IPv6　over　Low　Power　Wireless　Personal　Area　Networks　(6LoWPAN)　has　an　ample　share　in　the　Internet　of　Things.　Sensor　nodes　in　6LoWPAN　collect　vital　information　from　the　environment　and　transmit　to　a　central　server　through　the　public　Internet.　Therefore,　it　is　inevitable　to　secure　communications　and　allow　legitimate　sensor　nodes　to　access　network　resources.　This　paper　presents　a　lightweight　Authentication　and　Key　Exchange　(AKE)　scheme　for　6LoWPAN　using　an　authenticated　encryption　algorithm　and　hash　function.　Upon　successful　authentication,　sensor　nodes　and　the　central　server　can　establish　the　secret　key　for　secure　communications.　The　proposed　scheme　ensures　header　verification　during　the　AKE　process　without　using　IP　security　protocol　and,　thus,　has　low　communication　and　computational　overheads.　The　logical　correctness　of　the　proposed　scheme　is　validated　through　Burrows-Abadi-Needham　logic.　Furthermore,　automatic　security　analyses　by　using　AVISPA　illustrate　that　the　proposed　scheme　is　resistant　to　various　malicious　attacks　in　6LoWPANs.