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Gate polarity dependence of impact ionization probabilities in metal-oxide-silicon structures under Fowler–Nordheim stress
Solid-State Electronics 46 (2002) 279–285
Gate polarity dependence of impact ionization probabilities in metal-oxide-silicon structures under Fowler–Nordheim stress Piyas Samanta *, C.K. Sarkar Department of Electronics and Telecommunication Engineering, Jadavpur University, Calcutta 700 032, India Received 21 May 2001; accepted 27 June 2001
Abstract A theoretical investigation on the gate polarity dependence of Fowler–Nordheim (FN) tunneling electron initiated impact ionization probabilities in the bulk silicon dioxide (SiO2 ) films as well as hole injection from the anode material (nþ poly-Si gate or silicon substrate) in metal-oxide-silicon devices is presented. Our theoretical results of the gate polarity dependence of the probabilities of various impact ionization processes under constant current FN stress correlate the experimentally observed positive charge generation/trapping and stress-induced trap creation in thick to thin SiO2 films. Ó 2002 Published by Elsevier Science Ltd.
1. Introduction Albeit hot-electron induced impact ionization phenomenon is an old issue in explaining the dielectric instability and breakdown of metal-oxide-silicon (MOS) devices under high-field stress [1,2]. However, from the fundamentals of electron heating in the oxide conduction band and their interactions with the oxide lattice as well as the anode material, a consensus about the various types of impact ionization processes in MOS structures under high-field stress has only been reached in the last decade [3–5]. Hot-electron induced various ionization processes include band-to-band ionization (BTBI) [3], trap-to-band ionization (TTBI) [4] in the oxide bulk and hole injection from the anode, popularly known as anode hole injection (AHI) [5]. Positive charge and/or hole trapping in the bulk gate oxide is manifested by the negative shift of gate voltage VG or Fowler–Nordheim (FN) threshold voltage VFN under constant current FN stress. Experimental results [6,7] of temporal evolution of the gate voltage shift DVG at a given constant FN current density have shown a
*
Corresponding author. E-mail address: [email protected] (P. Samanta).
strong gate polarity hence FN injecting electrode dependence. Gate polarity dependence of DVG has not been explained so far in literature. Moreover, as can be seen from Fig. 1, the relative absolute magnitude of DVG in thicker samples is higher during negative gate injection compared to substrate injection (positive gate bias) at a given constant FN current level [6] while, gate electron emission results lower absolute DVG in thinner samples [7] compared to substrate electron emission at a given constant current density. DVG is directly proportional to the product of the concentration of oxide-trapped charges and their centroids measured from the gate electrode [8] by DVG ¼
Qþ x ox
DVG ¼
Qþ ðtox xÞ ox
for substrate injection;
for gate injection;
ð1aÞ
ð1bÞ
where Qþ is the areal density of trapped positive charge in C/cm2 , tox is the oxide thickness, ox is the permittivity of SiO2 and x is the location of the centroid of trapped positive charge measured from the gate/oxide interface during FN injection from either gate or substrate.
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2. Theoretical analysis Electronic conduction in thermally grown SiO2 films above 5 nm thick is limited by the emission of electrons via FN tunneling through the interfacial barrier Uec at oxide fields above 6 MV/cm subject to the condition Vox > Uec , where Vox is the voltage drop in the oxide. In FN tunneling regime, electrons are injected into the oxide conduction band from either the Fermi level of the metal/degenerately doped n-type poly-Si gate at negative gate bias or the first subband energy level [9] located at 0.2 eV from the bottom of the conduction band of nondegenerately doped silicon in the case of injection from quantized accumulation and/or inversion layer at negative gate bias. Irrespective of the injecting electrode [10], as long as the barrier at the injecting electrode/oxide interface remains triangular, FN tunneling current density j obeys the well-known expression Fig. 1. FN threshold voltage shift DVFN as a function of electron fluence injected from substrate (positive bias on poly-Si gate) and nþ poly-Si gate (negative bias) at constant current density. Symbols are from experiment (a) on MOS capacitors on (1 0 0) n-Si [6] and (b) on n-MOSFET having channel length/ width ratio 100/20 lm/lm [7].
Taking the symmetric [6] or asymmetric [3] distribution of trapped holes under FN injection with positive and negative gate bias in Eqs. (1a) and (1b), experimental data of DVFN at injected electron fluence Qinj upto 102 C/cm2 as shown in Fig. 1(a) indicate that in thick samples gate injection results higher hole generation rate compared to substrate injection. However, polarity dependence of hole generation/trapping rate in thinner samples as estimated from Fig. 1(b) is opposite to that in thicker samples. Gate polarity dependence of experimentally observed negative shift of gate voltage is a consequence of the polarity dependence of the probabilities of various impact ionization processes responsible for the generation of hole and/or positive charge in the bulk oxide. Albeit our above interpretation on hole generation and hence the impact ionization probability in explaining experimental observations in thick to thin oxide films seems elegant, no theoretical clarification has yet been available in the literature. Present communication is therefore, an attempt to theoretically investigate the gate polarity dependence of the individual probability of various impact ionization processes with an aim to correlate these theoretical results with experimentally observed polarity dependence of DVFN and breakdown in MOS devices with thick to thin oxide films. Devices studied here are nþ poly-Si gated MOS capacitors grown on (1 0 0) n-Si substrate having resistivity of 5–7 X cm or transistor structures grown on nondegenerately doped p-Si.
j ¼ AEc2 expðB=Ec Þ;
ð2Þ
where A and B are material dependent constants [9,11] related to the interfacial barrier energy Uec for electron. Fig. 2 shows the variation of FN current density as a function of cathode field during injection from the nþ poly-Si gate as well as from the quantized accumulation layer of n-Si. In poly-Si gate structures, at a given cathode field Ec , the higher value of FN current density j during emission from the accumulation layer of n-Si as shown in Fig. 2 is due to lower barrier height en-
Fig. 2. Variation of FN current density with cathode oxide field during injection from the quantized accumulation layer of (1 0 0) n-Si (positive gate bias) and nþ poly-Si gate (negative gate bias) at time t ¼ 0 in poly-Si gate MOS capacitors. Results are shown for triangular electron-energy barrier at the cathode/ oxide interface. Symbols are from experiment during injection from nþ poly-Si gate into the oxide conduction band [11] and curves are from theory.
P. Samanta, C.K. Sarkar / Solid-State Electronics 46 (2002) 279–285 Table 1 Barrier energies of hole Uha at the anode/oxide interface and electron Uec and Uea at the cathode/oxide and anode/oxide interfaces, respectively FN injection from
Uec (eV)
Uea (eV)
Uha (eV)
(1 0 0) n-Si substrate nþ poly-Si gate
2.9 3.22
3.22 3.1
4.68 4.8
countered by the electrons residing in the first subband energy level in contrast to the emission from the bottom of the conduction band in degenerately doped poly-Si gate where quantization effect do not arise. From the FN plots shown in Fig. 2, the estimated barrier energies of electron Uec at cathode/oxide interfaces are listed in Table 1. Assuming the band-gap energies of Si and SiO2 equal to 1.1 and 9 eV respectively, the estimated barrier energies of hole Uha at the anode/ oxide are given in Table 1. In estimating hole-barrier energy at the Si/SiO2 interface, we have taken a value 3.1 eV instead of 2.9 eV for the electron barrier height at the Si (anode)/SiO2 interface, because we are concerned with those holes generated in the bulk of the Si substrate by impact ionization. FN tunneling is accompanied by oxide electric field as high as 6–7 MV/cm. At such a high oxide electric field, electrons while transported towards the anode in the oxide layer become hot and equilibriate their energies by impact ionization in the oxide bulk as well as in the anode material after entering into the anode. In the following subsections, the various types of impact ionization processes in the context of present knowledge will be discussed.
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th where Pmf is the electron multiplication factor and Ebi is the threshold field for BTBI depending on the oxide thickness tox as outlined in Ref. [3]. The spatial weight function gðxÞ takes into account the electron heating distance or dead space kbi measured from the cathode/ oxide interface [10] and is given by gðxÞ ¼ 1 for x > kbi and gðxÞ ¼ 0 for x 6 kbi . HðuÞ ¼ 1 for u > 0 and 0 for u 6 0. The function HðuÞ takes care of the fact that beth low the threshold field Ebi electrons cannot be accelerated to energies high enough to cause BTBI in the oxide [10].
2.2. TTBI probability In thermally grown oxide films, neutral bridging ox- ygen vacancy centres V0O ðSi–OÞ3 Si Si ðO–SiÞ3 located at an energy 7 eV below the oxide conduction band edge act as the donor-like electron traps [4]. Under high-field stress, TTBI occurs when one of the bound electrons at an initially neutral oxygen vacancy V0O is released by the impact of hot-electron with kinetic energy greater than 7 eV from the oxide conduction band edge via V0O þ e ð> 7 eVÞ ! Vþ O þ 2e . The above TTBI model is consistent with the observed [3] hotelectron energy distribution in thermally grown oxide films under high-field stress. TTBI really occurs nearer th the anode at fields above the threshold Eem when the injected electron gains the threshold energy of 7 eV for impact emission beyond the threshold distance kem measured from the cathode/oxide interface [8,10]. Field ðEÞ and thickness ðtox Þ dependencies of TTBI probability an per injected electron are given by [10] an ¼ 2m 1;
m ¼ aN ðEÞ½tox kem ðEÞ :
ð4aÞ
2.1. BTBI probability At electric fields above 7 MV/cm in oxide films thicker than 20 nm, the acoustic phonon scattering alone can no longer stabilize the electron energy distributions leading to an electron runaway from phonon scattering and therefore, the high-energy tail of the hot-electron distribution extends beyond the 9 eV band gap of SiO2 as observed both theoretically and experimentally [3]. Electrons from these high-energy tails (>9 eV) of their distribution functions cause electron–hole pair generation by BTBI in the oxide. Holes generated via BTBI having lower mobility compared to electrons, while drifting towards the cathode get trapped in the existing neutral hole trap sites. Using the expression parameterized from the Monte Carlo simulation results of Arnold et al. [3], the field and thickness dependencies of the BTBI probability aI of an electron can be written as [10] 4 EðxÞ th aI ¼ Pmf ; ð3Þ 1 gðxÞH EðxÞ Ebi th Ebi
The spatial dependence of electron impact emission coefficient aN in cm1 is given by th ; ð4bÞ aN ðE; xÞ ¼ a0N exp ½ H =EðxÞ gðxÞH EðxÞ Eem where similar to the arguments given in Section 2.1, the weight functions gðxÞ and HðuÞ take into account the dead space kem and the threshold field criteria for the initiation of TTBI. Fitting Eqs. (4a) and (4b) with experimental data of Thompson and co-worker [4], we obtained the values of the constants a0N and H equal to 1598.54 cm1 and 28.77 MV/cm, respectively [8]. 2.3. AHI probability The electron tunneling into the conduction band of the oxide via FN mechanism, reaches the anode by gaining kinetic energy from the applied electric field and subsequently losing energy by scattering with oxide phonons beyond tunneling distance xt ð¼ Uec =Eox Þ measured from the cathode/oxide interface. The average
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energy gain hwi by the electron in the oxide can be calculated from the energy relaxation equation [5] dhwi hwi w0 ; ¼ qEox ðxÞ kðhwiÞ dx
ð5aÞ
where w0 ¼ 1:5kB T is the thermal energy, q is the magnitude of electronic charge, Eox is the oxide field, x is the spatial coordinate measured from the cathode/oxide interface, and kðhwiÞ is the energy-dependent mean free path in the oxide. Upon entering into the anode at x ¼ tox , each electron will acquire an additional potential energy equal to the anode barrier energy Uea for electron. Therefore, the total energy of the primary electron with respect to the bottom of the anode conduction band is given by e win e ¼ hwi þ Ua :
ð5bÞ
Assuming the oxide to be initially free from trappedoxide charges, average hot-electron energy hwi measured from the oxide conduction band as well as from the anode conduction band are shown in Fig. 3 as a function of constant FN current density during injection from the nþ poly-Si gate as well as the substrate. The energetic electron in the anode conduction band then loses its energy via direct interband ionizing collision with the bound electron in the valence band and thereby creates an electron–hole pair. The rest of the primary electron energy is shared between the generated pair. Assuming an equipartition of energy between the generated pair and neglecting the emitted phonon energy during impact ionization in the anode valence band, the hole energy wh from the top of the anode valence band can be written as [5]
wh ¼ 12ðwin e eI Þ;
where eI ¼ 1:1 eV [12] is the threshold energy for impact ionization in the anode material. Depending upon the energy wh of the generated hot-hole, hole injection into the oxide valence band occurs either over the interfacial anode barrier Uha or via tunneling through it. In our calculation, however, we have observed the hot-hole energy wh never exceeds the barrier height Uha at the anode/oxide interface irrespective of the gate bias polarity. Therefore, in our simulation hole injection into the oxide always occurs via tunneling through the interfacial barrier. Taking into account of the image force barrier lowering effect on the triangular hole-energy barrier and using WKB approximation, the hole tunneling probability Hh can be written as [5] ( ) pffiffiffiffiffiffiffiffiffiffiffi 3=2 4 2qmh ðUha wh Þ Hh ¼ exp vðyÞ ; ð7Þ a 3 hEox where mh ¼ 0:2m0 is the effective mass of hole in SiO2 with m0 as the free electron mass, vðyÞ is the image force correction term with normalized barrier lowering term y, a the h is the reduced Planck constantð¼ h=2pÞ, and Eox field in the oxide near the anode. In the absence of oxidetrapped charges at time t ¼ 0, oxide field remains unia form and Eox is equal to the cathode field Ec . According to the above physical model, the AHI probability ah per tunnel injected electron can be calculated from [5] ah ¼ Pii Hh ;
ð8Þ
where Pii is the probability of impact ionization or hole generation in the anode material (silicon substrate or poly-Si gate) by a tunneling primary electron. Incorporating soft threshold energy into lucky drift model, the probability of impact ionization Pii in silicon or poly-Si can be calculated according to the prescription of Marsland [12] as (
" 1 l0 ðakE l0 Þ1=3 Pii ¼ kE exp 1p ðkE kÞ kE kE !#
1=3 ðakE l0 Þ l0 Hi k exp kE k !#) " 1=3 1=3 ðakE l0 Þ ðakE l0 Þ Hi ; ð9aÞ 1p k k
a¼ Fig. 3. Average hot-electron-energy measured from the bottom of the oxide conduction band (panel (a)) and the anode conduction band edge (panel (b)) as a function of FN current density during electron emission from poly-Si gate and n-Si substrate.
ð6Þ
x h ; pð2n þ 1ÞqESi
kE ¼
qESi k2 ð2n þ 1Þ ; 2 hx
l0 ¼
eI ; qESi ð9bÞ
where hx ¼ 55 meV is the phonon energy, n is the Bose– Einstein number, p ¼ 0:030 is the softness parameter
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is the mean free path between collisions and k ¼ 70:2 A or mean free path for momentum relaxation [13], kE is the mean free path for energy relaxation, l0 is the dead space to acquire the threshold energy eI for impact ionization in silicon and ESi is the field in silicon and the related Airy function HiðzÞ is given by
Z 1 1 HiðzÞ ¼ p1 exp t3 þ zt dt: ð9cÞ 3 0
3. Results and discussion For a given constant FN current density, the cathode field Ec is higher during gate emission compared to substrate emission as evident from inset of Fig. 2. Therefore, from the solution to Eq. (5a), constant current gate emission results a higher value of average hotelectron energy in the bottom of the oxide conduction band than does substrate emission in poly-Si gated devices as shown in Fig. 3(a). Using the barrier energies Uea of electron at the anode/oxide interface from Table 1 into Eq. (5b), average hot-electron energy in the anode conduction band is observed smaller under constant current gate emission relative to its substrate injection counterpart (Fig. 3(b)). Under constant current FN stress, the gate bias dependence of the probabilities aI and an of bulk impact ionization processes BTBI and TTBI in the oxide is opposite to that of AHI probability ah as shown in Fig. 4. The probabilities aI and an of the bulk ionization processes BTBI and TTBI, respectively are increasing functions of the electric field in the oxide in accordance with Eqs. (3), (4a) and (4b). Gate polarity dependence of aI and an under constant current FN injection as shown in Fig. 4(a) and (b) arises through the different values of the cathode field Ec at a given j (see Fig. 2) via the FN Eq. (2). BTBI and TTBI being bulk processes, depend on the hot-electron energy in the oxide conduction band. The higher values of both aI and an under constant current gate emission compared to substrate electron emission as shown in panels (a) and (b) of Fig. 4 are consistent with the gate bias dependence of average hotelectron energy depicted in Fig. 3(a). In view of the above discussion, we propose that unlike constant current FN stress, constant voltage FN stress will not exhibit the gate polarity dependence of bulk ionization probabilities. The anomalous behavior of AHI probability ah on the gate polarity dependence under constant current FN stress as shown in Fig. 4(c) is explained below. According to our AHI model described in Section 2.3, AHI probability ah is the product of hot-electron induced impact ionization or hole generation probability Pii in the bulk of anode material and hole tunneling proba-
Fig. 4. Gate bias dependence of BTBI probability (aI ), TTBI probability (an ), and AHI probability (ah ) of an electron as a function of FN current density at t ¼ 0 under constant current injection from nþ poly-Si gate (– – –) as well as from the quantized accumulation layer of n-Si (—) in MOS capacitors having 35 nm thick gate oxide.
bility Hh through the anode/oxide barrier Uha . Hole generation probability Pii in the anode material is directly related to the oxide field near anode governed by Eq. (9b). At a given j, the higher value of cathode field Ec and hence the anode oxide field Ea results a higher value of impact ionization probability Pii in the anode (bulk silicon substrate) during constant current gate emission relative to substrate emission as shown in Fig. 5(a). The average electron energy win e in the anode conduction band is estimated smaller (Fig. 3(b)) while the holeenergy barrier Uha is higher (Table 1) during gate electron emission relative to substrate injection. Therefore, in accordance with Eq. (6), hot-hole energy wh becomes higher and consequently, the effective hole barrier height Uheff ¼ Uha wh at the anode/oxide interface becomes lower resulting a smaller value of hole tunneling probability Hh under constant current substrate emission than its gate emission counterpart as shown in Fig. 5(b). Gate polarity dependence of Hh being stronger than that of Pii , FN injection from poly-Si gate shows smaller AHI probability ah than does substrate emission at a given constant current density (Figs. 4(c) and 5(c)). According to Eq. (7), hole tunneling probability Hh is a an increasing function of oxide field Eox near anode and h is a decreasing function of Ueff . Therefore, the results shown in Fig. 5(b) and (c) indicate that work function difference of the anode material is the controlling factor of the gate bias dependence of AHI probability ah rather than the oxide field near anode. The above argument is further verified from Fig. 6 showing the gate polarity dependence of AHI probability under constant voltage
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Fig. 5. Variation of (a) impact ionization probability (Pii ) in the anode material, (b) hole tunneling probability Hh through the anode/oxide barrier and (c) AHI probability ah at t ¼ 0 with FN current density in a 8 nm thick SiO2 film during constant current injection from substrate and poly-Si gate.
Fig. 6. Gate polarity dependence of AHI probability (ah ) of an electron at t ¼ 0 as a function of initial applied oxide electric field under constant voltage FN stress in MOS capacitors with a 10 nm thick gate oxide.
FN stress similar to that under constant current FN stress depicted in Figs. 4(c) and 5(c). Due to the variation of work function of the anode material, unlike the bulk ionization probabilities, AHI probability ah exhibits the gate polarity dependence under constant voltage FN stress as depicted in Fig. 6. Fig. 7 shows the relative magnitude of the probabilities aI , an and ah of a FN tunnel injected electron as a function of initial applied oxide field in thick to thin SiO2 films during constant voltage FN injection from nþ
Fig. 7. Relative magnitudes of BTBI, TTBI and AHI probabilities of an electron at t ¼ 0 as a function of oxide electric field (a) in a 27 nm thick oxide and (b) in a 10 nm thick oxide during constant voltage FN injection from nþ poly-Si gate.
poly-Si gate at negative bias. Positive gate injection also exhibits similar relative trend of the various probabilities (not shown here). Theoretical results of the gate bias dependence of various ionization probabilities of a FN tunnel injected electron discussed above can be correlated with the MOS device degradation as follows. As evident from Fig. 7(a), relative to TTBI and AHI probabilities, BTBI probability of an electron is higher in a 27 nm thick SiO2 film at an applied field around 10 MV/cm corresponding to j ¼ 1 mA/cm2 . Therefore, our theoretical results of higher BTBI probability under gate electron emission compared to substrate electron emission at a given constant current density qualitatively explains experimentally observed [6] polarity dependence of DVG at low injected electron fluence in a 27 nm thick SiO2 . Recent [7], experimental gate voltage shift DVG data in a 10 nm thick SiO2 film at a 10 mA/cm2 constant current density have shown that the absolute magnitude of DVG is smaller during gate injection compared to substrate emission in contrast to thicker samples of Fazan and co-workers [6]. At 10 mA/cm2 constant current density, the estimated values of the cathode fields Ec from Eq. (2) are observed far below the thresholds for TTBI and BTBI in this thin sample. Therefore, only AHI contributes in hole trapping in a 10 nm thick SiO2 film at j ¼ 10 mA/cm2 . Taking a constant value of 5 nm for the trapped-hole centroid, experimental [7] DVG data indicate that hole generation/trapping rate during gate injection is smaller than that during substrate electron emission. Our theoretical results of gate polarity dependence of ah shown in Fig. 5(c) qualitatively explains
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the experimental results of gate voltage shift data with thin oxides of Park and Schroder [7]. Destructive dielectric breakdown in MOS devices under high-field stress is triggered by the stress-induced defect generation caused by two processes, viz., electron– hole recombination at injected electron fluence Qinj 6 103 C/cm2 and trap creation via hydrogen release by direct hot-electron impact beyond 103 C/cm2 of Qinj [3,14]. The rate of recombination of trapped hole and/or TTBI generated positive charge with free electron is proportional to the concentration of trapped holes [8,15]. Therefore, theoretically observed higher probabilities of hole and positive charge generation from BTBI and TTBI processes indicate higher recombination-induced trap creation during gate electron emission compared to substrate injection under constant current stress. In oxide films thicker than 20 nm, BTBI contributes more compared to other processes TTBI and AHI at fields above 9 MV/cm as shown in Fig. 7. Therefore, gate injection results higher stress-induced trap creation compared to substrate injection in thicker films in support with experimental results of Fazan et al. [6] at constant current FN stress. In MOS devices with oxide films as thin as 8 nm, AHI is the dominant hole generation mechanism. According to the recombination-induced trap creation model [14], our theoretically estimated results of polarity dependence of AHI probability as shown in Figs. 5(c) and 6 indicate higher trap creation in thin SiO2 films under substrate injection compared to its gate injection counterpart in either stressing modes (constant current and voltage) in contrast to thicker samples. Furthermore, in samples below 10 nm thick, breakdown is dominated by trap creation due to direct hot-electron impact and trap creation rate strongly depends on electron energy measured from the anode conduction band [14]. In view of the trap creation model [14], the higher value of the hotelectron energy (measured from the anode conduction band) as shown in Fig. 3(b) indicates that substrate injection stress generates more traps than does gate injection stress consistent with report of Park and Schroder [7].
4. Conclusion Our theoretical analysis of the polarity dependence of various ionization probabilities in the oxide bulk as well as in the anode material under constant current FN in-
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jection gives a physical insight of the experimentally observed anomalous behavior [6,7] of gate voltage shift at injected fluence as low as 102 C/cm2 in MOS devices with thick and thin SiO2 films. From our analysis, we propose that the probabilities of hole and/or positive charge generation dictated by bulk ionization processes viz., BTBI and TTBI in thicker samples do not depend on the gate bias polarity under constant voltage stress. However, in samples as thin as 8 nm, AHI is the dominant mechanism responsible for the origin of trapped holes. Furthermore, AHI is dependent on interfacial barrier energies of electron and hole [5]. Due to the variation of work function of the anode material, AHI probability ah exhibits the similar gate polarity dependence under both constant current and voltage FN stress. Polarity dependence of stress-induced trap creation in thin samples can be correlated with our theoretically predicted AHI probabilities during either constant current or voltage FN stress.
Acknowledgements Computational facilities provided in the Department of Physics, Jadavpur University is gratefully acknowledged.
References [1] DiStefano TH, Shatzkes M. Appl Phys Lett 1974;25:685. [2] Klein N, Solomon P. J Appl Phys 1976;47:4364. [3] Arnold D, Cartier E, DiMaria DJ. Phys Rev B 1994; 49:10278. [4] Thompson SE, Nishida T. J Appl Phys 1992;72:4683. [5] Samanta P. Solid-State Electron 1999;43:1677. [6] Fazan P, Dutoit M, Martin C, Ilegems M. Solid-State Electron 1987;30:829. [7] Park YB, Schroder DK. IEEE Trans Electron Dev 1998; ED-45:1361. [8] Samanta P, Sarkar CK. J Appl Phys 1998;83:2662. [9] Weinberg ZA. J Appl Phys 1982;53:5052. [10] Samanta P. PhD Thesis, Jadavpur University, 1999. [11] Kong S, Kwok CY, Wong SP. Proceedings of IEEE Tencon, Hongkong, 1995. p. 456. [12] Marsland JS. Solid-State Electron 1987;30:125. [13] Marsland JS. Semicond Sci Technol 1990;5:177. [14] DiMaria DJ. Solid-State Electron 1997;41:957. [15] Samanta P, Sarkar CK. Appl Phys Lett 2000;77:4350.
Gate polarity dependence of impact ionization probabilities in metal-oxide-silicon structures under Fowler–Nordheim stress Piyas Samanta *, C.K. Sarkar Department of Electronics and Telecommunication Engineering, Jadavpur University, Calcutta 700 032, India Received 21 May 2001; accepted 27 June 2001
Abstract A theoretical investigation on the gate polarity dependence of Fowler–Nordheim (FN) tunneling electron initiated impact ionization probabilities in the bulk silicon dioxide (SiO2 ) films as well as hole injection from the anode material (nþ poly-Si gate or silicon substrate) in metal-oxide-silicon devices is presented. Our theoretical results of the gate polarity dependence of the probabilities of various impact ionization processes under constant current FN stress correlate the experimentally observed positive charge generation/trapping and stress-induced trap creation in thick to thin SiO2 films. Ó 2002 Published by Elsevier Science Ltd.
1. Introduction Albeit hot-electron induced impact ionization phenomenon is an old issue in explaining the dielectric instability and breakdown of metal-oxide-silicon (MOS) devices under high-field stress [1,2]. However, from the fundamentals of electron heating in the oxide conduction band and their interactions with the oxide lattice as well as the anode material, a consensus about the various types of impact ionization processes in MOS structures under high-field stress has only been reached in the last decade [3–5]. Hot-electron induced various ionization processes include band-to-band ionization (BTBI) [3], trap-to-band ionization (TTBI) [4] in the oxide bulk and hole injection from the anode, popularly known as anode hole injection (AHI) [5]. Positive charge and/or hole trapping in the bulk gate oxide is manifested by the negative shift of gate voltage VG or Fowler–Nordheim (FN) threshold voltage VFN under constant current FN stress. Experimental results [6,7] of temporal evolution of the gate voltage shift DVG at a given constant FN current density have shown a
*
Corresponding author. E-mail address: [email protected] (P. Samanta).
strong gate polarity hence FN injecting electrode dependence. Gate polarity dependence of DVG has not been explained so far in literature. Moreover, as can be seen from Fig. 1, the relative absolute magnitude of DVG in thicker samples is higher during negative gate injection compared to substrate injection (positive gate bias) at a given constant FN current level [6] while, gate electron emission results lower absolute DVG in thinner samples [7] compared to substrate electron emission at a given constant current density. DVG is directly proportional to the product of the concentration of oxide-trapped charges and their centroids measured from the gate electrode [8] by DVG ¼
Qþ x ox
DVG ¼
Qþ ðtox xÞ ox
for substrate injection;
for gate injection;
ð1aÞ
ð1bÞ
where Qþ is the areal density of trapped positive charge in C/cm2 , tox is the oxide thickness, ox is the permittivity of SiO2 and x is the location of the centroid of trapped positive charge measured from the gate/oxide interface during FN injection from either gate or substrate.
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2. Theoretical analysis Electronic conduction in thermally grown SiO2 films above 5 nm thick is limited by the emission of electrons via FN tunneling through the interfacial barrier Uec at oxide fields above 6 MV/cm subject to the condition Vox > Uec , where Vox is the voltage drop in the oxide. In FN tunneling regime, electrons are injected into the oxide conduction band from either the Fermi level of the metal/degenerately doped n-type poly-Si gate at negative gate bias or the first subband energy level [9] located at 0.2 eV from the bottom of the conduction band of nondegenerately doped silicon in the case of injection from quantized accumulation and/or inversion layer at negative gate bias. Irrespective of the injecting electrode [10], as long as the barrier at the injecting electrode/oxide interface remains triangular, FN tunneling current density j obeys the well-known expression Fig. 1. FN threshold voltage shift DVFN as a function of electron fluence injected from substrate (positive bias on poly-Si gate) and nþ poly-Si gate (negative bias) at constant current density. Symbols are from experiment (a) on MOS capacitors on (1 0 0) n-Si [6] and (b) on n-MOSFET having channel length/ width ratio 100/20 lm/lm [7].
Taking the symmetric [6] or asymmetric [3] distribution of trapped holes under FN injection with positive and negative gate bias in Eqs. (1a) and (1b), experimental data of DVFN at injected electron fluence Qinj upto 102 C/cm2 as shown in Fig. 1(a) indicate that in thick samples gate injection results higher hole generation rate compared to substrate injection. However, polarity dependence of hole generation/trapping rate in thinner samples as estimated from Fig. 1(b) is opposite to that in thicker samples. Gate polarity dependence of experimentally observed negative shift of gate voltage is a consequence of the polarity dependence of the probabilities of various impact ionization processes responsible for the generation of hole and/or positive charge in the bulk oxide. Albeit our above interpretation on hole generation and hence the impact ionization probability in explaining experimental observations in thick to thin oxide films seems elegant, no theoretical clarification has yet been available in the literature. Present communication is therefore, an attempt to theoretically investigate the gate polarity dependence of the individual probability of various impact ionization processes with an aim to correlate these theoretical results with experimentally observed polarity dependence of DVFN and breakdown in MOS devices with thick to thin oxide films. Devices studied here are nþ poly-Si gated MOS capacitors grown on (1 0 0) n-Si substrate having resistivity of 5–7 X cm or transistor structures grown on nondegenerately doped p-Si.
j ¼ AEc2 expðB=Ec Þ;
ð2Þ
where A and B are material dependent constants [9,11] related to the interfacial barrier energy Uec for electron. Fig. 2 shows the variation of FN current density as a function of cathode field during injection from the nþ poly-Si gate as well as from the quantized accumulation layer of n-Si. In poly-Si gate structures, at a given cathode field Ec , the higher value of FN current density j during emission from the accumulation layer of n-Si as shown in Fig. 2 is due to lower barrier height en-
Fig. 2. Variation of FN current density with cathode oxide field during injection from the quantized accumulation layer of (1 0 0) n-Si (positive gate bias) and nþ poly-Si gate (negative gate bias) at time t ¼ 0 in poly-Si gate MOS capacitors. Results are shown for triangular electron-energy barrier at the cathode/ oxide interface. Symbols are from experiment during injection from nþ poly-Si gate into the oxide conduction band [11] and curves are from theory.
P. Samanta, C.K. Sarkar / Solid-State Electronics 46 (2002) 279–285 Table 1 Barrier energies of hole Uha at the anode/oxide interface and electron Uec and Uea at the cathode/oxide and anode/oxide interfaces, respectively FN injection from
Uec (eV)
Uea (eV)
Uha (eV)
(1 0 0) n-Si substrate nþ poly-Si gate
2.9 3.22
3.22 3.1
4.68 4.8
countered by the electrons residing in the first subband energy level in contrast to the emission from the bottom of the conduction band in degenerately doped poly-Si gate where quantization effect do not arise. From the FN plots shown in Fig. 2, the estimated barrier energies of electron Uec at cathode/oxide interfaces are listed in Table 1. Assuming the band-gap energies of Si and SiO2 equal to 1.1 and 9 eV respectively, the estimated barrier energies of hole Uha at the anode/ oxide are given in Table 1. In estimating hole-barrier energy at the Si/SiO2 interface, we have taken a value 3.1 eV instead of 2.9 eV for the electron barrier height at the Si (anode)/SiO2 interface, because we are concerned with those holes generated in the bulk of the Si substrate by impact ionization. FN tunneling is accompanied by oxide electric field as high as 6–7 MV/cm. At such a high oxide electric field, electrons while transported towards the anode in the oxide layer become hot and equilibriate their energies by impact ionization in the oxide bulk as well as in the anode material after entering into the anode. In the following subsections, the various types of impact ionization processes in the context of present knowledge will be discussed.
281
th where Pmf is the electron multiplication factor and Ebi is the threshold field for BTBI depending on the oxide thickness tox as outlined in Ref. [3]. The spatial weight function gðxÞ takes into account the electron heating distance or dead space kbi measured from the cathode/ oxide interface [10] and is given by gðxÞ ¼ 1 for x > kbi and gðxÞ ¼ 0 for x 6 kbi . HðuÞ ¼ 1 for u > 0 and 0 for u 6 0. The function HðuÞ takes care of the fact that beth low the threshold field Ebi electrons cannot be accelerated to energies high enough to cause BTBI in the oxide [10].
2.2. TTBI probability In thermally grown oxide films, neutral bridging ox- ygen vacancy centres V0O ðSi–OÞ3 Si Si ðO–SiÞ3 located at an energy 7 eV below the oxide conduction band edge act as the donor-like electron traps [4]. Under high-field stress, TTBI occurs when one of the bound electrons at an initially neutral oxygen vacancy V0O is released by the impact of hot-electron with kinetic energy greater than 7 eV from the oxide conduction band edge via V0O þ e ð> 7 eVÞ ! Vþ O þ 2e . The above TTBI model is consistent with the observed [3] hotelectron energy distribution in thermally grown oxide films under high-field stress. TTBI really occurs nearer th the anode at fields above the threshold Eem when the injected electron gains the threshold energy of 7 eV for impact emission beyond the threshold distance kem measured from the cathode/oxide interface [8,10]. Field ðEÞ and thickness ðtox Þ dependencies of TTBI probability an per injected electron are given by [10] an ¼ 2m 1;
m ¼ aN ðEÞ½tox kem ðEÞ :
ð4aÞ
2.1. BTBI probability At electric fields above 7 MV/cm in oxide films thicker than 20 nm, the acoustic phonon scattering alone can no longer stabilize the electron energy distributions leading to an electron runaway from phonon scattering and therefore, the high-energy tail of the hot-electron distribution extends beyond the 9 eV band gap of SiO2 as observed both theoretically and experimentally [3]. Electrons from these high-energy tails (>9 eV) of their distribution functions cause electron–hole pair generation by BTBI in the oxide. Holes generated via BTBI having lower mobility compared to electrons, while drifting towards the cathode get trapped in the existing neutral hole trap sites. Using the expression parameterized from the Monte Carlo simulation results of Arnold et al. [3], the field and thickness dependencies of the BTBI probability aI of an electron can be written as [10] 4 EðxÞ th aI ¼ Pmf ; ð3Þ 1 gðxÞH EðxÞ Ebi th Ebi
The spatial dependence of electron impact emission coefficient aN in cm1 is given by th ; ð4bÞ aN ðE; xÞ ¼ a0N exp ½ H =EðxÞ gðxÞH EðxÞ Eem where similar to the arguments given in Section 2.1, the weight functions gðxÞ and HðuÞ take into account the dead space kem and the threshold field criteria for the initiation of TTBI. Fitting Eqs. (4a) and (4b) with experimental data of Thompson and co-worker [4], we obtained the values of the constants a0N and H equal to 1598.54 cm1 and 28.77 MV/cm, respectively [8]. 2.3. AHI probability The electron tunneling into the conduction band of the oxide via FN mechanism, reaches the anode by gaining kinetic energy from the applied electric field and subsequently losing energy by scattering with oxide phonons beyond tunneling distance xt ð¼ Uec =Eox Þ measured from the cathode/oxide interface. The average
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energy gain hwi by the electron in the oxide can be calculated from the energy relaxation equation [5] dhwi hwi w0 ; ¼ qEox ðxÞ kðhwiÞ dx
ð5aÞ
where w0 ¼ 1:5kB T is the thermal energy, q is the magnitude of electronic charge, Eox is the oxide field, x is the spatial coordinate measured from the cathode/oxide interface, and kðhwiÞ is the energy-dependent mean free path in the oxide. Upon entering into the anode at x ¼ tox , each electron will acquire an additional potential energy equal to the anode barrier energy Uea for electron. Therefore, the total energy of the primary electron with respect to the bottom of the anode conduction band is given by e win e ¼ hwi þ Ua :
ð5bÞ
Assuming the oxide to be initially free from trappedoxide charges, average hot-electron energy hwi measured from the oxide conduction band as well as from the anode conduction band are shown in Fig. 3 as a function of constant FN current density during injection from the nþ poly-Si gate as well as the substrate. The energetic electron in the anode conduction band then loses its energy via direct interband ionizing collision with the bound electron in the valence band and thereby creates an electron–hole pair. The rest of the primary electron energy is shared between the generated pair. Assuming an equipartition of energy between the generated pair and neglecting the emitted phonon energy during impact ionization in the anode valence band, the hole energy wh from the top of the anode valence band can be written as [5]
wh ¼ 12ðwin e eI Þ;
where eI ¼ 1:1 eV [12] is the threshold energy for impact ionization in the anode material. Depending upon the energy wh of the generated hot-hole, hole injection into the oxide valence band occurs either over the interfacial anode barrier Uha or via tunneling through it. In our calculation, however, we have observed the hot-hole energy wh never exceeds the barrier height Uha at the anode/oxide interface irrespective of the gate bias polarity. Therefore, in our simulation hole injection into the oxide always occurs via tunneling through the interfacial barrier. Taking into account of the image force barrier lowering effect on the triangular hole-energy barrier and using WKB approximation, the hole tunneling probability Hh can be written as [5] ( ) pffiffiffiffiffiffiffiffiffiffiffi 3=2 4 2qmh ðUha wh Þ Hh ¼ exp vðyÞ ; ð7Þ a 3 hEox where mh ¼ 0:2m0 is the effective mass of hole in SiO2 with m0 as the free electron mass, vðyÞ is the image force correction term with normalized barrier lowering term y, a the h is the reduced Planck constantð¼ h=2pÞ, and Eox field in the oxide near the anode. In the absence of oxidetrapped charges at time t ¼ 0, oxide field remains unia form and Eox is equal to the cathode field Ec . According to the above physical model, the AHI probability ah per tunnel injected electron can be calculated from [5] ah ¼ Pii Hh ;
ð8Þ
where Pii is the probability of impact ionization or hole generation in the anode material (silicon substrate or poly-Si gate) by a tunneling primary electron. Incorporating soft threshold energy into lucky drift model, the probability of impact ionization Pii in silicon or poly-Si can be calculated according to the prescription of Marsland [12] as (
" 1 l0 ðakE l0 Þ1=3 Pii ¼ kE exp 1p ðkE kÞ kE kE !#
1=3 ðakE l0 Þ l0 Hi k exp kE k !#) " 1=3 1=3 ðakE l0 Þ ðakE l0 Þ Hi ; ð9aÞ 1p k k
a¼ Fig. 3. Average hot-electron-energy measured from the bottom of the oxide conduction band (panel (a)) and the anode conduction band edge (panel (b)) as a function of FN current density during electron emission from poly-Si gate and n-Si substrate.
ð6Þ
x h ; pð2n þ 1ÞqESi
kE ¼
qESi k2 ð2n þ 1Þ ; 2 hx
l0 ¼
eI ; qESi ð9bÞ
where hx ¼ 55 meV is the phonon energy, n is the Bose– Einstein number, p ¼ 0:030 is the softness parameter
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is the mean free path between collisions and k ¼ 70:2 A or mean free path for momentum relaxation [13], kE is the mean free path for energy relaxation, l0 is the dead space to acquire the threshold energy eI for impact ionization in silicon and ESi is the field in silicon and the related Airy function HiðzÞ is given by
Z 1 1 HiðzÞ ¼ p1 exp t3 þ zt dt: ð9cÞ 3 0
3. Results and discussion For a given constant FN current density, the cathode field Ec is higher during gate emission compared to substrate emission as evident from inset of Fig. 2. Therefore, from the solution to Eq. (5a), constant current gate emission results a higher value of average hotelectron energy in the bottom of the oxide conduction band than does substrate emission in poly-Si gated devices as shown in Fig. 3(a). Using the barrier energies Uea of electron at the anode/oxide interface from Table 1 into Eq. (5b), average hot-electron energy in the anode conduction band is observed smaller under constant current gate emission relative to its substrate injection counterpart (Fig. 3(b)). Under constant current FN stress, the gate bias dependence of the probabilities aI and an of bulk impact ionization processes BTBI and TTBI in the oxide is opposite to that of AHI probability ah as shown in Fig. 4. The probabilities aI and an of the bulk ionization processes BTBI and TTBI, respectively are increasing functions of the electric field in the oxide in accordance with Eqs. (3), (4a) and (4b). Gate polarity dependence of aI and an under constant current FN injection as shown in Fig. 4(a) and (b) arises through the different values of the cathode field Ec at a given j (see Fig. 2) via the FN Eq. (2). BTBI and TTBI being bulk processes, depend on the hot-electron energy in the oxide conduction band. The higher values of both aI and an under constant current gate emission compared to substrate electron emission as shown in panels (a) and (b) of Fig. 4 are consistent with the gate bias dependence of average hotelectron energy depicted in Fig. 3(a). In view of the above discussion, we propose that unlike constant current FN stress, constant voltage FN stress will not exhibit the gate polarity dependence of bulk ionization probabilities. The anomalous behavior of AHI probability ah on the gate polarity dependence under constant current FN stress as shown in Fig. 4(c) is explained below. According to our AHI model described in Section 2.3, AHI probability ah is the product of hot-electron induced impact ionization or hole generation probability Pii in the bulk of anode material and hole tunneling proba-
Fig. 4. Gate bias dependence of BTBI probability (aI ), TTBI probability (an ), and AHI probability (ah ) of an electron as a function of FN current density at t ¼ 0 under constant current injection from nþ poly-Si gate (– – –) as well as from the quantized accumulation layer of n-Si (—) in MOS capacitors having 35 nm thick gate oxide.
bility Hh through the anode/oxide barrier Uha . Hole generation probability Pii in the anode material is directly related to the oxide field near anode governed by Eq. (9b). At a given j, the higher value of cathode field Ec and hence the anode oxide field Ea results a higher value of impact ionization probability Pii in the anode (bulk silicon substrate) during constant current gate emission relative to substrate emission as shown in Fig. 5(a). The average electron energy win e in the anode conduction band is estimated smaller (Fig. 3(b)) while the holeenergy barrier Uha is higher (Table 1) during gate electron emission relative to substrate injection. Therefore, in accordance with Eq. (6), hot-hole energy wh becomes higher and consequently, the effective hole barrier height Uheff ¼ Uha wh at the anode/oxide interface becomes lower resulting a smaller value of hole tunneling probability Hh under constant current substrate emission than its gate emission counterpart as shown in Fig. 5(b). Gate polarity dependence of Hh being stronger than that of Pii , FN injection from poly-Si gate shows smaller AHI probability ah than does substrate emission at a given constant current density (Figs. 4(c) and 5(c)). According to Eq. (7), hole tunneling probability Hh is a an increasing function of oxide field Eox near anode and h is a decreasing function of Ueff . Therefore, the results shown in Fig. 5(b) and (c) indicate that work function difference of the anode material is the controlling factor of the gate bias dependence of AHI probability ah rather than the oxide field near anode. The above argument is further verified from Fig. 6 showing the gate polarity dependence of AHI probability under constant voltage
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Fig. 5. Variation of (a) impact ionization probability (Pii ) in the anode material, (b) hole tunneling probability Hh through the anode/oxide barrier and (c) AHI probability ah at t ¼ 0 with FN current density in a 8 nm thick SiO2 film during constant current injection from substrate and poly-Si gate.
Fig. 6. Gate polarity dependence of AHI probability (ah ) of an electron at t ¼ 0 as a function of initial applied oxide electric field under constant voltage FN stress in MOS capacitors with a 10 nm thick gate oxide.
FN stress similar to that under constant current FN stress depicted in Figs. 4(c) and 5(c). Due to the variation of work function of the anode material, unlike the bulk ionization probabilities, AHI probability ah exhibits the gate polarity dependence under constant voltage FN stress as depicted in Fig. 6. Fig. 7 shows the relative magnitude of the probabilities aI , an and ah of a FN tunnel injected electron as a function of initial applied oxide field in thick to thin SiO2 films during constant voltage FN injection from nþ
Fig. 7. Relative magnitudes of BTBI, TTBI and AHI probabilities of an electron at t ¼ 0 as a function of oxide electric field (a) in a 27 nm thick oxide and (b) in a 10 nm thick oxide during constant voltage FN injection from nþ poly-Si gate.
poly-Si gate at negative bias. Positive gate injection also exhibits similar relative trend of the various probabilities (not shown here). Theoretical results of the gate bias dependence of various ionization probabilities of a FN tunnel injected electron discussed above can be correlated with the MOS device degradation as follows. As evident from Fig. 7(a), relative to TTBI and AHI probabilities, BTBI probability of an electron is higher in a 27 nm thick SiO2 film at an applied field around 10 MV/cm corresponding to j ¼ 1 mA/cm2 . Therefore, our theoretical results of higher BTBI probability under gate electron emission compared to substrate electron emission at a given constant current density qualitatively explains experimentally observed [6] polarity dependence of DVG at low injected electron fluence in a 27 nm thick SiO2 . Recent [7], experimental gate voltage shift DVG data in a 10 nm thick SiO2 film at a 10 mA/cm2 constant current density have shown that the absolute magnitude of DVG is smaller during gate injection compared to substrate emission in contrast to thicker samples of Fazan and co-workers [6]. At 10 mA/cm2 constant current density, the estimated values of the cathode fields Ec from Eq. (2) are observed far below the thresholds for TTBI and BTBI in this thin sample. Therefore, only AHI contributes in hole trapping in a 10 nm thick SiO2 film at j ¼ 10 mA/cm2 . Taking a constant value of 5 nm for the trapped-hole centroid, experimental [7] DVG data indicate that hole generation/trapping rate during gate injection is smaller than that during substrate electron emission. Our theoretical results of gate polarity dependence of ah shown in Fig. 5(c) qualitatively explains
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the experimental results of gate voltage shift data with thin oxides of Park and Schroder [7]. Destructive dielectric breakdown in MOS devices under high-field stress is triggered by the stress-induced defect generation caused by two processes, viz., electron– hole recombination at injected electron fluence Qinj 6 103 C/cm2 and trap creation via hydrogen release by direct hot-electron impact beyond 103 C/cm2 of Qinj [3,14]. The rate of recombination of trapped hole and/or TTBI generated positive charge with free electron is proportional to the concentration of trapped holes [8,15]. Therefore, theoretically observed higher probabilities of hole and positive charge generation from BTBI and TTBI processes indicate higher recombination-induced trap creation during gate electron emission compared to substrate injection under constant current stress. In oxide films thicker than 20 nm, BTBI contributes more compared to other processes TTBI and AHI at fields above 9 MV/cm as shown in Fig. 7. Therefore, gate injection results higher stress-induced trap creation compared to substrate injection in thicker films in support with experimental results of Fazan et al. [6] at constant current FN stress. In MOS devices with oxide films as thin as 8 nm, AHI is the dominant hole generation mechanism. According to the recombination-induced trap creation model [14], our theoretically estimated results of polarity dependence of AHI probability as shown in Figs. 5(c) and 6 indicate higher trap creation in thin SiO2 films under substrate injection compared to its gate injection counterpart in either stressing modes (constant current and voltage) in contrast to thicker samples. Furthermore, in samples below 10 nm thick, breakdown is dominated by trap creation due to direct hot-electron impact and trap creation rate strongly depends on electron energy measured from the anode conduction band [14]. In view of the trap creation model [14], the higher value of the hotelectron energy (measured from the anode conduction band) as shown in Fig. 3(b) indicates that substrate injection stress generates more traps than does gate injection stress consistent with report of Park and Schroder [7].
4. Conclusion Our theoretical analysis of the polarity dependence of various ionization probabilities in the oxide bulk as well as in the anode material under constant current FN in-
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jection gives a physical insight of the experimentally observed anomalous behavior [6,7] of gate voltage shift at injected fluence as low as 102 C/cm2 in MOS devices with thick and thin SiO2 films. From our analysis, we propose that the probabilities of hole and/or positive charge generation dictated by bulk ionization processes viz., BTBI and TTBI in thicker samples do not depend on the gate bias polarity under constant voltage stress. However, in samples as thin as 8 nm, AHI is the dominant mechanism responsible for the origin of trapped holes. Furthermore, AHI is dependent on interfacial barrier energies of electron and hole [5]. Due to the variation of work function of the anode material, AHI probability ah exhibits the similar gate polarity dependence under both constant current and voltage FN stress. Polarity dependence of stress-induced trap creation in thin samples can be correlated with our theoretically predicted AHI probabilities during either constant current or voltage FN stress.
Acknowledgements Computational facilities provided in the Department of Physics, Jadavpur University is gratefully acknowledged.
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