EECS 5 Sprig 4, Lecture 9 Lecture 9: Diffusio, Electrostatics review, ad Capacitors EECS 5 Sprig 4, Lecture 9 Cotext I the last lecture, we looked at the carriers i a eutral semicoductor, ad drift currets I this lecture, we will cotiue to study trasport--- the motio of carriers due to diffusio, ad the ifluece of charge distributios o the electric fields Review of Electrostatics Diffusio IC MIM Capacitors I the ext lecture, we will study P-N diodes
EECS 5 Sprig 4, Lecture 9 Electrostatics Review The force betwee ay two charges is give by Coulomb s law r qq F eˆ 4πr Where ê is uit vector i the directio away from the other charge. Sice Maxwell s equatios are liear, we ca add up all the forces from other charges, ad defie the electric field at a poit: r E r q F ˆ e q q πr 4 EECS 5 Sprig 4, Lecture 9 Electrostatic fields Sice we are goig to be dealig with large umbers of charges, we ca use a cotiuum model: The charges are give as a smooth desity: ρ( r v, t) Coulombs/cm 3 The electric field is a smooth vector field which diverges from positive charge, ad coverges o egative charges r r E(, t) v ρ ( r, t) Which is oe of Maxwell s equatios -
EECS 5 Sprig 4, Lecture 9 Gauss s Law Gauss s Law state that the total amout of E field (flux) leavig a volume is equal to the et charge eclosed E dv dv V Recall: V ρ Q E ds Q / Q E dv E ds V S EECS 5 Sprig 4, Lecture 9 Electrostatics i D I EE5, we are almost always goig to use -D models, so it simplifies: de ρ ρ E de x ρ( x') E( x) E( x) ' x Cosider a uiform charge distributio Zero field boudary coditio ρ(x) ρ x x x x E x ρ ( ') ρ ( ) ' x E(x) ρ x x 3
EECS 5 Sprig 4, Lecture 9 Electrostatic Potetial The electric field (force) is related to the potetial (eergy): dφ E The E field is the (-) slope of the potetial! Negative sig says that field lies go from high potetial poits to lower potetial poits (egative slope) Note: Electros float to a high potetial poit: F e dφ qe e φ F e dφ e e φ EECS 5 Sprig 4, Lecture 9 More Potetial Itegratig this basic relatio, we have that the potetial is the itegral of the field: r φ( x) φ( x) E dl I D, this is a simple itegral: φ( x) φ( x) x x C E( x') ' φ( x ) dl r φ(x) Sice the derivative of the E field is the charge, we ca itegrate agai to get Poisso s equatio i D: d φ ( x) ρ( x) E 4
EECS 5 Sprig 4, Lecture 9 Boudary Coditios Potetial must be a cotiuous fuctio. If ot, the fields (forces) would be ifiite Electric fields eed ot be cotiuous. We have already see that the electric fields diverge o charges. I fact, across a iterface we have: E ds ES ES Qiside x E ( ) E ( ) S Q iside x ES ES Field discotiuity implies charge desity at surface! E E EECS 5 Sprig 4, Lecture 9 Boudary coditios Metals The presece of metals greatly simplifies boudary coditios. Iside a metal, the E fields are very small (otherwise the curret would be very large) The iside of a metal has o free charges (if it had free charges, they are free to move ad would very rapidly scatter to the edges of the metal So all of the et charge o a metal occurs o its surface, ad The surface of a metal is therefore all early at the same potetial (exceptio: log wires coductig a curret) 5
EECS 5 Sprig 4, Lecture 9 Note: Bad edge diagrams We will ofte draw a diagram of the valece ad the coductio bad edges as a fuctio of positio. The eergy at the bad edge correspods to the potetial eergy that a electro has (which is the egative of the electrostatic potetial). Thus the slope of the bad edge with distace is the electric field. --- --- --- (Silico) E field Force o electros Eergy Distace P type N type EECS 5 Sprig 4, Lecture 9 Diffusio Diffusio occurs whe there exists a cocetratio gradiet I the figure below, imagie that we fill the left chamber with a perfume at temperate T If we suddely remove the divider, what happes? The perfume will fill the etire volume of the ew chamber. How does this occur? 6
EECS 5 Sprig 4, Lecture 9 Diffusio Eve though there is o force actig o the perfume molecules, because there are more o the left tha o the right, their radom motios take more from the left to the right tha are goig from right to left. Electros ad holes do the same thig, but sice they are charged, they carry curret with them. Diffusio moves particles i additio to the motio from electric forces EECS 5 Sprig 4, Lecture 9 Diffusio (cot) The et motio of gas molecules to the right chamber was due to the cocetratio gradiet Diffusio will lead to a et flow of particles as log as the distributio of particles is ot uiform Diffusio causes a flow of particles from places of high cocetratio to places of lower cocetratio. 7
EECS 5 Sprig 4, Lecture 9 Diffusio If the electros move log distaces without collisios, they would quickly spread out, diffusio would be large. diffusio is proportioal to λ, the mea free path If the larger the particles thermal velocities are, the the faster diffusio will be. diffusio is proportioal to v th, the mea uidirectioal thermal velocity Diffusio is also proportioal to the rate of chage of the umber of particles with distace diffusio is d Flux v thλ d EECS 5 Sprig 4, Lecture 9 Diffusio Equatios Assume that the mea free path is λ Fid flux of carriers crossig x plae ( λ) ( λ)v th λ () (λ) λ ( λ)v th Where the diffusio costat is defied: Flux vth ( ( λ) ( λ) ) d d Flux vth () λ () λ d Flux vthλ d Flux D Flux D v th λ d 8
Relatio betwee carrier Diffusio ad carrier Mobility EECS 5 Sprig 4, Lecture 9 The motio of particles uder the ifluece of a force (like a E field) ad the motio due to cocetratio gradiets are related to each other. For example, lets look at a semicoductor i a electric field, but also at thermal equilibrium: Coductio bad Fermi Level Valece bad EECS 5 Sprig 4, Lecture 9 Thermal Equilibrium A couple fudametal priciples about thermal equilibrium: The eergy that electros are filled up to (the Fermi level) is the same everywhere. The curret is zero at all poits. So whe we look at this: Coductio bad Fermi Level Valece bad We kow that the electros are feelig a force due to the electric field, but there is also diffusio which cotributes a exactly cacelig amout of curret! This meas the diffusio costat ca always be foud i terms of mobility 9
EECS 5 Sprig 4, Lecture 9 The Eistei relatio (diffusio) Sice the diffusio process has a fudametal relatioship to the mobility i a electric field, we ca fid the diffusio costat i terms of the mobility µ. kt D µ q We ca do a rough derivatio for the case of our simple particle model (ext slide) EECS 5 Sprig 4, Lecture 9 Eistei Relatio The average thermal velocity is give by kt * m v th λ v τ th c kt τ vthλ vthτ c kt m c * kt q Mea Free Time qτ m c * J d kt qvthλ q µ q kt D µ q d
EECS 5 Sprig 4, Lecture 9 Total Curret ad Boudary Coditios The total curret is give by the sum of drift ad diffusio: d J J drift J diff qµ E qd I resistors, the carriers are approximately uiform ad the secod term is early zero I metals, there are a very large umber of carriers, i very uiform cocetratio, ad the coductio curret is quite liear with E (ohmic) EECS 5 Sprig 4, Lecture 9 IC MIM Capacitor Bottom Plate Top Plate Bottom Plate Thi Oxide Cotacts (Side view) Q CV (Top view) By formig a thi oxide ad metal (or polysilico) plates, a capacitor is formed Cotacts are made to top ad bottom plate Parasitic capacitace exists betwee bottom plate ad substrate (Parasitic meas that it s there whether we wat it or ot!)
EECS 5 Sprig 4, Lecture 9 V s Review of Capacitors Gauss s Law The total E field from the top plate comes from its charge Q E ds The total E field ito the bottom plate comes from its charge Q E ds E dl E t ox V s E V t s ox Q CV s Q Vs Q E ds EA A t ox C A t ox EECS 5 Sprig 4, Lecture 9 Capacitor Q-V Relatio Q V s y Q(y) y Q CV s The total charge o each plate is liearly related to voltage (Q o top plate, -Q o bottom plate) Charge desity is a delta fuctio at surface (for metals)
EECS 5 Sprig 4, Lecture 9 A No-Liear Capacitor Q y V s Q(y) y Q f ( V s ) We ll soo meet capacitors that have a o-liear Q-V relatioship If plates are ot metal, the charge desity ca peetrate ito surface EECS 5 Sprig 4, Lecture 9 Itroductio to small sigal models For a o-liear capacitor, we have Q f ( V s ) CV s We ca t idetify a capacitace Imagie we apply a small sigal o top of a bias voltage: Q f ( V v ) s s df ( V ) f ( Vs ) dv V Vs v Costat charge s The icremetal charge is therefore: Q Q q df ( V ) f ( Vs ) dv V Vs v s 3
EECS 5 Sprig 4, Lecture 9 Small Sigal Capacitace Break the equatio for total charge ito two terms: Q Q q Costat Charge Icremetal Charge df ( V ) f ( Vs ) dv V Vs v s df ( V ) q dv V Vs v s C v s df ( V ) C dv V V s EECS 5 Sprig 4, Lecture 9 Example of No-Liear Capacitor Next lecture we ll see that for a PN juctio, the charge is a fuctio of the reverse bias: V Q j ( V ) qnaxp φ b Voltage Across NP Juctio Charge At N Side of Juctio Costats Small sigal capacitace: dq j C j ( V ) dv qnax φ b p V φ b C j V φ b 4
EECS 5 Sprig 4, Lecture 9 Carrier Cocetratio ad Potetial I thermal equilibrium, there are o exteral fields ad we thus expect the electro ad hole curret desities to be zero: J q µ E d µ oe D qd q kt do dφ o o kt dφ q d o V th d EECS 5 Sprig 4, Lecture 9 Carrier Cocetratio ad Potetial () We have a equatio relatig the potetial to the carrier cocetratio kt dφ q If we itegrate the above equatio we have φ ( x) φ ( x d ) V o ( x) l ( x ) We defie the potetial referece to be itrisic Si: th V th d φ ( x ) ( x ) i 5
EECS 5 Sprig 4, Lecture 9 Carrier Cocetratio Versus Potetial The carrier cocetratio is thus a fuctio of potetial x e φ ( x) / V th ( ) i Check that for zero potetial, we have itrisic carrier cocetratio (referece). If we do a similar calculatio for holes, we arrive at a similar equatio p x e φ ( x) / V th ( ) i Note that the law of mass actio is upheld φ ( x)/ V ( x)/ V ( x) p( x) i e th φ th e i EECS 5 Sprig 4, Lecture 9 The Dopig Chages Potetial Quick calculatio aid: For a p-type cocetratio of 6 cm -3, the potetial is -36 mv N-type materials have a positive potetial with respect to itrisic Si Due to the log ature of the potetial, the potetial chages liearly for expoetial icrease i dopig: ( x) ( x) ( x) φ( x) Vth l 6mV l 6mV l log ( x ) ( x ) i ( x) φ( x) 6mV log p( x) φ( x) 6mV log i 6
EECS 5 Sprig 4, Lecture 9 PN Juctios: Overview The most importat device is a juctio betwee a p-type regio ad a -type regio Whe the juctio is first formed, due to the cocetratio gradiet, mobile charges trasfer ear juctio Electros leave -type regio ad holes leave p-type regio These mobile carriers become miority carriers i ew regio (ca t peetrate far due to recombiatio) Due to charge trasfer, a voltage differece occurs betwee regios This creates a field at the juctio that causes drift currets to oppose the diffusio curret I thermal equilibrium, drift curret ad diffusio must balace p-type N A N D -type V EECS 5 Sprig 4, Lecture 9 PN Juctio Currets Cosider the PN juctio i thermal equilibrium Agai, the currets have to be zero, so we have J do qµ E qd do qµ E qd do D E µ dpo Dp E µ p kt q kt q p d dp 7
EECS 5 Sprig 4, Lecture 9 PN Juctio Fields p-type -type N A N D p N a p ( x) E J diff p N i d x p x N d N i a E J diff Trasitio Regio EECS 5 Sprig 4, Lecture 9 Total Charge i Trasitio Regio To solve for the electric fields, we eed to write dow the charge desity i the trasitio regio: ρ ( x) q( p N d N a I the p-side of the juctio, there are very few electros ad oly acceptors: ρ x) q( p N ) < x ( a Sice the hole cocetratio is decreasig o the p- side, the et charge is egative: ) x p < N a > p ρ ( x) < 8
EECS 5 Sprig 4, Lecture 9 Charge o N-Side Aalogous to the p-side, the charge o the -side is give by: ρ( x ) q( Nd ) < x < x The et charge here is positive sice: N d > ρ ( x) > N d N i a E J diff Trasitio Regio 9