▁Sub script ▁ 0 s ▁indicate ▁equilibrium ▁concentr ations . ▁ ▁The ▁electrons ▁and ▁the ▁holes ▁re comb ine ▁with ▁the ▁car rier ▁lifetime ▁ τ . ▁ ▁We ▁define ▁▁▁▁ ▁so ▁the ▁upper ▁equations ▁can ▁be ▁re written ▁as : ▁ ▁In ▁a ▁simple ▁approximation , ▁we ▁can ▁consider ▁the ▁electric ▁field ▁to ▁be ▁constant ▁between ▁the ▁left ▁and ▁right ▁electro des ▁and ▁neglect ▁ ∂ E / ∂ x . ▁However , ▁as ▁electrons ▁and ▁holes ▁diff use ▁at ▁different ▁spe eds , ▁the ▁material ▁has ▁a ▁local ▁electric ▁charge , ▁indu cing ▁an ▁in hom ogeneous ▁electric ▁field ▁which ▁can ▁be ▁calculated ▁with ▁G auss ' s ▁law : ▁ ▁where ▁ ε ▁is ▁perm itt ivity , ▁ ε 0 ▁the ▁perm itt ivity ▁of ▁free ▁space , ▁ ρ ▁is ▁charge ▁density , ▁and ▁e 0 ▁elementary ▁charge . ▁ ▁Next , ▁change ▁variables ▁by ▁the ▁substitution s : ▁▁ ▁and ▁suppose ▁ δ ▁to ▁be ▁much ▁smaller ▁than ▁. ▁The ▁two ▁initial ▁equations ▁write : ▁ ▁Using ▁the ▁Ein stein ▁relation ▁, ▁where ▁ β ▁is ▁the ▁inverse ▁of ▁the ▁product ▁of ▁temperature ▁and ▁the ▁Bol tz mann ▁constant , ▁these ▁two ▁equations ▁can ▁be ▁combined : ▁ ▁where ▁for ▁D *, ▁ μ * ▁and ▁ τ * ▁holds : ▁, ▁ ▁and ▁▁ ▁Consider ing ▁n ▁>> ▁p ▁or ▁p ▁→ ▁ 0 ▁( that ▁is ▁a ▁fair ▁approximation ▁for ▁a ▁sem icon duct or ▁with ▁only ▁few ▁holes ▁inject ed ), ▁we ▁see ▁that