Determination of electron and hole effective masses in thermal oxide utilizing an n-channel silicon MOSFET

Transcription

1 IOSR Journal of Appled Physcs (IOSR-JAP) e-issn: Volume 6, Issue 3 Ver. I (May-Jun. 2014), PP Determnaton of electron and hole effectve masses n thermal oxde utlzng an n-channel slcon MOSFET Rav Kumar Chanana Department of Electrcal and Electroncs Engneerng, Galgota s College of Engneerng and Technology, Afflated to Uttar Pradesh Techncal Unversty, Lucknow, Inda. 1, Insttutonal Area, Knowledge Park-II, Greater Noda , Inda. Abstract: The gate tunnellng electron current and the substrate hole current obtaned from carrer separaton n an n-channel S MOSFET are used to determne the parabolc electron and hole effectve masses n the thermal oxde. The oxde voltages for electron and hole conducton n the n-mosfet n nverson are formulated, and the carrer effectve masses of 0.42m for electron and 0.58m for hole for a free Ferm gas model of carrers at the emttng electrode are determned usng the Fowler-Nordhem tunnellng characterstcs. These carrer masses are related to the band offsets n SO 2 /S<100> MOS devces accurately through the MOSFET model for the frst tme. The carrer effectve masses are predcted to be the same for all thckness of oxde. The technque can be extended to other nsulatng materals as well. Also, the 1/E model of the anode hole njecton over the hole barrer of 4.6 ev completely explans the oxde breakdown n thn oxdes of 5 to 10nm at hgh electrc felds, and havng a slope constant of 516 MV/cm. Keywords: effectve mass, FN-tunellng, band offsets, metal-nsulator-semconductor I. Introducton The knowledge of electron and hole effectve masses and conducton and valence band offset values n a metal-oxde-semconductor (MOS) devce can facltate smulaton of Fowler-Nordhem (FN) tunnellng currents through a MOS devce at hgh felds. The FN onset feld and the delectrc breakdown feld can also be determned. The FN onset feld can be obtaned for a mnmum dsplacement current densty of A/cm 2, and the delectrc breakdown feld can be obtaned for a current densty of 10-4 A/cm 2, wth the above knowledge. The onset feld s the upper lmt to whch the oxde can work as a good nsulator wthout njecton and trappng of carrers n t. The trappng of carrers causes degradaton of the oxde and reduces ts relablty. The above knowledge of carrer effectve masses and band offsets can also be utlzed to determne oxde thckness to wthn 0.3nmof other physcal technques of thckness measurement [1]. In vew of the above, the determnaton of the electron and hole effectve masses and band offset values n a MOS devce s of utmost mportance. These tunnellng parameters can be determned from the FN tunnellng equaton. The equaton models the current-voltage characterstcs across a MOS devce at hgh felds [2]. The electron effectve mass n the thermal oxde as nsulator has been determned earler to be 0.42m, where m s the free electron mass based on the free Ferm gas model for the carrers at the emttng electrode of a MOS devce n accumulaton [2, 3]. In the present study, the carrer effectve masses n the thermal oxde are determned utlzng the gate tunnellng electron current and the substrate hole current versus the oxde voltage characterstcs of a MOSFET. These currents are obtaned from carrer separaton n an n-channel slcon MOSFET devce n nverson [2], wth the oxde bandgap value taken as 8.9eV [4, 5]. The carrer mass values obtaned are 0.42m for the electron effectve mass and 0.58m for the hole effectve mass for a free Ferm gas model of carrers at the emttng electrode. However, f quantum confnement of carrers s consdered at the emttng electrode, where the conducton or valence band edge forms a trangular potental well when based, then these values have to be corrected. Nearly 70% populaton of carrers s calculated to be resdng at the ground state subband energy level n the potental well [6], whch reduces the electron barrer to oxde conducton band by 0.2eV [7], and the hole barrer to oxde valence band by 0.16eV [8] n S<100> MOS devces. The corrected electron and hole barrer heghts due to quantum confnement results n the electron and hole effectve masses n the thermal oxde to 0.51m and 0.65m respectvely. It needs to be mentoned here that quantum confnement whch causes bandgap broadenng at the semconductor-oxde nterface and affects the carrer effectve mass calculatons, s vald for extremely thn channels n the MOSFETs when the peak of the electron concentraton le several angstroms away from the S/SO 2 nterface and thus reduces the saturaton current. To the author s knowledge to date, quantum correcton models, such as the trangular potental well model used by Wenberg [7] has not been able to reproduce the reduced saturaton current characterstcs n the MOSFETs havng thn channels. 1 Page

2 I. Theory: FN electron and hole tunnellng has been observed n S and SC MOS devces and n organc lght emttng dodes [9-12]. The FN equaton models the current-voltage characterstcs across a MOS devce at hgh felds. It s gven by the classcal equaton [7,10]: ( ) (1) where J s the current densty across the MOS devce n A/cm 2,E s the oxde electrc feld n V/cm, and the preexponent A and the slope constant B are gven by: ( ) ( ), (2) ( ) ( )(3) In A and B constants, e s the electronc charge, m s the free electron mass, m ox s the electron or hole mass n the oxde, s Planck's constant and s the electron or hole barrer heght expressed n electron volts. A plot of ln(j/e 2 ) versus 1/E, called an FN plot, gves the value of the slope constant B, from whch ( ) product can be obtaned. Then, wth a known effectve mass, can be calculated, and wth a known, the effectve mass n the oxde can be calculated. The slope constant B s very senstve to the oxde feld as t s n the exponental and therefore precse determnaton of the oxde feld s absolutely crtcal n the evaluaton of the tunnellng parameters. The ln(j/e 2 ) term s relatvely much less senstve to the oxde feld as t s n the natural logarthm. A. Formulaton of the oxde voltage n a n-channel-mosfet: In an deal MOS dode, f the appled voltage across the dode s V, then s the voltage across the oxde [13]. For an n-channel MOSFET Q d s negatve, gvng a postve voltage across the oxde nsulator. The threshold voltage for strong nverson, V T for a practcal MOS devce s expressed as: ( ) ( ) ; (4) where, s the metal-semconductor work functon dfference, Q s the oxde nsulator charge densty, C s the oxde nsulator capactance per unt area, Q d s the depleton charge densty n the semconductor, and s twce the bulk potental n the semconductor at whch strong nverson occurs for the MOS devce and s the polyslcon depleton potental. For an appled voltage V T across the MOS devce, deally -Q d /C wll fall across the oxde. So, the voltage across the oxde -Q d /C can be expressed as: ; (5) where ( ) s the flatband voltage. Next, f the appled voltage s, then the expresson for the oxde voltage becomes:. (6) Here, s the reverse bas appled to the substrate of a MOSFET. It s used to control the threshold voltage n a MOSFET. Ths s the same equaton as developed by Depas and co-researchers [14], except that V sb s also part of the equaton for a MOSFET. In terms of the threshold voltage V T, the equaton can be wrtten as: ; (7) For an n-channel MOSFET n nverson, V sb s negatve, V T s postve, and Q d s negatve. Also, Q d gets modfed due to the applcaton of V sb. Therefore, utlzng the absolute voltages for an n-channel MOSFET, the expresson for the oxde voltage for electron tunnellng across the deal oxde that does not contan any postve charges, can be wrtten as:. (8) The electron tunnellng across the oxde s represented by the gate current n the n-channel MOSFET [9]. The substrate hole current n the n-channel MOSFET [9] s beleved to be due to the back njecton of hot holes from the polyslcon anode havng the hole barrer of 4.6eV from the valence band of the polyslcon anode to the oxde valence band [15]. Once the hot holes come nto the valence band over the barrer [15], the hole current follows the exp(-b/e) dependence of the FN tunnellng equaton (1) through the oxde. An n- channel MOSFET havng substrate hole conducton s dentcal to a p-channel MOSFET havng hole njecton followed by conducton wth the and of p-mosfet changed as and of n-mosfet. Ths can be observed from Fg.1(a) and (b). Fg. 1(a) shows an n-mosfet and Fg.1 (b) shows a p-mosfet. It can be observed that and of the p-mosfet can be changed to and for the n-mosfet respectvely makng the n-mosfet wth hole conducton smlar to p-mosfet wth hole conducton. 2 Page

3 Therefore, the oxde voltage across the n-mosfet for substrate hole current conducton can be obtaned from the oxde voltage across the p-mosfet. In a p-mosfet, the threshold voltage equaton (4) gven above has negatve V T, negatve and postve Q d, and Bas Poly VB -qv ox E F -qψ s -qv -qψ poly + Bas -qψ poly -qv E F Poly CB - qv ox -qψ s 4.6 (a) (b) n + Poly SO 2 p-s p + Poly SO 2 n-s Fg. 1. Energy-band dagram for a (a) n-mosfet wth n + polyslcon gate and for a (b) p-mosfet wth p + polyslcon gate. Both the devces are based n nverson. assumng that the magntude of V T for n- and p-mosfets s the same. Therefore, for postve oxde charges, the threshold voltage equaton (4) for the p-mosfet can be wrtten as: ; (9) Interchangng and and multplyng throughout by -1, gves: ; (10) Here, for V T as the appled voltage, Q d /C s the deal drop across the oxde. Ths drop n voltage for an appled voltage V T can be wrtten as V ox. Therefore, for as the appled voltage, the oxde voltage drop across the n-mosfet wth hole conducton can be wrtten as: ; (11) For the sake of smplfed ntermedate steps n formulaton of the corrected oxde voltage, the V fb for n- and p-mosfet s assumed to be the same. Now, from equaton (4) for V T across an n-mosfet, where Q d s negatve, s postve, and wth the oxde havng postve charges; ; (12) Substtutng equaton (12) n equaton (11) gves the equaton for V ox n n-mosfet havng hole conducton as:. (13) Thus, for an n-channel MOSFET devce, equaton (8) represents the corrected oxde voltage for electron conducton through the oxde, and equaton (13) represents corrected oxde voltage for substrate hole conducton through the oxde. The addton of V T for hole conducton n equaton (13) suggests that for electron conducton becomes for hole conducton. The oxde voltage for electron conducton s therefore dfferent from the oxde voltage for hole conducton. Ths pont s also dscussed n the authors s earler work [2]. It s shown that the oxde voltage has to be corrected by the flatband voltage for MOS devces n accumulaton. In 3 Page

4 the present study utlzng an n-channel MOSFET, the oxde voltages are modfed by the threshold voltage. II. Sample Calculatons and Results: A sample calculaton for the electron and hole effectve masses s performed here, utlzng the currentvoltage data on an n + -polyslcon gated n-channel S MOSFET. Two ponts of current and voltage are taken n the FN regme for the gate current representng electron tunnellng current and the substrate hole current from fg. 2 of reference [9], and are presented n Table I below. Table I. Current-voltage data n the FN regon for gate electron current and substrate hole current of the MOSFET. Current Type I 1 I 2 Uncorrected B (V) (A) (V) (A) (MV/cm) Gate Electron Current Data ( ) 10-8 ( ) Substrate Hole Current Data ( ) 10-9 ( ) For the calculaton of corrected oxde voltages, the threshold voltage V T s taken as 1.25V. The substrate bas adjusts the threshold voltage of the MOSFET. As the magntude of the substrate bas V sb ncreases from 1V to 16V, V T also ncreases from 1.25V to 3.25V as presented n fg. 43 of reference [16]. Thus, V T s 1.25V for the substrate bas of -1V, n the present study. The V T values presented n the fg.43 of the above reference [16] are consdered more accurate because they are calculated expermentally usng the MOSFET channel conductance. The other desred parameter values are calculated and presented n Table II below. Table II. MOSFET parameters for the calculatons of corrected oxde voltages. Substrate dopng N A= /cm 3 Intrnsc carrer conc. n = 1.45 x /cm 3 Surface potental at strong nverson 0 = x F/cm S n + poly S gate and p-s work functon dfference ms V r =3.9 Oxde thckness d = 85 x 10-8 cm Depleton charge densty Q 2 qn V 2kT N s 2 B ln q n Oxde capactance C 0 r d = 4.0 x 10-7 F/cm 2 d S A s sb d 0.19 V C = 7.56 x 10-8 C/cm 2 Oxde fxed charge densty Flatband voltage Q q. 2 x C/cm 2 V fb ms Q Q C = -1.1 V A 0.7 V Utlzng the values of the above parameters, the corrected oxde voltages can be calculated. The corrected slope constant B s determned from the I-V characterstcs usng equaton (1) gven above. Frst, ( ) ( )s calculated by takng at least two ponts on the I-V characterstcs n the FN regme at hgh felds gven n Table I. For ths, the sum of the gate voltage and substrate bas voltage s used for a partcular gate current or substrate current n case of the n-channel MOSFET devce. Ths yelds the uncorrected B presented n Table I. Next, the corrected B s calculated by dvdng ( ) by ( ), where ( ) s the same that was used for the calculaton of uncorrected B, but the ( )employs the corrected oxde voltages. Ths s because ( ) does not change wth the oxde feld, as t s n the natural logarthm, and the oxde voltages are corrected for the same current densty J as for uncorrected B. The corrected oxde voltages for the gate electron current and the substrate hole current are gven n Table III. These values of oxde voltages are obtaned by usng equaton (8) and (13). The oxde voltage dvded by the oxde thckness gves the corrected oxde feld. Fnally, from the corrected B value, m ox or can be determned f one of them s known usng equaton (3). The corrected oxde voltages, the corrected slope constant B, the electron and hole barrer heghts used, and the calculated effectve masses m ox, for electron and hole are presented n Table III below. 4 Page

5 Table III. Corrected oxde voltages and tunnellng parameters for gate electron and substrate hole currents n the MOSFET. Current Type Corrected V ox1 (V) Corrected V ox2 (V) Corrected B (MV/cm) (ev) (determned) Gate Electron Current Data m Substrate Hole Current Data m The calculated electron effectve mass n the thermal oxde for an electron barrer of 3.2eV from the S conducton band to oxde conducton band s 0.42m. The hole effectve mass n the oxde for an anode hole barrer of 4.6eV s determned to be 0.58m. For quantum confnement at the emttng electrode when based, the reduced electron and hole barrer heghts gve electron effectve mass n the oxde as 0.51m and the hole effectve mass as 0.65m. The carrer effectve mass values for a free Ferm gas model for carrers at the emttng electrode are the same as that obtaned n a recent study usng n- and p-4h-sc MOS devces n accumulaton wth SO 2 /4H-SC devces havng vastly dfferent band offsets than SO 2 /S<100> devces [2]. Ths reaffrms the values and also suggests that an n-channel S MOSFET devce that allows carrer separaton can be used to determne the carrer effectve mass values accurately. III. Dscusson FN tunnellng of carrers occurs n thck oxdes greater than 4nm [1,14,17] n whch the applcaton of hgh electrc feld results n carrer tunnellng as a wave nto the oxde conducton or valence band through the trangular barrer and then travels through the nsulator as a partcle wth an effectve mass to reach the opposte electrode. In oxdes havng thckness of 4nm or less, the electron or hole drectly tunnels across the trapezodal barrer of the MOS devce to the opposte electrode as a wave wthout gong to the oxde conducton or valence band. Thus, the carrer propagates as a wave-partcle dual n FN tunnellng, and as a wave n drect-tunnellng across the MOS devce. In ether case the propagaton s through the oxde. Therefore, the author predcts that the carrer effectve masses wll reman the same for all the thcknesses of the oxde or nsulatng materals. A recent report has modelled the drect and FN-tunnelng of electrons utlzng a thckness ndependent tunnel mass of 0.42m [18]. Another report presents the electron effectve mass n a thn tunnel oxde of 3.5nm as 0.42m [19], whch s the same as that n thck oxdes[2,3]. The orgn of the substrate current n n-mosfets havng thn oxde from 5 to 10nm s under debate [9, 15, 20, 21]. Etan and Kolodny were one of the frst to observe substrate hole current n n-channel MOSFET havng 8.5nm oxde, and attrbuted the hole current to the valence electron tunnellng from the S substrate havng a barrer of 4.3eV from the S valence band to the oxde conducton band [9]. DMara et al. proposed later that these holes orgnate as hot holes from the polyslcon anode. The energy for the hot holes are provded by the FN tunnellng electrons from the cathode, whch have a threshold average energy of about 5eV from the bottom of the oxde conducton band. These hot holes are concluded to be back njected from the anode over the hole barrer at the S anode for thn oxdes of 5 to 10nm. In these thn oxdes, the electron transport s quasballstc, so that the maxmum electron energy at the anode s ndependent of the oxde thckness gvng a thckness ndependent threshold [15]. More recently, the substrate holes are expermentally shown to be generated by FN-nduced photons n the polyslcon gate [20]. A report after the above, refuted ths process of hole generaton and concluded that the generaton effcency of photons wth energy above the S bandgap energy s 10-4 tmes smaller than that of the electron-hole pars by mpact onzaton [21]. The above studes, leads the author to beleve that the hot holes from the anode are back njected nto the oxde over the barrer [15, 22]. After comng to the oxde valence band, the hole current follows the exp(-b/e) dependence of the FN tunnellng equaton (1). The calculated hole effectve mass based on ths dependence s exactly the same as that determned from the FN tunnellng of holes observed n p-4h-sc MOS devces n accumulaton [2]. Ths value of hole effectve mass s 0.58m for a free Ferm gas model of carrers at the emttng electrode. A value of 0.57m has been used earler as a fttng parameter n a hgh frequency tunnel emtter transstor model [23]. The value s also consstent wth the evdence of lght holes near the top of the oxde valence band [24]. The valence band offset of 4.6eV (E g, oxde - E g, S - S CB offset) results n the hole effectve mass value of 0.58m n the oxde. Ths valence band offset has also been reported recently n expermental studes on remote plasma grown dry SO 2 of 0.9 to 2.2nm, and 2nm dry thermal SO 2 utlzng soft x-ray photoemsson spectroscopy on SO 2 /S(100) test samples. Table I of the reference [25] presents the valence band offset for the remote plasma SO 2 /S(100) sample as ev, and the more recent study on 2nm dry thermal SO 2 /S(100) sample presents the valence band offset as ev [26]. The resultng hole mass of 0.58m s the same as that obtaned for p-4h-sc MOS devce n accumulaton undergong FN tunnellng of holes [2], where the Schottky barrer lowerng does not change the tunnellng dstance. Ths mples that Schottky barrer lowerng n thn oxdes of 5 to 10nm, n whch the hot holes are njected nto the oxde valence band by thermonc emsson over the barrer s absent [15, 22]. Ths s due to the fact that the generated hot holes due to mpact onzaton at the polyslcon anode of the electrons arrvng from the cathode also have a thckness 5 Page

6 ndependent energy. Ths elmnates any Schottky barrer lowerng at the valence band whch s feld-dependent [22] and therefore thckness-dependent. The oxde breakdown n thn oxde flms of 5 to 10nm s ntmately related to the substrate current due to hot holes, whch s proportonal to exp(-b/e) at hgh electrc felds wth the slope constant B of 516 MV/cm presented n Table III. An n-channel slcon MOSFET when based n nverson as shown n Fg.1a, results n FN tunnellng of electrons from the cathode nto the oxde due to the smaller electron barrer to oxde conducton band of 3.2 ev. They then arrve at the polyslcon anode, where they mpact onze and create hot holes. The hot holes nject over the hole barrer of 4.6 ev at the anode [15] and cause the substrate hole current through the oxde. Some holes are trapped n the oxde causng ncrease n the feld at the cathode [2]. Ths results n larger electron current njecton from the cathode nto the oxde by FN tunnellng followed by ncreased substrate current and hole trappng n the oxde. Ths postve feedback results n the oxde breakdown. The tme-to-breakdown s therefore proportonal to exp(-b/e) wth the slope constant B of 516 MV/cm nstead of 350 MV/cm reported earler [27]. Ths 1/E model of the anode hole njecton [27] completely explans the oxde breakdown n thn oxde flms of 5 to 10nm at hgh electrc felds. Several MOS devce confguratons can be used to determne electron and hole effectve masses or band offsets n thermal oxde and n other vable hgh-k delectrcs [26, 28, 29]. Sx of them are lsted below: 1. A p-4h-sc or p-6h-sc MOS devce can be used n accumulaton to determne hole effectve mass from the hole current versus voltage characterstcs [2]. 2. An n-4h-sc or n-6h-sc MOS devce can be used n accumulaton to determne electron effectve mass from electron current versus voltage characterstcs [2]. 3. An n-s MOS devce can be used n accumulaton to determne electron effectve mass from the electron current versus voltage characterstcs. 4. A p-s MOS devce can be used n accumulaton to determne hole effectve mass from the hole current versus voltage characterstcs. Here, all the metals avalable as gate contact have a lower electron barrer to oxde conducton band as compared to the hole barrer of 4.6eV at the S anode. Therefore the p-mos devce n accumulaton wll not have the hole tunnellng current as the domnant current. The choce of gate materal should be p+ poly slcon carbde. Ths wll ensure a 6eV valence electron barrer to the oxde conducton band and the domnant FN hole tunnellng from the S anode havng a 4.6eV hole barrer wll occur at an onset feld of about 14MV/cm. 5. An n-channel n+ polyslcon gated S MOSFET havng a thn oxde of 5 to 10nm can be used to determne electron effectve mass from the gate tunnellng current and the hole effectve mass from the substrate current. Ths s demonstrated n the present study. 6. A p-4h-sc based MOS devce n nverson can be used to fnd electron effectve mass from the domnant electron current versus voltage characterstcs. Ths method s complex because strong nverson n SC does not take place wthout the applcaton of temperature or shnnng UV lght. Ths s due to the fact that the ntrnsc carrer concentraton n SC s low because of ts wde band gap. IV. Concluson: The parabolc electron and hole effectve masses n the thermal oxde are determned to be 0.42m and 0.58m for a free Ferm gas model of carrers at the emttng electrode utlzng a S n-channel MOSFET devce n nverson that allows carrer separaton. These carrer masses are related to the band offsets n SO 2 /S<100> MOS devces accurately though the MOSFET model and s reported for the frst tme. For quantum confnement of carrers at the emttng electrode when based, the electron and hole effectve masses n the oxde corrects to 0.51m and 0.65m, respectvely, keepng n vew that quantum correcton s vald for extremely thn channels n MOSFETs. The effectve masses are predcted to be the same for all thcknesses of the oxde. The devce confguratons dscussed can be utlzed wth other nsulatng materals as well for the determnaton of carrer masses. Also, the 1/E model of the anode hole njecton over the hole barrer of 4.6 ev completely explans the oxde breakdown n thn oxdes of 5 to 10nm havng a slope constant of 516 MV/cm. References [1]. H.S.Chang, H.D.Yang, H.Hwang, H.M.Cho, H.J.Lee andd.w.moon, Measurement of the physcal and electrcal thckness of ultrathn gate oxdes, J.Vac.Sc.Technol., vol. 20,pp , Sep [2]. R.K. 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