Plan Bipolar junction transistor Elements of small-signal analysis Transistor Principles: PETs and FETs Field effect transistor Discussion

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1 Physics of silicon transistors Plan Bipolar junction transistor homojunction and heterojunction Doping considerations Transport factor and current gain Frequency dependence of gain in a microwave transistor Base shrinkage and finite output conductance Ebers-Moll model and small-signal equivalent circuit Elements of small-signal analysis Two port; common terminal configurations Power gain, Mason orem Transistor Principles: PETs and FETs Field effect transistor Two-dimensional channel; sheet conductance Conduction Laws by dimensional analysis Gated diode Short channel effects MOS structure: relevant energies Band bending; threshold evaluation Inversion layer structure Pinch-off effect in MOSFET Concept of quasi-fermi level (imref) Discussion Why silicon???

2 Bipolar Junction Transistors Junction Transistor n p n V eb V cb Figure 1: Schematic diagram of a npn transistor in equilibrium and under applied bias. By Kirchhoff law: I E = I C + I B I C Neglecting recombination in base and parasitic injection of holes emitter, collector current flows through a much larger impedance emitter current, whence power gain. into than In a homojunction transistor, injection of holes into emitter is suppressed compared to n p 0 N useful injection of electrons into base only by factor D (emitter) =. p n 0 N A (base) This means that base must be lightly doped compared to emitter and hence base resistance is a concern. The fundamental trade-off: thicker base for lower base resistance, thinner base for faster diffusion across base.

3 Bipolar Junction Transistors I e I c n p n I b Φ 1 V eb Φ 2 V cb Ε V Heterojunction bipolar transistor (HBT) Figure 2: Homojunction and heterojunction npn transistor In homojunction transistor at base-emitter junction (e ) J E (h ) J E e e β Φ1 β Φ2 N A (base) = N D (emitter) J E (h ) J E (e ) Emitter efficiency η J E (e ) (e ) (h ) J E + J 1 doping ratio. E In heterojunction transistor: N A (base) = N D (emitter) J E (h ) J E (e ) e β E V

4 Bipolar Junction Transistors Back to (homo)junction transistor. We understand why base doping must be much lower doping. Now why collector doping must be much lower doping? N D (emitter) >> N A (base) >> N D (collector) than than emitter emitter Threefold answer: For W B to have little dependence on V cb (need high output impedance) to lower base-collector capacitance C cb to lower field in base-collector junction (increase breakdown voltage) I e I c n p n I b Φ 1 Φ 2 V ce I c (ma) I b = 0.6 ma V ce Figure 3: Common-emitter transistor characteristics. Base current is stepped up by increment of 0.1 ma and emitter current increases by much larger amounts. Here current gain β 100.

5 Bipolar Junction Transistors Base transport factor and current gain. From continuity equation: n n = 0 L D 2 x n (x ) = A e L D + B e x L D Boundary conditions at x and at W : n (0) = n (W ) = n i 2 N A 2 n i N A e β V eb e β V cb 0 n (x ) = A sinh (W x ) L n (0) sinh (W x ) L D D = sinh ( W L D ) J n (x ) = e D n = x _ e n (0) DL D cosh (W x ) L D sinh ( W L D ) The base transport factor alpha: α J n (W ) = J n (0) 1 1 cosh ( W L D ) 2 L 2 D W 2 J(0) J(W) J B Current gain: β = _ I C. By Kirchhoff s law, β = I B α 1 α Combined with emitter efficiency η, we take instead of α product α η.

6 Bipolar Junction Transistors Frequency dependence of current gain. β (db) static gain slope 10 db/dec f T frequency f (GHz) Figure 5: High-frequency gain. At high frequencies current gain rolls-off at 10 db/per decade (i.e. as 1 f ). This behavior is quite universal and has nothing to do with eir recombination or emitter efficiency. The characteristic cut-off frequency f T is defined by condition of unity current gain ( β 1 ) and is mainly due to propagation delay τ of minority carriers through base. f T = 1 2π τ In general, τ = W v, where v is average velocity of minority carrier propagation. For diffusive transport, W 2 τ D = << τ C. 2 D Do not confuse this τ with minority carrier lifetime (which is typically much longer). Let us denote minority-carrier lifetime by τ C ("capture" time).

7 Bipolar Junction Transistors Let us derive an expression for α (f ). Begin with continuity equation: _ n = D _ 2 n n t x 2 τc Take n (t ) = n 0 + δ n e i ωt where n 0 = n 0 (x ) is static solution and δn = δn (x ) is harmonic variation amplitude at frequency ω 2π f. Both static equation for n 0 and dynamic equation for δn are of similar form: where 2 n 0 x 2 L 2 D n 0 = 0 _ 2 δn x 2 L 2 δn = 0 L D 2 D τ C L 2 = L 2 D (1 + i ω τ C ) ωτ C >> 1 _ i ω D The solutions to both equations are similar too (in form): whence we find (in analogy to α 0 = n 0 (x ) = A sinh W x L D δn (x, ω) = B sinh W x L α (ω) 1 cosh ( W L D ) _ J (W, ω) = J (0, ω) 1 cosh ( W L )

8 Bipolar Junction Transistors At sufficiently high ω s (nothing "spectacular", just ω τ C >> 1 ) we have α (ω) = _ W L = 2i ω τ D where τ D = 2D W 2 = 2i ω τ D remember i = e i 1 = cosh ( W L ) π 4 = 1 + i 2 ( 1 + i ωτ D ) 1 for ωτ D << 1 ωτ 2 e D e i ωτ D for ωτ D > 1 Im modern transistors, typically, W < 1,000 A and D 50 cm 2 s 1. Therefore, at frequencies easily accessible to measurement (up to, say 20 GHz) one has, typically, τ D < s and ωτ D << 1. α (ω) 1 e i τ 1 + i ωτd e i ω τ D β (ω) = α = = 1 α i ω τ 1 e D β (ω) = 1 2 sin ( ωτ D 2) ω D i ωτ D 2 i e 2 sin ( ωτ D 2) 1 ω τ D Uisng heterostructures, it is possible to design an HBT such that α (ω) does not spiral in significantly, even for large ω s remaing close to circle exp (i ωτ) for phase angles φ = ω τ as large as φ = 2π. Such coherent transistors, are capable of " life after death", showing current gain above f T and power gain above f max. A. A. Grinberg and S. Luryi, " Coherent transistor", IEEE Trans. Electron Devices ED-40, pp (1993). S. Luryi, A. A. Grinberg, and V. B. Gorfinkel, " Heterostructure bipolar transistor with enhanced forward diffusion of minority carriers", Appl. Phys. Lett. 63, pp (1993).

9 Bipolar Junction Transistors Base shrinkage and finite output conductance. Emitter V ce Base Collector I c (ma) I b = 0.6 ma Early voltage V ce V A Figure: Base shrinkage and Early effect. Neglecting recombination, injected minority current must base,. J = 0. If current is by diffusion only, n be constant in _ dn = const = dx _ n (0) n (W ) W n p 0 e βv eb W W shrinks with collector bias. Hence, increasing collector current (Early effect) at a fixed base current, one has an I C 1 + V cb V A Finite output conductance is detrimental; we live better without it. The Early voltage depends on base width W and Gummel number n G = N A W. For W >> than BC junction depletion width, one has V A e N A W 2 e n G W ε ε

10 Bipolar transistor, continued Ebers-Moll model α I I R α N I F E i E i C C R Ex I F I R R Cx C Cx i B R Bx B E B C Figure 1: Dashed line indicates "intrinsic" portion of device, excluding "parasitic" extrinsic elements. The model for diode blocks can be furr specified to include internal junction capacitance.

11 Bipolar transistor, continued Small-signal equivalent circuit of an abrupt junction HBT C E α B i E α C E i E CC i C C R Ex R E R Cx R B C Cx i B R Bx B Ε V Figure 2: This model is good for "ballistic" propagation of carriers across base. For diffusive propagation, intrinsic portion must be adjusted, A. A. Grinberg and S. Luryi, IEEE Trans. Electron Devices ED-40, pp (1993).

12 Bipolar transistor, continued Elements of small-signal analysis All variables A (t ) are considered varying harmonically in a small range about a dc point: A (t ) = A 0 I (t ) = I 0 V (t ) = V 0 + δ A e i ω t + δ I e i ω t + δ V e i ω t Many alternative notations, e.g., i and v instead of δi and δv. The δa s are complex quantitites, may be position-dependent fields, δa (x ). The relationship between different δa s, e.g. between δv and δi (generalized impedances or admittances) depend on chosen dc point. 2 -PORT δ I 1 δ I 2 δv 1 δv 2 common terminal Figure 3: General two-port. Transformers are 2-ports ("passive"). From small-signal point of view, transistors are two-port amplifiers. Admittance matrix: δi 1 δi = 2 y 11 y 12 δv 1 y 21 y 22 δv 2

13 Bipolar transistor, continued Admittance matrix: δ I 1 δ I = 2 y 11 y 12 δ V 1 y 21 y 22 δ V 2 For example: y 11 d I 1 d V, input admittance 1 V 2 Impedance matrix: δ V 1 δ V = 2 z 11 z 12 δ I 1 z 21 z 22 δ I 2 For example: z 22 d V 2 d I, output impedance 2 I 1 Hybrid matrix (h-parameters): δ V 1 δ I = 2 h 11 h 12 δ I 1 h 21 h 22 δ V 2 For example: h 21 d I 2 d I, forward current gain 1 V 2 Definition of se parameters essentially involves specification of boundary condition at one or anor port. Thus h 22 is output admittance for open-circuit input port. y 22 is output admittance for short-circuit input port. h 21 is forward current gain for short-circuit output port, etc. Each set of parameters ( z -parameters, y -parameters, h -parameters) is complete in sense that it can be used to derive or sets unambiguously.

14 Bipolar transistor, continued Common terminal configurations δ I 1 δ I 2 δv 1 δv 2 common terminal δ I 1 δ I 2 δv 1 E B C δv 2 common base δ I 2 δ I 2 δ I 1 C δ V 2 δ I 1 E δ V 2 δ V 1 B δ V 1 B E C common emitter common collector Figure 4: Different common-terminal configurations give parameters. Thus, short-circuit current gains are rise to very different h 21 e β h 21 b α and hence h 21 e = b 1 h 21 h 21 b

15 Bipolar transistor, continued Indefinite parameters All of parameter sets corresponding to different configurations (common- base, common-emitter, common-collector) are derivable from one anor. A convenient trick (works best in y -parameter representation) is to disregard common reference and treat third terminal as an additional port: δ I 1 δ I 2 δv 1 δv 2 δ I 3 δv 3 Figure 5: indefinite matrix: δ I 1 δ I 2 δ I 3 = y 11 y 21 y 31 y 12 y 22 y 32 y 13 y 23 y 33 δ V 2 δ V 3 δ V 1 From Kirhhoff s Law and fact that matrix should work for arbitrary set of { δ V i } it follows that sum of all columns (or rows) in indefinite matrix is zero. Thus, if we assume a short circuit at ports 1 and 2, fact that sum of all currents must be zero implies that Σ of y -parameters in third column vanishes, and so on. To prove that Σ must vanish in each row, we note that if all three { δ V i } are equal no ac current can flow at any port. It is exceedingly simple to transform from one common-terminal configuration to anor. Thus, if we know y -matrix in common base configuration, corresponding common-emitter matrix is: b y 11 b y 21 y 31 y 12 b y 22 b y 32 y 13 y 23 y 33 y 11 y 21 y 31 y 12 y 22 y 32 y 13 y 23 y 33 y 11 y 21 y 31 y 12 y 11 e y 21 e y 13 e y 12 e y 22

16 Bipolar transistor, continued Power gain definitions: Power gain G is ratio of power delivered to load to power input into network. It depends on both input and load circuits. Maximum available gain (MAG) is maximum gain achievable from a particular transistor without external feedback. MAG equals value of forward gain G which results when both input and output are simultaneously matched in an optimum way. For example, realization of MAG requires that load resistance be matched to output resistance Re (z 22 ). Unilateral gain U is maximum available power gain of a device after it has been made unilateral by adding a lossless reciprocal feedback circuit. This means that lossless network around amplifier (inductances and capacitances) is adjusted so as to set reverse power gain to zero. Unilateral gain is independent of common-lead configuration! The unilateral gain U can be calculated from any of following equivalent expressions: U = = = z 21 z 12 2 ; 4 [ Re (z 11 ) Re (z 22 ) Re (z 12 ) Re (z 21 ) ] y 21 y 12 2 ; 4 [ Re (y 11 ) Re (y 22 ) Re (y 12 ) Re (y 21 ) ] h 21 + h 12 2, 4 [ Re (h 11 ) Re (h 22 ) + Im (h 12 ) Im (h 21 ) ] where z i j, y ij, and h ij are impedance, admittance, parameters of a transistor, respectively, for any configuration. and hybrid This remarkable result (Mason s orem) is main reason for wide-spread use of U. See S. J. Mason, " Power gain in feedback amplifiers", IRE Trans. Circuit Theory CT-1, pp (1954).

17 Bipolar transistor, continued Small-signal model of an abrupt junction HBT C E α B i E α C E i E CC i C C R Ex R E R Cx R B C Cx i B R Bx B Ε V Figure 6: Small-signal analysis of this simple model, including frequency dependence of power gain in both ballistic and diffusive regimes, has been carried out by Grinberg and Luryi (1993). A. A. Grinberg and S. Luryi, " Coherent transistor", IEEE Trans. Electron Devices ED-40, pp (1993). A. A. Grinberg and S. Luryi, " Dynamic Early effect in heterojunction bipolar transistors", IEEE Electron Device Lett. EDL-14, pp (1993). Quasi-static (Ebers-Moll-like) model of abrupt-junction HBT can be found in A. A. Grinberg and S. Luryi, " On rmionic-diffusion ory of minority transport in heterostructure bipolar transistors", IEEE Trans. Electron Devices ED-40, pp (1993).

18 Bipolar transistor, continued TRANSISTOR PRINCIPLES : FETs & PETs Field Effect: Screening - Q Q v Potential Effect: Control of a cathode work function δφ Φ V CE

19 Field Effect and Potential Effect Transistors FETs: - Q I = Q v L Q v L "Biblical" principle: Q for Q I for I "Transit time" limitation : τ > Q I in out = L v

20 Field Effect and Potential Effect Transistors PETs δφ Φ V CE Ι δφ ~ ~ e Φ /kt δ Q in τ ~ Ι -1 Speed increases with current until exponential law fails at high currents PET FET τ (space-charge effect) limited by transit time across

21 Field effect transistors Two Dimensional Channel V G Conductance of 3D sample: e n g di = [ e N dv [ cm 2 ] = C A V G µ ] A, e N µ L Ω. 1 cm Conductance of 2D sample: g di = [ e n µ ] dv _ W, e n µ L 1 Ω W v L Note: resistance Ω of a square independent of its size (contrast with a cube!)

22 Field effect transistors Knowing " resistance per square" 1 e n µ one can simply count squares: 1/3 3 Current density per unit width [ A cm 1 ] J I = en v W = en µ F = en µ V L conductance per unit channel width g = en µ L [ ms mm ] Transconductance (also per unit width) g m J V G J = en v g m = V D _ (en ) n = C v [ ms mm ] V G A Figure of Merit ("FOM"): C = _ L ( delay time g ) m v

23 Field effect transistors Conduction Laws by dimensional analysis In CGS system conductivity has units of a velocity. Taking this velocity to be an effective carrier velocity v, we can write a generic expression for diode current in form I ε 4π v V _ A L 2, (bulk) _ W. (film) I ε v V 4π L where L is length, A = D. W cross-sectional area, and W width of diode. The relative permittivity ε ε ε0 of material enters because it scales space-charge potential in Poisson s equation. The actual current-voltage dependence (up to a numerical coefficient) can be derived from above equation whenever conduction process involves a dominant transport mechanism, which provides a unique scaling relationship between v and V. Thus, for free electron motion, velocity scales as v 2 ( e m ) V and one obtains laws appropriate for ballistic transport, e.g. for bulk case Child s law of vacuum electronics: I = ζ _ ε e 1 2 4π m V 3 2 _ A L 2 where ζ = _ 4 2 [ Child 9 ] For case when electron velocity is saturated, take v = v S. For case of constant mobility µ, velocity scales as v µ V L, which leads to following expressions: I I = ζ ε 3 4π = ζ ε 2 4π µv 2 A, (bulk) where ζ 3 = _ 9 8 L 3 µv 2 W. L 2 (film) [ Mott Gurney ] S. Luryi, " Device building blocks", Chap. 2 in High-Speed Semiconductor Devices, ed. by S. M. Sze, Wiley Interscience (1990) pp Connection to international units is obtained by replacing dimensionless factor ε 4π with ε in farads per meter.

24 Field effect transistors Field-effect transistor characteristics Ι en µw/l Ι V D V G3 V G2 V G1 V D

25 Field effect transistors Transistor channel versus thin film diode V G V D pinch off Potential diagram along channel "gated diode" V D

26 Luryi Serge transistors effect Field effects channel Short V D pinch off The shorter. becomes channel left, moves point off pinch As decreasing L Recall conductance) output (finite current increasing to leads bipolars in effect Early Ι V D V G1 V G2 V G3 less channel, to gate closer length, channel given a For effects short-channel are important

27 Field effect transistors MOS structure Equilibrium φ (x) V bi E F φ B Flatbands E F φ B V FB Figure 1: Built-in voltage V bi and flatband voltage V FB. These quantities are not identical. V bi is electrostatic potential drop in equilibrium: it equals sum of potential drops in semiconductor and oxide. V FB is voltage that must be applied to gate to flatten bands. Since "voltage" means Fermi level difference, V FB equals difference in work functions of semiconductor and metal. Not same equilibrium. thing as electrostatic potential difference, which can exist even in

28 Field effect transistors vacuum level χ S W S W M E C E i E F φ B V FB E V E F Figure 2: Relevant energies in MOS system at flatbands. W S : work function of semiconductor W M : work function of metal V FB = W S W M : flatband gate voltage χ S : electron affinity of semiconductor φ B E i E F : bulk doping characteristic, φ B = kt ln ( N A n i ). W S = χ S + E G E F = χ S + (E C E V ) (E F E V ) = χ S + E C E F

29 Field effect transistors F = 0 φ = 0 F (x) φ (x) φ s depletion boundary W F s Figure 3: Evaluation of band bending in MOS structure Poisson s equation: φ e (N A p ) = ε p = N A e βφ φ = en A ε 1 e βφ (1) Note a trick: φ _ d φ = d φ _ dφ = _ df F = _ 1 df 2 dx dφ dx dφ 2 dφ multiply Poisson s equation (1) by dφ and integrate from x = φ = 0 F = 0 to x = surface φ = φs F = F S using φ S 0 1 e βφ d φ = _ kt e βφ S + e βφ S 1

30 Field effect transistors Exact: _ 1 F 2 2 S = kt N A ε Approximate ( βφ S >> 1, i.e., φ S >> kt e ): F S 2 = 2kT N A βφ S = ε βφ S + e βφ S 1 2 e N A ε The same result is obtained in depletion approximation: en A W φ S = 2 F S = 2 ε _ en A W ε where W is depletion width φ S F ox F = 0 φ = 0 φ (x) V ox d ox E i F s φ B φ s V G W Figure 4: Threshold gate voltage evaluation: Given bulk doping N A e φ B = kt ln ( N A n i ) at threshold φ S = 2 φ B 4 F S = N A φ B εs ε S F S = ε OX F OX F OX = εox ε S 4 N _ εs e A φ B V OX d OX F OX V T = e φ S + V OX + V FB

31 Field effect transistors Above Treshold we need to include charge of this is a quantum mechanical problem, let us go back to basic semiconductor physics inversion layer itself; E L E Γ E X L Λ Γ X hh E so lh so Figure. Important extremal points in band structure of cubic semiconductors. The schematic picture (drawn not to scale) is appropriate for a direct-gap III-V semiconductor. For GaAs at room temperature indicated energies are: E Γ = 1.42 ev, E L = 1.71 ev, E X = 1.90 ev, E so = 0.34 ev. In silicon lowest conduction band point is in direction, 85% of way to X point. The indicated energies for silicon at 300 K are: E Γ = 4.08 ev, E L = 1.87 ev, E = 1.13 ev, E so = 0.04 ev. In Ge lowest conduction band point is at L but Γ point is not far away: E Γ = 0.89 ev, E L = 0.76 ev, E = 0.96 ev, E so = 0.29 ev. Bands: degenerate/nondegererate isotropic/anisotropic parabolic/nonparabolic In vicinity of a nondegenerate extremal point it is convenient 1 dispersion relation E n (k) using effective mass tensor M n : E n ( k 0 + k) E n (k 0 ) = (h _2 2) k. 1 M. h _ 2 n k 2 Σ 3 i, j = 1 to describe 1 M n k ij i k j, 1 where components of M n can be written down in terms of free electron mass and parameters of lattice potential ( periodic V (x, y, z ) characteristic of point k 0.

32 Field effect transistors In vicinity of a nondegenerate band extremum, surfaces of equal energy are 1 ellipsoids, as evident from Eq. (1). The symmetric tensor M n has, most generally, six independent components. The coordinate axes can always be chosen so as to diagonalize this tensor, i.e. along ellipsoid s principal directions: 1 m M 1 n = 0 1 m m 3 In general, energy ellipsoid is determined by six independent parameters: three 1 diagonal values of M n and three directions of principal axes. However, number of parameters can be often reduced by symmetry considerations. The ellipsoid symmetry is determined uniquely by symmetry of extremal point k 0. For an extremum located on a crystal symmetry axis, one of principal directions of energy ellipsoid coincides with symmetry axis. If latter is an axis of 3-fold, 4-fold, or 6-fold symmetry, n ellipsoid is an ellipsoid of revolution (m 1 = m 2 m t, m 3 m l ). If more than one such axis intersects at k 0, n ellipsoid anisotropy disappears and energy surfaces become spherical (m 1 = m 2 = m 3 m ). Such is situation in conduction band at Γ point of cubic semiconductors: E (k ) = _ h _2 k 2. 2 m In silicon conduction band minima are on 4-fold rotation axes and low- energy isoenergetic surfaces are ellipsoids of revolution, ir long axes being along <100> directions. There are six symmetry related minima. Similar local minima exist in conduction band of germanium, but true conduction band minima in Ge are located at L points. There are only four symmetry-related ellipsoids of constant energy in vicinity of conduction band edge of Ge. It is convenient to picture se ellipsoids as eight half-ellipsoids joined toger on opposite faces by translations through suitable reciprocal lattice vectors. In each ellipsoid, band curvature is least in direction along rotation axis and highest in transverse directions. This means that longitudinal mass is heavier than transverse mass. The anisotropy is particularly high in Ge, m l m t = 20, but it is also considerable in Si, where m l m t 5. In most cases, it is a reasonable approach to describe uniform electric field by effective mass equation. electronic motion in a

33 Field effect transistors Figure. Surface of constant energy in vicinity of conduction band edge in silicon represents six ellipsoids of revolution, extended along <100> directions. The band curvature is least in longitudinal direction (heavy mass) and highest in transverse direction (light mass) The effective mass ratio m l m t 5. If band edge in Si were at zone boundary rar than at a general point in direction ( 85 % toward X), n re would be only three ellipsoids. Location away from zone boundary of conduction band edge in Si and local minimum in Ge is related to inversion symmetry of diamond structure.

34 Luryi Serge transistors effect Field Figure edge band conduction of vicinity in energy constant of Surface : along extended revolution, of ellipsoids four represents germanium in <111> large, very is ratio mass effective The directions. m l m t = each that so 20, sausage. a like looks really ellipsoid for cell primitive a chose to have would we ellipsoid full a exhibit to order In each picture, zone Brillouin In internal. be would point L some which to shifted is half equivalent and boundary by two in cut is ellipsoid vector. lattice reciprocal a by face opposite

35 Field effect transistors Inversion layer in a silicon MOS structure The schematic cross-section of a silicon MOS structure is illustrated on next page along with band-bending near Si/SiO2 interface under a sufficiently large positive gate bias. Let us look in more detail at band structure of 2DEG in an inversion layer on {100} surface. In a (roughly triangular) quantum well formed near Si/SiO2 interface under a positive gate bias, ellipsoids oriented differently with respect to surface give rise to a quite different subband structure. Let us specify actual {100} Si surface as a (100) crystal plane. Electrons in two ellipsoids elongated in [100] direction possess heavy mass m l in z direction and an isotropic light mass m t in any direction lying in (100) plane. These electrons give rise to subbands whose bottom-edge energies are denoted by E n. The or four ellipsoids, whose longitudinal axes lie in (100) plane, correspond to light mass m t in [100] direction, and ir subbands are denoted by E n. Because quantum-well energy levels scale with 1 m, one has E 0 < E 0, and so inversion-layer electrons in ir ground subband have an isotropic light mass. The order of higher-lying subbands can be established only on basis of self-consistent numerical calculations. The subband energies depend not only on field and background doping but also on temperature, which affects relative population of higher-lying subbands and, hence, self-consistent field. Notation: equivalent crystallographic planes, e.g., (100), (01 0), etc., are collectively denoted by {100}. Similarly, equivalent (symmetry-related) directions in reciprocal lattice, e.g., [100], [01 0], etc., are collectively referred to as <100> direction.

36 Field effect transistors E 1 E 0 (2) (2) (4) (2) E 4 E 3 E 2 E 1 E 0 Figure: The order of seven lowest subbands, calculated for an inversion layer at a Si-{100} surface with n S = cm 2 in a lightly-doped ( N A = cm 3 ) p -type material at room temperature. The levels E 0 and E 1 are close in energy and, in fact, change ir order at lower T and/or n S.

37 Field effect transistors Pinch off in a MOSFET Recall gradual channel approximation: Treat potential diagram locally at any channel cross-section ( x ) by ignoring voltage difference between source and drain, but taking channel to be not not at ground voltage but at V = V ch (x ), i.e., by replacing V G V G V ch (x ) The V ch (x ) is imref ( quasi-fermi level) which varies monotonically from E F = 0 in source to E F = V D in drain. E F imref E C V D pinch off V G V ch < V T E F

38 Field effect transistors Imref The term Fermi level in semiconductor physics is potential ; in equilibrium it is defined by n = E 0 synonymous de g (E ) f (E E F ) N C e ( E F E C ) kt, with chemical where f (E ) = [ exp ( E kt ) + 1 ] 1 is Fermi function and approximation is valid for nondegenerate semiconductors; here we may have E C = E C (x ) and n = n (x ), but chemical potential E F = const In presence of a current flow concept of Fermi level is not defined. However, let us define n = N C e ( E F E C ) kt where n = n (x ) E C = E C (x ) E F = E F (x ) The drift diffusion equation n reduces to J = e ( n µ F + D n ) e n µ E F n (x) E (x) C E (x) F quasi-fermi level ( imref )

39 Field effect transistors Example: use of imref to distinguish "pictorially" different transport mechanisms. Thermionic (Be) Schottky diodes. versus diffusion (Schottky) mechanism of conduction in Schottky diode (forward bias) Semiconductor Be Schottky Metal Imref varies little where net current is much smaller than eir diffusion or drift components. These regions are " approximately in equilibrium".

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