6 THE COUPLED-DIODE MODEL BWx()BBPxW∆=0BP0BP0()0BO BPPx+∆ =()BPx Figure 6-1: Emitter-base junction is forward biased and collector base junction is forward biased.2 THE COUPLED DIODE MODEL Chapter 6 BWx0BP0BP0()0BO BPPx+∆ =()BPx Figure 6-2: Emitter-base junction is forward biased and collector base junction is grounded. BWx()BBPxW∆=0BP0BP0()BPx Figure 6-2: Emitter-base junction is grounded and collector base junction is forward biased. ()()001EBqVnnnkTnnpx p pxepp=− ∆ ===− (6.1) ()()1CBqVnnnbkTnnpxW p pxWepp=− ∆ ===− (6.2)Electronics of Semiconductor Devices 3 pnpnnpCBEBxW=0x=pnpnnpCBEBxW=0x =saturationactive modeEBVCBVpnpnnpCBEBxW=0x =pnpnnpCBEBxW=0x=invertedcut-off()00EBqVkTBnPx Pe==()0CBqVkTBBnPxW Pe==()00EBqVkTBnPx Pe==()0BBPxW==()00BPx∆==()0CBqVkTBBnPxW Pe==()00BPx==()0BBPxW== EBERS-MOLL EQUATIONS The one dimensional npn transistor can be represented as two pn junction diodes connected back-to-back with common p−region, named base as shown in figure 6-20. In the active mode of operation, the emitter base junction is forward biased. The emitter diode current FI which crosses the base-emitter junction is exponential function of the emitter base bias voltage,EBV . A large portion of this current FFIαreaches to collector. Fαis the forward common-base current gain. In the inverted mode of operation, the collector base junction is forward bias. The diode collector4 THE COUPLED DIODE MODEL Chapter 6 currentRI, which crosses the collector base bias junction, is also exponential function of the collector base bias voltageCBV . A large portion of this current , RRIαreaches to emitter, Rαis the reverse common base current gain. FIRIRRIαFFIαBICIEICBE Figure 6-20: Circuit diagram of pnp transistor of Ebber-Moll model 1EBqVkTFESIIe=− (6.3) 1CBqVkTRCSIIe=− (6.4) EFRRIIIα=− (6.5) CFFRIIIα=− (6.6) ()()11BEC FF RRIII I Iαα=−=− +− (6.7) () ()1111CBEBqVqVkT kTBFES RCSIIe Ieαα=− −+− − (6.8) 11CBEBqVqVkT kTEES RCSIIe Ieα=−− − (6.9)Electronics of Semiconductor Devices 5 11CBEBqVqVkT kTCFES CSIIe Ieα=−−− (6.10) For BBWL>> Then ESI and CSI are simply of diode 00EBES E BEBDDIqA n pLL=+ (6.11) 00CBCS C BCBDDIqA n pLL=+ (6.12) In general ()/011sinhEBqV kTEES BEBDIqA p eWLL=− ()/011sinhqVCB kTBBBBBDqA P eLWL−− (6.13) ()/00coth 1EBqV kTCBBCS C BCB BDDWIqA n p eLL L=+ − ()/00coth 1EBqV kTCBBCBCB BDDWqA n p eLL L−+ − (6.14) 0cscBBFES RCS BBBDWIIqApLLαα== (6.15) FDCαα= (6.16) For BBWL>>6 THE COUPLED DIODE MODEL Chapter 6 201112FDCEEB BBBO E BnDW WDP L Lαα==++ (6.17) in general 000cschcothBBBBBFEB BEBBB BDWqA pLLDD WqA n pLL Lα=+ (6.18) 001cosh sinhFEBEB BBBEB BnWDL WLLLP Lα= +   (6.19) ()()000001EBqVBBBkTBBpx p pepp=− ∆==− (6.20) ()()0001CBqVBBBBBkTBBpxW p pWepp=− ∆==− (6.21) For active mode common base 0EBV > , 0CBV< 11CBEBqVqVkT kTEES RCSIIe Ieα=−− − (6.22) 1EBqVkTEES R CSIIe Iα=−+ (6.23) 1EBqVkTCFES CSIIe Iα=−+ (6.24) 1EBqVERCSkTESIIeIα−−= (6.25) ()1CFE FRCSIIIααα=+− (6.26)Electronics of Semiconductor Devices 7 0CDCECBIIIα=+ (6.27) where ()1CBO R F CSIIαα=− (6.28) FIRIRRIαFFIαBICIEICBE Figure 6-21: Circuit diagram of npn transistor of Ebber-Moll model 1BEqVkTFESIIe=− (6.29) 1BCqVkTRCSIIe=− (6.30) EFRRIIIα=− (6.31) CFFRIIIα=− (6.32) BECIII=− (6.33) Rα= Reverse gain Fα= Forward gain FES R CSIIαα= where the N= Normal mode I=Inverted mode. Iα= Reverse gain Fα= Forward gain8 THE COUPLED DIODE MODEL Chapter 6 ESI = The emitter saturation current when 0CBV=. CSI = The collector saturation current when 0EBV=. 11BCBEqVqVkT kTEES RCSIIe Ieα=−− − (6.34) 11BCBEqVqVkT kTCFES CSIIe Ieα=−−− (6.35) () ()1111CBEBqVqVkT kTBEC FES RCSIII Ie Ieαα=−=− −+− − (6.36) 00cothEB BES E BEB BDD WIqA n pLL L=+ (6.37) 00cothCBBCS C BCB BDDWIqA n pLL L=+ (6.38) 0cscBBFES RCS BBBDWIIqApLLαα== (6.39) 00sinhpBBbpBpBFnEEnEDPqAWLLDqA nLα=+ (6.40) ()()000001EBqVBBBkTBBpx p pepp=− ∆==− (6.41) ()()0001CBqVBBBBBkTBBpxW p pWepp=− ∆==− (6.42) The Diode Connections of Bipolar Junction TransistorElectronics of Semiconductor Devices 9 EBCAVIEBAVI(2)(1)II(6)(5)II(4)(3)p+np+−p+npp+np+−p+p+p+nnnpppAV+−AV+−AV+−AV+− EBAVIp+np+−CBEBIEII=AV+− 1), , and 0EAEB CII V V I== = 110CBEBqVqVkT kTCFES CSIIe Ieα=−−−= (6.43) then: 11CBEBqVqVkT kTFES CSIe Ieα−=− (6.44)10 THE COUPLED DIODE MODEL Chapter 6 11CBEBqVqVFESkT kTCSIeeIα−=− (6.45) 11EB EBqV qVNESkT kTEES RCSCsIII I e I eIαα == −− −   (6.46) ()11AqVkTERFESII I eαα==− − (6.47) ()0011EB AqV qVBkT kTBPeeP∆=−= − (6.48) ()()0000011EB AqV qVBB BkT kTBBppxpeepp∆=−==−=− (6.49) ()000AqVkTBBpx pe== (6.50) ()0111CBEB EBqVqV qVBBFESkT kT kTRBCSpWIeeepIαα∆=−= −= − (6.51) ()01AqVBBkTRBpWepα∆=− (6.52) ()()0001AqVBB B B BkTRBBpW p xW peppα∆=−==− (6.53) ()011AqVBBkTRBpxWepα==−+ (6.54) ()001AqVkTBBBR BpxW p e pα== −+ (6.55) ()0AqVkTBBRBPxW Peα== (6.56)Electronics of Semiconductor Devices 11 1a) If AkTVq>> ()00AqVkTBBpx pe== (6.57)