), the current goe to zero. Obviouly, any parallel conduction or leakage current i neglected in thi imple model. The entire current i aumed to be only due to the channel charge. 33

47 (a) (b) Figure 3. 3: (a) The I Dat -V G and (b) the I D -V D characteritic of the p-channel SiGe MOS-gate HEMT, uing the analytical model of Jain et al [16]. 34

48 The I D -V D plot i obtained uing equation 3. and the I Dat -V G plot. Thi plot i hown in Figure 3.3(b). It hould be noted that in thi model, the variation of aturation current with drain to ource voltage (V DS ) i ignored. Thu, the current in the aturation region i contant for a fixed gate voltage. A hole mobility value of 500 cm /V- i ued in the calculation of current voltage characteritic. Thi value i etimated uing [8]. In [8], the hole mobility value i almot 1000 cm /V, which i obtained experimentally for a deltadoped HEMT device that conit of a trained Ge channel grown on a cubic Si 0.3 Ge 0.7 buffer. A delta-doped HEMT ha higher mobility than a regular HEMT. Thu, it ha been aumed that for the regular-doped HEMT an approximate mobility value of 500 cm /V- can be ued. The hole aturation velocity of 3X10 7 cm/ i ued in the calculation of Jain et al [16]. Thi value i definitely an overetimation. A hole aturation velocity of 4X10 6 cm/ ha been ued in thi thei. Ue of thi value provide a cloer agreement between the current-voltage characteritic calculated from the analytical model and that obtained by the Medici TM imulation The tranconductance (g m ) and cutoff frequency (f T ) The tranconductance g m i an important parameter of any tranitor, which determine it mall ignal (ac) performance. The expreion that ha been ued to calculate g m i 35

49 given below: g m = V G V THp I Dat v L + ( )ln(1 to ) µ Uing thi relation, g m ha been calculated for different gate voltage. The g m v. V G plot i hown in Figure 3.4(a). It i clear that a the gate voltage goe down from the threhold voltage, the aturation tranconductance goe up abruptly. At around a gate voltage value of volt, the value of g m tart to aturate and it remain baically contant over the ret of the gate voltage range. It hould be noted that in practical cae, g m value aturate and then tart going down for gate voltage le than V Gmin. The current-voltage model ued in thi thei conider current due to channel conductance only and thereby ignore any parallel conduction path uch a the pacer layer. A mentioned before, for a gate voltage value le than V Gmin, parallel conduction take place in the pacer layer. Thu, the g m calculated here i baically the aturation tranconductance due to the channel current only. But, for normal operation, gate bia i uually limited within the range: V Gmin <V G < V THp. Thu, although, g m value calculated here i not very accurate for V G < V Gmin, it i not a big concern. The cutoff frequency i another important figure of merit. Due to it high mobility, the HEMT ha quite high cutoff frequency. The ue of MOS gate in HEMT, further enhance 36

50 (a) (b) Figure 3. 4: (a) The g m v. V G and (b) the f T v. L plot of the p-channel SiGe MOSgate HEMT, uing the analytical model of Jain et al [16]. 37

51 gate length, thu higher f T value can be obtained. 3. The Modified Model 3..1 The threhold voltage expreion In the exiting model, the threhold voltage expreion ha been derived completely ignoring the effect of ilicon cap layer. A modified model for threhold voltage will be propoed, which will include the effect of the ilicon cap layer and thu provide a more accurate etimation of the threhold voltage. Figure 3.5 how the device chematic, charge ditribution and band diagram for the modified model. The derivation tart by conidering the charge neutrality condition, q ( x d ) = qρ + qn W..3.5 m i Ge Ge A the Ge layer i unintentionally doped, ρ >> N Ge W Ge. Thu, equation 3.5 can be written a, q( x m d ) = qρ i ρ x m = + N d i Again, Wdep = d d + d + d i x m

52 Figure 3. 5: Device chematic, charge ditribution and energy band diagram of the p-channel MOS-gate HEMT, ued in the modified model. 39

53 Uing equation 3.4, Wdep = d + d - d ρ N. 3.8 Again, the gate voltage i given by equation 3.1 a, V G Q ψ D = VFB Cox Finding ψ : Conidering the electric field and potential in the direction normal to the gate oxide/si interface and uing Gau law, dξ dx qn i =, 0 ε i < x < d dξ qn =, d dx ε < x < W dep.3.30 In the region, d <x<w dep, qn ξ ( x ) = x + A ε where, A i the contant of integration At, x=w dep, ξ ( x) = 0. Uing thi boundary condition in 3.7 yield, 40

54 qn A = Wdep..3.3 ε Subtituting the value of A in equation 3.7, + At x = d, qn ξ ( x) = ε ( W dep x) qn = + + ξ ( x d ) = ( W dep d ) ε Now, for 0<x<d, qn ξ ( x ) = x + B.3.35 ε where, B i the contant of integration. At x = d, the electric flux D ha to be continuou, D( d + i ) ε ξ ξ ( d D( d = ) ) = ε ξ ( d ( d ε ) = ε ξ ( d + i + ) ) ξ ( d ) = qn ε ( W d + dep i )

55 Subtituting thi value in equation 3.31 yield, qn qn i B = ( Wdep d ) + d ε ε i i Thu, the electric field can be expreed a, qn qn i ξ (x) = ( Wdep d ) + ( d x), 0<x<d ε ε i i qn and, ξ ( x) = ( W dep x), d <x<w dep ε The potential function can be found by integration of equation 3.38 and 3.39, ψ (x) = qn ε i ( W dep d ) x qn ε i i ( d x x ) + K1, 0<x<d qn x ψ (x) = ( Wdep x ) + ε where, K 1 and K are the contant of integration. K, d <x<w dep At x=w dep, the potential become zero, ψ ( Wdep ) = Uing equation 3.4 with equation 3.41 yield, K qn = W dep ε 4

56 Thu, in the range, d <x<w dep, qn x ψ (x) = ( Wdep x ) + ε qn ε W dep At x=d, qn d ψ ( d ) = ( Wdepd ) + ε qn ε W dep Now, potential ha to be continuou acro the Si/SiGe interface. Thu, ubtituting the value of ψ d ) in equation 3.36 and performing ome algebraic manipulation, the ( following expreion i obtained, qn qn qn K d ) d i = ( Wdep d ) + d + ( Wdep ε ε i ε i It i clear from Figure 3.5 that, K = ψ (0) = ψ Combining equation 3.46and equation 3.47, the expreion for ψ i found a, ψ = qn ε ( W dep d ) qn + ε i i d qn + ε i ( W dep d ) d.3.48 Finding V FB : The expreion for the flat band voltage V FB i given by equation 3.4 a, 43

57 V = FB Q ox φm. Cox Again, the difference between the metal and the emiconductor work function i given by, φ m = m φi φ where, φi i the work function of ilicon. The ilicon ued in the ilicon cap layer i normally lightly doped p-type material. It can be hown with the help of the band diagram that the work function of the p-type Si 0.5 Ge 0.5 i almot equal to the work function of the lightly doped ilicon layer. Thu φ m can be expreed a, φm φm φ where, φ i the work function of Si 0.5 Ge 0.5. Thu, uing equation 3.5, E g E E v f φm φ m [ χ + ( Vd )] q q q The flatband voltage can be written a, V FB E E g v f = φm [ χ + ( Vd )] q q E q Q C ox ox

58 Finding Q d : From Figure 3.5, it i clear that the depletion charge due to the gate voltage, Q d i given by, Q d = qn d + qn ( W d ) i dep The Threhold Voltage: Subtituting the value of Ψ, V FB and Q d from equation 3.48,3.5,3.53 into equation 3.1 yield, VG = qn ε ( W dep d ) qn + ε i i d qn + ε i ( W dep d ) d + φ m χ E g q E + ( q v V d E f q Q ) C ox ox qn id + qn ( Wdep d ) C ox Now, uing depletion approximation, it can be hown that, V d = qρ qρ di ε N ε Uing the value of V d in equation 3.54 yield, VG = qn ε ( W dep d ) qn + ε i i d qn + ε i ( W dep d ) d + φ m χ E g q E + q v qρ ( ε N qρ d + ε i E f q Q C ox ox qn id + qn ( Wdep d ) C ox Uing the linear approximation, E f can be expreed a, 45

59 E f = E + aqρ fo where, E fo i the y-axi intercept of the E f v.ρ plot. a i the lope of the plot. Subtituting thi value of E f in equation 3.56 and performing ome algebraic manipulation, the final charge control expreion i obtained, C ρ = q eq [ VG VTHp The equivalent or total capacitance C eq i given by, ] C eq = d d + di + d ε d + ε i 1 + C ox where, ε a d =. q The modified expreion for threhold voltage V THp i given by, V THp = φ m χ Eg q E + q v E q fo Q C ox ox qn + ε d d qn i + ε i d qn d qn qn + + d + d d d i ε i Cox Cox d 3.. Calculation of threhold voltage uing modified model Now, the threhold voltage for the p-channel HEMT hown in Figure 3.1 will be calculated, uing the modified threhold voltage expreion. The device and material parameter ued to calculate V THp i hown in Table 3.. It hould be noted that, in thi cae, a d d value of 5 nm i ued intead of 30 nm. Actual value of the width of the SiGe 46

60 Table 3. : Device and material parameter for the calculation of V THp uing modified model. Parameter Value Unit Φ m 4.1 ev χ 4.05 ev q 1.6X10-17 Coulomb E g.7755 ev E v.1 ev E 0 ev f0 Q ox 1X10 11 Coulomb/cm t ox 50X10-7 cm a 0.15X10-1 V-cm - d d 5X10-7 cm d 5X10-7 cm d i 4X10-7 cm ε ε 0 ε X10-14 F/cm ε oxide 3.90 ε 0 ε i 11.7 ε 0 N 5X10 17 cm -3 N i 5X10 8 cm -3 47

61 layer i 5 nm in both cae. But, in the Jain model, a the effect of the ilicon layer have been ignored, the width of the ilicon layer (5 nm in thi cae) had to be included, in the d d value. In the modified model, a the width of the SiGe pacer layer and the ilicon layer have been defined by the term d d and d repectively, the d d value of 5 nm can be ued. The addition of d d and d (d d +d ), pecifie the total ditance from the gate oxide to the upply/pacer interface. In ection 3.1., the method of calculating parameter like χ, E v, E fo etc have already been dicued. A mentioned before, the p-type ilicon-cap layer i uually very lightly doped. Here, a doping concentration value of 5X10 8 cm -3 ha been elected for the ilicon layer. The width of the ilicon layer i 5 nm. The main reaon of the ue of the ilicon layer i to grow the gate oxide (SiO ). Thu, it i expected that mot of the ilicon layer originally grown on the pacer layer i conumed in growing SiO. Still, there exit a definite layer of ilicon (few nanometer) that have been taken into account and termed it width a d. Later, it will be hown that the threhold voltage change with the variation in the value of d. Uing the parameter pecified in Table 3., the threhold voltage ha been calculated a, V THp =.4839 V A it will be hown later, thi value i very cloe to the threhold voltage value obtained from Medici TM imulation. 48

62 3..3 Calculation of V Gmin The value of V Gmin can be calculated uing equation 3.0, qρ V Gmin = C o eq + V THp An expreion will be derived that can help to predict ρ o accurately. From equation 3.55 V d can be expreed a, qρ qρ d V d = + ε N ε i Aρ + Bρ V = d where, A= q ε N, B= qd i. ε Now, from Figure 3.5, it i clear that, V d can be expreed a, V d = E E E E ) v f ( F v E F -E v i the difference between the Fermi level and the maximum point of the valence band in SiGe. Thi difference i calculated uing the following relation, E F N Ev = kt ln( ) N v From equation 3.57, E f can be expreed a, E f = E + aqρ. fo 49

63 Uing equation 3.57, 3.6 and 3.63in equation 3.61and performing ome algebraic manipulation, the expreion for ρ i found a, ρ = ε N q [ E v E fo N + kt ln( N v )] + N ( d i + d) N ( d i + d).3.64 Again, from equation 3.9, Ceq ρ = [ VG VTHp ]. q V Gmin i the gate bia at which the gate depletion and the channel depletion of the SiGe pacer layer jut touch each other, but they do not interpenetrate. For thi condition, the equilibrium channel charge ρ o can be found uing the imultaneou olution of equation 3.9 and Thu, the value of ρ o can be found uing equation 3.64 and the V Gmin value can be found uing equation 3.55, qρ V Gmin = C o eq + V THp The ρ o value ha been calculated uing equation The calculated value i X10 11 cm -. Uing thi value in equation 3.55 yield a V Gmin value of, V Gmin = V. 50

64 3..4 Current-voltage characteritic The current-voltage characteritic for the modified model are calculated uing [] a before. The ame equation and parameter are ued; except the value of threhold voltage V THp i different here. The I Dat v. V G plot i hown in Figure 3.6(a). The ame plot for the previou model i alo hown in the ame figure. Comparion of thee two plot how that, the exiting model overetimated the current value. The I D -V D plot for different V G value, for both model are hown in Figure 3.6(b). Again, it i evident from the comparion that the exiting model overetimated the current value Tranconductance (g m ) and cutoff frequency (f T ) The g m v. V G plot uing both model are hown in Figure 3.7(a). For gate voltage cloe to the threhold voltage, the value of g m uing exiting model i higher than the modified model. But for lower voltage, where g m value aturate, both model yield almot ame 51

65 (a) (b) Figure 3. 6: (a) The I Dat -V G and (b) the I D -V D characteritic, uing exiting and modified model. 5

66 (a) (b) Figure 3. 7: (a) The g m v. V G and (b) The f T v. L plot, uing exiting and modified model. 53

67 value. It i evident that in thi region, g m value i not very enitive to the threhold voltage value. The plot of f T v. L for both model are hown in Figure3.7 (b). It i clear from the plot that the exiting model lightly overetimate the value of the cutoff frequency for all gate length. 3.3 Effect of Device and Material Parameter on V THp and V Gmin In thi ection, the effect of ome device parameter uch a doping concentration (N ), ilicon cap layer width (d ), pacer layer width (d i ) and oxide thickne on the value of threhold voltage V THp and minimum gate voltage V Gmin will be analyzed. Alo, the effect of the mole fraction value x of Si 1-x Ge x on V THp and V Gmin will be invetigated. All analye will be performed uing the modified model Effect of doping concentration The threhold voltage and the minimum gate voltage can be changed by varying the doping of the SiGe pacer layer. Figure 3.8(a) how the N v. V THp plot. The threhold voltage increae almot linearly with the increae in the doping concentration. The linear variation can be expected by looking at the expreion of V THp, which contain only firt order term of N. For a doping denity variation from 4X10 17 cm -3 to 1X10 18 cm -3, the 54

68 (a) (b) Figure 3. 8: (a) N v. V THp plot and (b) N v. V Gmin plot of the p-channel SiGe MOS-gate HEMT. 55

69 V THp value change from 1.8 V to 5.6 V. Thu, it can be concluded that V THp i trongly dependent on the doping denity of the pacer layer. The plot of N v. V Gmin i hown in figure 3.8(b). Similar to V THp, V Gmin alo varie linearly with doping denity. The total change in V Gmin i almot ame a the total change in V THp. Thu, the operating voltage range of the HEMT (V THp ~V Gmin ) remain almot contant with the variation of the doping denity Effect of pacer layer width The pacer layer width d i v. V THp plot i hown in Figure 3.9(a). The expreion of V THp doen t include any term conit of d i. Thu, the threhold voltage remain practically indifferent to the change of the pacer layer width. On the other hand, the plot of d i v. V Gmin in Figure 3.9(b) how that V Gmin i weakly dependent on the value of d i. The expreion for V Gmin contain the term C eq. C eq i dependent on d i ; thu making the V Gmin value dependent on d i. 56

70 (a) (b) Figure 3. 9: (a) d i v. V THp plot and (b) d i v. V Gmin plot of the p-channel SiGe MOS-gate HEMT. 57

71 3.3.3 Effect of ilicon layer Width The incluion of the effect of ilicon cap layer in the expreion of V THp i the main goal of the modified model. A a reult, uing the modified model, the effect of the variation of the width of the ilicon cap layer can be analyzed. A mentioned before, the main purpoe of the ilicon cap layer i to grow the SiO gate inulator, which i an inevitable part of the MOS tructure. It i expected that, mot of the ilicon layer would be conumed in thi oxidation proce. In mot cae, there remain a thin layer of ilicon after the oxidation i done. Depending on the thickne of thi layer it might have a ignificant or negligible effect on V THp and V Gmin value. Figure 3.10 how the d v. V THp plot. The variation i almot linear. The expreion of V THp include econd order term of d. But a the coefficient of the econd order term i Ni, which i much maller than N, the econd order term ha almot negligible effect in the variation of V THp. It i evident from the plot that the variation of d moderately affect the value of V THp. For a large doping denity of the ilicon layer, the variation would be larger. The variation of V Gmin hown in Figure 3.10(b) demontrate a weak dependence of V Gmin on the value of d. The operating voltage range i lightly varied with the variation of d. 58

72 (a) (b) Figure 3. 10: (a) d v. V THp plot and (b) d v. V Gmin plot of the p-channel SiGe MOS-gate HEMT. 59

73 3.3.4 Effect of oxide thickne The incluion of the gate oxide make the MOS-gate HEMT different from the Schottkygate HEMT. The t ox v. V THp plot i hown in Figure 3.11(a). The threhold voltage how a trong dependence on the election of the oxide thickne. The dependence i almot linear. The expreion of V THp doe not contain any term of t ox. But it contain term with 1/C ox. 1/C ox i directly proportional to t ox. Thi explain the linear dependence. Figure 3.11(b) how the t ox v. V Gmin plot. It i evident that, V Gmin i not o trongly dependent on t ox a the threhold voltage. The V Gmin expreion conit of a term -1/C ox. Thi term compenate part of the change that occur due the change in V THp. It i clear from the comparion of Figure 3.11(a) and (b) that increaing the oxide thickne can reult in effective increae in the operating gate voltage range. On the other hand, exceive increae in t ox reult in poor tranconductance value. Thu, an optimum value of t ox hould be elected while deigning a MOS-gate HEMT, o that an wide operating voltage range, high cutoff frequency a well a reaonable gate tranconductance can be achieved. One hould alo keep in mind that the minimum value of t ox i determined by the breakdown electric field. 60

74 (a) (b) Figure 3. 11: (a) t ox v. V THp plot and (b) t ox v. V Gmin plot of the p-channel SiGe MOS-gate HEMT. 61

75 3.3.5 Effect of mole fraction in Si x Ge 1-x The p-channel MOS-gate HEMT device ued in the imulation conit of Si 0.5 Ge 0.5 pacer and upply layer. The variation of the mole fraction x in Si 1-x Ge x reult in the change of bandgap, valence band dicontinuity ( E v ) and other parameter. The effect of the choice of the mole fraction x on V THp and V Gmin will be invetigated in thi ection. Figure 3.1(a) how the dependence of the threhold voltage on the mole fraction. V THp how an almot linear and moderate dependence on x. A the pacer layer material approache Ge (higher value of x), the threhold voltage increae. A it i evident from Figure 3.1(b), V Gmin how an oppoite reaction to the increae in x value. V Gmin value goe down a the x value i increaed. Thu, for higher value of x, larger operating voltage range can be achieved. 3.4 Concluion In thi chapter, the exiting model, the modified model and the comparion between them have been dicued. It i evident from the dicuion that the modified model include more device and material parameter and thu more accurately predict device behavior. Uing the modified model, the effect of ome device and material parameter on the threhold voltage, V Gmin and the operating voltage range have been hown. Thee imulation reult can be really handy to a device deigner at the initial phae of 6

76 (a) (b) Figure 3. 1: (a) Mole fraction x v. V THp plot and (b) mole fraction x v. V Gmin plot of the p-channel SiGe MOS-gate HEMT. 63

77 the deign. Certainly, to obtain a more accurate etimation of the threhold voltage and other parameter, ome kind of numerical analyi oftware or device imulator (uch a Medici TM, Davinci TM, ATLAS TM ) ha to be ued. 64

78 Chapter 4 The Temperature Model In the previou chapter, the p-channel MOS-gate HEMT behavior uing the modified model ha been dicued. All the analye have been performed at room temperature (300 K). In thi chapter, the effect of temperature variation on the behavior of the HEMT will be invetigated. The HEMT ha a wide range of application that include low temperature electronic a well a high temperature application. Uing the modified model, effort will be made to theoretically predict the effect of temperature change on the threhold voltage, tranconductance and current-voltage characteritic. 4.1 Effect of Temperature on Threhold Voltage According to the modified model, the threhold voltage expreion i given by, V THp = φ m χ Eg q E + q v E q fo Q C ox ox qn + ε d d qn i + ε i d + qn d d d qn i qn + d + ε i Cox Cox d d The effect of the temperature variation on different term in thi expreion will be invetigated and combining the effect, the temperature dependence of the threhold voltage will be predicted. 65

79 4.1.1 Temperature dependence of doping concentration The temperature range under conideration i from 100 K to 450 K. 100 K i much higher than the freeze out temperature of Si, Ge or SiGe. Thu, for the entire temperature range (100 K< T < 450 K), it can be aumed that the doping concentration N and N i are almot contant Temperature dependence of metal work function (ϕ m ) Uing the free electron model, it can be hown that the temperature dependence of the metal Fermi level can be expreed a [3], E Fm π kt ( T ) = EFmo[1 ( ) ].4. 1 E Fmo Here, E Fmo i the metal Fermi level at 0 K, which can be evaluated uing following relation, h 3n 3 EFmo ( )( ) 8me π = where, m e i the ma of electron, h i Planck contant, n i the total number of valence electron in the metal/unit volume. Normally, the value of E Fmo i much larger than the value of kt. A a reult, the metal Fermi level how extremely weak temperature dependence. 66

80 Auming the vacuum level being contant, it can be concluded that the change in the metal Fermi level i equal to the change in the metal work function. Obviouly, the ign of the change i oppoite. Thu, if the change in the metal Fermi level with temperature can be calculated, the change of the work function can be found. Uing the material parameter of Al, the change of the metal Fermi level ha been calculated for a temperature change of 0 K to 300 K. The change i very mall (few mv). Thu, in thi preent analyi, it can be eaily aumed that the metal work function i eentially contant Temperature dependence of emiconductor electron affinity (χ ) The applied gate bia voltage V G i independent of temperature. Two component of the gate voltage, ψ and Q d are independent of temperature variation. A a reult, the third component ϕ m ha to be contant alo. It ha been already proved that the metal work function i eentially contant. Thi make ϕ independent of temperature variation. The temperature independence of ϕ implie that the Fermi level of the emiconductor will be almot contant over the temperature range. Thu, it can be aumed that the change in the electron affinity of the SiGe upply layer i half of the change of it bandgap. Of coure, the ign of the change i oppoite. Thi relation can be expreed in an equation a, 67

81 1 χ ( 300) χ ( T ) = [ Eg ( T ) E g (300) ]..4.4 The value of the electron affinity of the SiGe layer at 300 K i known. Thu, uing equation 4.4, the temperature dependence of χ can be calculated Temperature dependence of bandgap The temperature dependence of the bandgap of the trained Si 1-x Ge x pacer layer i calculated uing equation 3.13 and 3.14 [9] mentioned in the previou chapter, E g (x, 4. K) = x x E g (x, T) = Eg(x,0 K) T T Temperature dependence of E v A mentioned in the previou chapter, the valence band dicontinuity E v i calculated uing the People and Bean [11] equation. The model doe not pecify the temperature dependence of E v. The other available model [10] to calculate E v of a trained ytem alo do not how any conideration of temperature variation. In the preent analyi, it i aumed that E v i temperature independent. 68

82 4.1.6 Temperature dependence of E fo There i no accurate model available to predict the temperature variation of E fo. But it i known that at 300 K, E fo =0 and at 77 k E fo =0.05 ev. It have been aumed that a linear variation of E fo between thee two temperature. With thi aumption, E fo (T) can be expreed a, E fo ( T ) = T Calculation of temperature dependence of V THp The other parameter in the threhold voltage equation can be aumed independent of the temperature variation. Thu, taking into conideration the temperature dependence of the parameter mentioned in the previou ection, the temperature dependence of the threhold voltage can be calculated. A MATLAB program ha been ued to calculate the temperature dependence of V THp. Figure 4.1 how the T v. V THp plot. A temperature range of 100 K to 450 K i conidered which fall within the extrinic temperature range of the emiconductor. A a reult, n 0 N. It i evident from the plot that the threhold voltage how weak dependence on temperature. The variation of V THp i almot linear. Over a temperature change of 350 K 69

83 Figure 4. 1: Temperature variation of the threhold voltage. (100K to 450 K), the threhold voltage change only 8 mv. In practical cae, a little more variation might be oberved, due to the variation of ome other device and material parameter that have not been included in thi analyi. 4. Effect of Temperature on Hole Mobility (µ h ) The hole mobility of the trained Ge channel varie with temperature. With increment in temperature, the mobility value goe down. At thi point, no accurate analytical model i available to decribe the temperature dependence of the hole mobility in the trained Ge 70

84 channel. The experimental data for temperature dependence of the hole hall mobility in the trained Ge channel reported in [8] will be ued in thi analyi. It hould be noted that thi data i obtained uing trained Ge channel on cubic Si 0.3 Ge 0.7. In the device under dicuion, the virtual ubtrate i not Si 0.3 Ge 0.7, but Si 0.5 Ge 0.5. Still, it can be aumed that the temperature dependence of mobility can be approximated uing the available data. Alo, the data i for a delta-doped HEMT tructure, which ha higher mobility than the regular-doped HEMT. A mobility value of 1000 cm /V- at 300 K i reported in [8]. In the preent analyi, the mobility value at 300 K i 500 cm /V-. Thu, the experimental mobility value at other temperature can be caled down by the factor of to ue in the preent analyi. The mobility value ued in the calculation are hown in Table 4.1. It hould be noted that the mobility value at 400 K wa not available in the experimental data. It ha been attempted to etimate the value uing linear approximation. Table 4. 1: The hole hall mobility at different temperature a etimated from the experimental plot. Temperature (K) Hole Hall Mobility (cm /V-)

85 4.3 The Current-Voltage Characteritic The current-voltage characteritic for different temperature have been calculated uing the Chang-Fetterman model []. It ha been aumed that the only two temperature dependent parameter in the Chang-Fetterman current-voltage characteritic are the threhold voltage and hole mobility. Figure 4.(a) how the I Dat v. V G plot. It i evident that the aturation current decreae a the temperature goe up. The current value at 100 K and 00 K are very cloe. A the temperature goe up from 00 K to 300 K, bigger reduction in the current value are noticed. Again, in the temperature range of 300 K to 400 K, the change in the current value are even larger. I D -V D characteritic for different temperature at gate voltage 1 V are hown in Figure 4. (b). Due to the high mobility at low temperature, the current increae very quickly with the increment of V DS in the linear region. A the temperature goe up, the mobility decreae due to advere effect uch a optical phonon cattering. A a reult, the current doe not increae a fat a in the cae of low temperature. In the aturation region, the current value remain contant, a the channel length modulation effect ha not been taken into conideration. 4.4 Tranconductance Figure 4.3 how the g m v. V G plot for different temperature. A the temperature goe up, tranconductance goe down. For the gate bia value cloe to the threhold voltage, g m i enitive to both temperature and threhold voltage. At lower gate voltage, g m 7

86 (a) (b) Figure 4. : Temperature variation of (a) the I Dat -V G and (b) the I D -V D characteritic. 73

87 Figure 4. 3: g m v. V G plot of different temperature. become almot contant. For lower temperature, thi aturation occur early. At high temperature, aturation take place lowly. A g m aturate, it become le enitive to temperature. 4.5 Concluion In thi chapter, a imple temperature model ha been preented. Uing thi model, the temperature dependence of the threhold voltage, current voltage characteritic and tranconductance have been predicted. It hould be noted that the aturation velocity (v ) 74

88 ha been aumed to be independent of the temperature variation. Thi i definitely not true in practical cae. No accurate analytical model or experimental data i available that could help predicting the temperature variation of v in the trained Ge channel. If it i aumed that v i weakly temperature dependent, thi temperature model hould be able to give u a rough idea of the temperature dependency. 75

89 Chapter 5 Analytical Modeling of Delta-Doped HEMT In a delta-doped HEMT, the whole upply layer i not doped. Intead, a very thin layer in the upply layer i doped. There are quite a few advantage of the delta-doped HEMT over the regular-doped HEMT that include increae in mobility and cutoff frequency. By varying the poition and width of the doped layer, one can control the cutoff frequency, the threhold voltage, the minimum gate voltage and the tranconductance. In the beginning of thi chapter, the exiting threhold voltage model for the delta-doped MOSgate HEMT will be preented and it limitation will be dicued. Then, a modified model will be propoed that i valid for any width of the doped layer. Alo, an expreion for the minimum gate voltage V tl will be derived. In the lat part of thi chapter, the effect of different material and device parameter on V THp and V tl are invetigated. 5.1 The Exiting Threhold Voltage Model The threhold voltage expreion An expreion of the threhold voltage for the p-channel delta-doped SiGe MOS-gate HEMT ha been propoed by Gokhale et al. [4]. The expreion of V THp i given by, V E E E Q g v fo ox d d 1 THp = φm χ q q q Cox Cox ε qn qn d 76

90 where, χ i the electron affinity of the emiconductor, Φ m i the metal work function, E v i the valence band dicontinuity at the Si 0.5 Ge 0.5 /Ge Interface, E g i the band gap of Si 0.5 Ge 0.5, n d i the charge heet denity in the delta doping of the upply layer, d 1 i the ditance of the doped plane from the oxide/si interface Limitation of the exiting model The exiting model for the threhold voltage i applicable only when the width of the doped layer i extremely thin. In other word, the expreion of the threhold voltage i derived with the aumption that the width of the doped layer (the width can be called δ) i negligible compared to the ditance of the doped layer from the oxide/si interface (d 1 ). Thi aumption i definitely not valid when the delta-doped layer i placed very cloe to the oxide layer. Alo, it i poible to tune the threhold voltage and the minimum gate bia by varying δ. The exiting model require that the δ value be quite thin and negligible compared to d 1. Thu, the exiting model i fallible when the δ value i varied freely and it become comparable to d 1. 77

91 5. The Modified Threhold Voltage Model 5..1 Need for the modified model A threhold voltage model i required that can overcome the limitation of the exiting model. Thi modified model hould be valid for any value of δ; tarting from almot negligible width to the whole upply layer width (when δ = d d = width of the upply layer). A a reult, the new model would enable u to include the effect of change in δ on the threhold voltage and the minimum gate voltage. Thu, thi model include one more dimenion to the exiting model. A new model for the threhold voltage will be propoed, which i a modified verion of the exiting model. 5.. The threhold voltage expreion Figure 5.1 how the device tructure ued to derive the expreion of the threhold voltage. Thi device i very imilar to the one that ha been ued in chapter 3. In thi cae, the upply layer i delta doped and there i no pacer layer in thi tructure. Figure 5. how the device chematic, band diagram and charge ditribution diagram ued in the derivation. The depletion charge in the upply layer i confined in a very narrow width. The part of the depletion charge i due to the gate voltage. The width of thi charge layer i δ 1. The width of the charge layer due to the channel charge i δ. A there i no pacer layer preent in thi tructure, the ditance d hould be equal or greater than the typical pacer layer width. 78

92 Figure 5. 1: The device tructure of the delta-doped MOS-gate SiGe HEMT. 79

93 Figure 5. : The device chematic, charge ditribution and band diagram ued to derive V THp expreion. 80

94 The ilicon cap layer ued here i undoped. The expreion of V THp ha been derived neglecting the effect of the ilicon layer. It can be aumed that for the delta-doped tructure and undoped ilicon layer, the effect of ilicon layer width i negligible. The derivation of the threhold voltage expreion i imilar to the derivation of the threhold voltage expreion of the regular-doped MOS-gate HEMT. The threhold voltage expreion i given by, V E E E Q qn δ qn δ qn d δ g v fo ox 1 THp = φm χ q q q Cox Cox ε ε Now, it can be hown that thi equation converge to the exiting expreion for V THp a δ δ become very thin. For a thin doped layer, << d1. If it i defined that n d = the charge heet denity in the delta doping of the upply layer = N δ, equation 5. become, V E E g v fo ox d d 1 THp = φm χ + + +, q q q Cox Cox ε E Q qn qn d which i baically equation Calculation of threhold voltage The threhold voltage for the delta-doped MOS-gate HEMT hown in Figure 5.1 i calculated uing the device and material parameter pecified in Table 5.1. The calculated threhold voltage value i, V THp = V. 81

95 Table 5. 1: Device and material parameter ued to calculate V THp of the deltadoped MOS-gate HEMT. Parameter Value Unit Φ m 4.1 ev χ 4.05 ev q 1.6X10-19 C E g ev E v 0.1 ev E f0 0 ev Q ox 1X10 11 C/cm t ox 50X10-7 cm a 0.15X10-1 V-cm - d 1 15X10-7 cm d 14X10-7 cm δ 1X10-7 cm ε ε 0 ε X10-14 F/cm ε oxide 3.9 ε 0 N 1X10 18 cm -3 8

96 Thi reult i very cloe to the experimental meaurement of the threhold voltage of the delta-doped MOS-gate HEMT with imilar tructure, found in the work of Murakami et al. [4]. 5.3 The Expreion for V tl The term V tl for a delta-doped HEMT i imilar to the term V Gmin for a regular-doped HEMT. The minimum gate bia V tl of a p-channel delta-doped MOS-gate HEMT i the gate voltage at which the top of the valence band of the upply layer almot touche the Fermi level. A a reult, real pace hole tranfer occur in the upply layer. Thu, a channel parallel to the Ge channel i formed. The conduction due to thi parallel channel harply reduce the device tranconductance. Thu, for normal operation of the deltadoped HEMT, the minimum gate voltage i retricted to equal or greater than V tl. Thu, the normal operating gate-voltage range of a delta-doped HEMT i: V tl < V G < V THp. An expreion for V tl ha been derived uing it definition. At V G = V tl, the ditance between the valence band maximum and the Fermi level, E FV become almot zero. The expreion for V tl i given by, V tl + V N N q[ [ Ev E fo + kt ln( )] + N ( d i + d) N ( d i q N v = C THp eq + d)]

97 The expreion for the total equivalent capacitance C eq i given by, C eq 1 = d + d 1 + C ox where, a d =. q Uing the parameter pecified in Table 5.1, the value of V tl ha been calculated. The calculated value i, V tl = V. The operating gate voltage range V G i given by, V G = V THp Vtl = ( ) V = V. 5.4 Effect of d 1 and δ on V THp and V tl A dicued earlier, the value of the threhold voltage and the minimum gate bia can be tuned by varying the ditance of the delta-doped layer from the oxide/si interface and the width of the delta-doped layer. The effect of thee two device parameter for different doping concentration will be invetigated in thi ection. 84

98 5.4.1 Effect of d 1 Figure 5.3(a) how the d 1 v. V THp plot for different doping concentration. A expected by inpection of equation 5., V THp varie linearly with the variation of d 1. For low doping concentration uch a 5x10 17 cm -3, the threhold voltage increae very lowly a the ditance of the delta-doped layer from the oxide/silicon interface i increaed. At higher doping concentration uch a 5x10 18 cm -3, the change of V THp due to the change in d 1 i teeper. Figure 5.3(b) how the d 1 v. V tl plot for different doping concentration. A the deltadoped plane i moved away from the oxide/silicon interface, the V tl value goe down. The variation i nonlinear, a could be anticipated from equation 5.3. For low doping concentration uch a 5x10 17 cm -3, the d 1 v. V tl plot look almot linear. On the other hand, at higher doping concentration uch a 5x10 18 cm -3, the curvature of the plot become more dominant. From the comparion of Figure 5.3 (a) and (b), it i evident that the operating voltage range V G goe up a the delta-doped layer i moved away from the gate oxide. Alo, for a fixed d 1 value, the higher the doping concentration, the higher V G i. On the other hand, if the delta-doped layer i placed very cloe to the channel, the mobility will be reduced. Thu, appropriate poitioning of the delta-doped layer i very important. 85

99 (a) (b) Figure 5. 3: (a) d 1 v. V THp plot and (b) d 1 v. V tl plot of the p-channel delta-doped SiGe MOS-gate HEMT. 86

100 5.4. Effect of δ Figure 5.4 (a) how the δ v. V THp plot for three different doping concentration. For low doping concentration, V THp i weakly dependent on the δ value. On the other hand, a the doping concentration increae, the threhold voltage how ignificant enitivity to the width of the delta-doped layer. Thu, for delta-doped MOS-gate HEMT with high doping, δ i a very effective deign parameter that can be ued to control the threhold voltage value. Figure 5.4 (b) how the δ v. V tl plot for different doping concentration. The dependence of V tl on δ i imilar to that of V THp. For higher doping concentration, V tl become quite enitive to the value of δ. It hould be noted that the nature of the δ v. V tl plot differ from that of the δ v. V THp plot a the δ v. V tl plot for different doping concentration interect with each other which i not true for the δ v. V THp plot. 5.5 Concluion In Chapter 3, the dependence of V THP and V Gmin of the regular-doped HEMT on the device and material parameter ha been dicued. In the cae of a delta-doped HEMT, the device deigner ha two more parameter to play with: d 1 and δ. It i evident from the imulation reult preented in thi chapter that both d 1 and δ can be very helpful deign 87

101 (a) Figure 5. 4: (a) δ v. V THp plot and (b) δ v. V tl plot of the p-channel delta-doped (b) SiGe MOS-gate HEMT. 88

102 parameter to achieve certain threhold voltage and operating voltage range for a deltadoped MOS-gate HEMT. 89

103 Chapter 6 Medici TM Simulation of MOS-Gate HEMT In thi chapter, the numerical imulation performed uing the device imulator Medici TM will be dicued. Some of the imulation reult will be directly compared to the reult obtained form the analytical model preented in the previou chapter. The other imulation are ued to invetigate the device behavior and examine the dependence of device and material parameter on device performance. 6.1 Brief Introduction To Medici TM Medici TM i an indutry-tandard device imulator that can predict the electrical characteritic of arbitrary two-dimenional tructure under uer pecified operating condition. It i capable of imulating a broad range of device including diode, BJT, MOSFET, JFET, MESFET, HBT, HEMT, IGBT, CCD and GTO. Medici TM can help u to: Calculate and plot I-V characteritic, gain and peed of a device. Undertand device phyic and operation through calculating and plotting potential, field, band diagram, carrier concentration, mobility, etc. Calculate ionization rate and current denity ditribution. Analyze and undertand breakdown mechanim. 90

104 Achieve optimal performance by refining device parameter. Study failure mechanim, uch a leakage path and hot electron effect. Invetigate tranient radiation effect, uch a ingle event and doe rate upet. Medici TM i particularly uitable for device that have tructural variation in any two dimenion. 6. V THp and V Gmin for the MOS-gate HEMT Figure 6.1 how the gate voltage v. drain current plot for mall drain-ource voltage (V DS =-0.1V). Thi imulation reult ha been obtained from Medici TM uing a HEMT tructure imilar to the SiGe MOS-gate HEMT hown in Figure 3.1. From Figure 6.1, it i evident that the value of the threhold voltage and the minimum gate voltage are.45 V and 0.4 V, repectively. The threhold voltage and the minimum gate voltage obtained from the modified analytical model are.48 V and 0.57 V, repectively. Thu, the modified analytical model i capable of very cloely predicting V THp and V Gmin at room temperature for MOS-gate HEMT. 6.3 Current Voltage Characteritic of the MOS-Gate HEMT Figure 6. (a) how the comparion between the V G v. I Dat plot obtained from Medici imulation and the one obtained uing the modified analytical model. It i evident that 91

105 Figure 6. 1: Medici TM imulation reult ued to determine V THp and V Gmin. 9

106 (a) (b) Figure 6. : (a) Comparion of I Dat -V G plot obtained from analytical model and Medici Simulation and (b) I D -V D plot from Medici TM. 93

107 the aturation current etimated by the analytical model are very cloe to the current etimated by Medici TM imulation. It hould be noted that the current-voltage data obtained from Medici TM i ued to plot the I Dat -V G characteritic in Matlab. It enable u to plot both reult (analytical and Medici) on the ame figure. Figure 6. (b) how the I D -V D characteritic obtained from Medici TM. Thi I D -V D characteritic differ from the I D -V D characteritic found uing analytical model by both hape and magnitude. A a reult, they have not been plotted in the ame figure. The difference between the I D -V D characteritic might be explained by the following reaon: In the analytical model, a mobility value etimated from experimental reult for trained-ge channel ha been ued. On the other hand, while performing Medici TM imulation, bulk SiGe and Ge have been ued to contruct the device. The mobility value ued by Medici TM ha been examined and found to be lower than the value ued in the calculation from analytical model. The aturation velocity value ued in the calculation from analytical model i not very accurate. It i an etimated value. Thi etimated value ha been ued due to lack of a uitable analytical expreion or experimental data for the aturation velocity in the trained-ge channel. 94

108 The Chang-Fetterman equation ued in the analytical model conider only channel current. On the other hand, Medici TM i capable of calculating leakage current beide the channel current. The baic Chang-Fetterman model ha been ued to calculate the current-voltage characteritic, which doe not include the channel-length modulation effect. Alo, the Chang-Fetterman equation doe not include poition-dependent mobility variation. Incluion of the mobility variation along the direction normal to the heterointerface would improve the model. 6.4 Doping Concentration Dependence of V THp Figure 6.3 how the I D -V G characteritic (at low V DS ) of the MOS-gate HEMT for different doping concentration of the upply layer. It i evident that with the increae in doping concentration, the threhold voltage how ignificant increment. The V Gmin value alo increae almot the ame amount. Thu, the operating voltage range (V THp ~V Gmin ) remain almot contant with variation in the doping level. Thee reult obtained from Medici TM imulation match quite cloely with the reult obtained from the analytical model. It hould be noted that the ubthrehold current become ignificant for high doping concentration uch a 8X10 17 cm

109 Figure 6. 3: I D -V G characteritic of the p-channel SiGe MOS-gate for different upply layer doping concentration. 96

110 6.5 The Temperature Variation of the Threhold Voltage Figure 6.4 how the V G v. I D characteritic calculated by Medici TM at different temperature. The threhold voltage at different temperature can be determined from thi plot. The V THp value at 100 K, 00 K, 300 K and 400 K are.15 V,.5 V,.45 V and.6 V, repectively. Thu, Medici TM predict a total of 450 mv variation in the threhold voltage a temperature change from 100 K to 400 K, a oppoed to only 90 mv change predicted by the analytical temperature model. Thi larger threhold voltage variation obtained from the Medici TM imulation might be the reult of uing bulk SiGe and Ge. The mall threhold voltage variation predicted by the analytical model appear more realitic. Similar work on temperature behavior modeling of InGaA HEMT [5] conider weak variation of threhold voltage with temperature. 6.6 The Temperature Dependence of the I D -V D Characteritic Figure 6.5 how the I D -V D Characteritic at different temperature (at V G =1.0 V), obtained from the Medici TM imulation. Again, the I D -V D Characteritic obtained from the Medici TM imulation differ ignificantly from the I D -V D Characteritic obtained from the analytical temperature model. At very low temperature uch a 100 K, Medici TM current value are ubtantially higher (almot 4 time) than the current value at 00 K, which i not true for the I D -V D Characteritic of the analytical temperature model. It ha been aumed in the analytical temperature model that the aturation velocity i 97

111 Figure 6. 4: Temperature Variation of V THp meaured from I D -V G plot at different temperature. 98

112 Figure 6. 5: Comparion of I D -V D characteritic at different temperature, obtained from Medici TM imulation. 99

113 temperature independent, which i not true in practical cae. The aturation velocity normally goe up at low temperature. The Medici TM imulation have certainly included the temperature dependence of the aturation velocity and thu, it calculated much higher current at low temperature. Alo, Medici TM ha ued quite high mobility value at low temperature. 6.7 V THp and V tl for Delta-Doped MOS-Gate HEMT Figure 6.6 how the V G v. I D plot for mall drain-ource voltage (V DS =-0.1V) obtained from the Medici TM imulation for the delta-doped MOS-gate HEMT. The device tructure i imilar to the one ued in the calculation from the analytical model. The threhold voltage value obtained from the figure i almot 0.5 V. Thi value i very cloe to the value (-0.46 V) etimated by the analytical model calculation. On the other hand, the V tl, value extracted from the Medici plot i almot 1.8 V. The calculated value of the V tl from the analytical model i. V. Thu, the V tl value calculated from the analytical model i lightly overetimated. In the derivation of the analytical expreion of V tl, it ha been aumed that the real pace hole tranfer occur when the maximum of the valence band touche the Fermi level. In practical cae, the hole tranfer tart to occur when the ditance between the maximum of the valence band and the Fermi level become comparable to the kt value. Thi i the reaon for the difference between the minimum gate voltage value etimated by the analytical model and the Medici TM imulation. Although the prediction of the V tl 100

114 Figure 6. 6: Medici TM imulation reult ued to determine V THp and V tl for the p- channel delta-doped SiGe MOS-gate HEMT. 101